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Research ArticleRobust Adaptive Fuzzy Control for a Class ofUncertain MIMO Nonlinear Systems with Input Saturation
Shenglin Wen and Ye Yan
College of Aerospace Science and Engineering National University of Defense Technology Changsha 410073 China
Correspondence should be addressed to Shenglin Wen wenshenglin1108gmailcom
Received 11 November 2014 Revised 30 January 2015 Accepted 3 February 2015
Academic Editor Yan-Jun Liu
Copyright copy 2015 S Wen and Y Yan This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper studies the robust adaptive fuzzy control design problem for a class of uncertain multiple-input and multiple-output(MIMO) nonlinear systems in the presence of actuator amplitude and rate saturation In the control scheme fuzzy logic systems areused to approximate unknown nonlinear systems To compensate the effect of input saturations an auxiliary system is constructedand the actuator saturations then can be augmented into the controllerThemodified tracking error is introduced and used in fuzzyparameter update laws Furthermore in order to deal with fuzzy approximation errors for unknown nonlinear systems and externaldisturbances a robust compensation control is designed It is proved that the closed-loop system obtains119867
infintracking performance
through Lyapunov analysis Steady and transient modified tracking errors are analyzed and the bound of modified tracking errorscan be adjusted by tuning certain design parameters The proposed control scheme is applicable to uncertain nonlinear systemsnot only with actuator amplitude saturation but also with actuator amplitude and rate saturation Detailed simulation results ofa rigid body satellite attitude control system in the presence of parametric uncertainties external disturbances and control inputconstraints have been presented to illustrate the effectiveness of the proposed control scheme
1 Introduction
In most practical control applications such as those inrobot manipulation and aerospace industry the performanceof the controller is directly related to the accuracy of themathematical model and external disturbances However itis difficult to establish an appropriate mathematical modelfor a large number of nonlinear systems when the systemsare complex and highly coupled nonlinear with structureduncertainties and external disturbances [1] To tackle withthis problem fuzzy logic systems and neural networks havebeen extensively used in complex and ill-defined nonlinearsystems due to their approximation ability of dealing with thenonlinear smooth functions [2] Many adaptive fuzzy controland adaptive neural network control schemes have beendeveloped for single-input and single-output (SISO) non-linear systems [3ndash7] MIMO nonlinear systems [8ndash17] andSISOMIMO nonlinear systems with immeasurable states[18ndash20] respectively Generally these adaptive fuzzy andneural network control approaches can achieve nice control
performance without control saturation If physical actuatorssaturation such as magnitude and rate constraints is consid-ered the adaptive intelligent control approaches mentionedabove can not be implemented [21]
As we know in many practical dynamic systems physicalactuators saturation on hardware indicates an inevitableconstraint of themagnitude and rate limitations of the controlsignal For example due to physical limitation momentumexchange devices or thrusters as actuator for the satelliteattitude control system fail to render infinite control torqueand thus the actuator can only provide limited control torqueswithin a limited rate [22] Control saturation is one ofthe most common nonsmooth nonlinearity that should beexplicitly considered in the control design The controllersthat ignore actuator limitations may give rise to undesirableinaccuracy severely degrade the performance of system oreven damage the stability of system [23]Hence the controllerdesign subjected to the control saturation while simulta-neously achieving higher performance is a very practicalproblem
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 561397 14 pageshttpdxdoiorg1011552015561397
2 Mathematical Problems in Engineering
The design of tracking controllers for uncertain MIMOnonlinear systems with actuator constraints is a challengingproblem During the past decades there have been extensiveresearches on the control of nonlinear systems with variousconstraints Analysis and design of control systems with con-trol saturation have been widely studied in [24ndash42] Farrell etal [24ndash26] have presented an adaptive backstepping approachfor unknown nonlinear systems with known magnituderate and bandwidth constraints on intermediate states oractuators without disturbance To tackle with the physical sat-uration an auxiliary systemwith the same order as that of theplant was constructed to compensate the effect of saturationThe control input saturation is investigated through onlineapproximation based control for uncertain nonlinear systemsin [26] An adaptive control and the constrained adaptivecontrol in combination with the backstepping technique areproposed in [29] A direct adaptive fuzzy control approachfor uncertain nonlinear systems with input saturation ispresented in [30] in which a Nussbaum function is usedto compensate for the nonlinear term arising from theinput saturation In [31] an adaptive fuzzy output feedbackcontrol algorithm for a class of output constrained uncertainnonlinear systems with input saturation is developed byemploying a barrier Lyapunov function and an auxiliarysystem Adaptive backstepping tracking control based onfuzzy neural networks is investigated for unknown chaoticsystems in [32] and with control input constraints in [33]Neural network based adaptive control schemeswith externaldisturbances and actuator saturations are presented in [35]in which auxiliary systems are added to attenuate the effectsof input saturation It is apparent that the presence of inputsaturation constraint substantially increases the complexityof control system design for uncertain MIMO nonlinearsystems In the constrained adaptive control the key problemis how to handle the constraint effect of the actuatorrsquos physicalconstraints To this end we introduce an auxiliary designsystem to handle the constraint effect in this paper Basedon the states of the auxiliary design system constrainedadaptive control is investigated for a class of uncertainMIMOnonlinear systems with input constraints using robust fuzzycontrol technique
In this paper a robust adaptive fuzzy tracking controlscheme is presented to handle the external disturbances andactuator physical constraints for uncertain MIMO nonlinearsystems In control design fuzzy logic systems are used toapproximate unknown nonlinear systems Note that inputsaturations are nonsmooth functions but the adaptive fuzzycontrol technique requires all functions differentiable [6]To compensate the effect of input saturations an auxiliarysystem is constructed and the actuator saturations then canbe augmented into the controller The modified trackingerror is introduced and used in fuzzy parameter update lawsBesides in order to deal with fuzzy approximation errorsfor unknown nonlinear systems and external disturbancesa robust compensation control is designed It is proved thatthe proposed control approach can guarantee that all thesignals of the resulting closed-loop system are bounded andthe closed-loop system obtains 119867
infintracking performance
through Lyapunov analysis The transient modified tracking
errors performance is derived to be explicit functions ofdesign parameters and thus bounds of modified trackingerrors can be adjusted by tuning design parameters
The rest of this paper is organized as followsThe descrip-tion of the uncertain MIMO nonlinear system under consid-eration and necessary preliminaries are given in Section 2 InSection 3 the robust adaptive fuzzy control without controlsaturation is firstly designed When the actuators have phys-ical limitations this approach may not be able to be success-fully implemented In order to solve this problem a robustadaptive fuzzy control scheme with input saturation is inves-tigated The simulation results of satellite attitude controlare presented to demonstrate the effectiveness of proposedcontroller in Section 4 Section 5 contains the conclusion
2 Problem Formulation and Preliminary
21 Problem Formulation Consider a class of uncertainMIMO affine nonlinear system
x = f (x) +
119898
sum
119894=1
g119894(x) 119906119894+ d1015840
1199101= ℎ1(x)
119910119898
= ℎ119898
(x)
(1)
where x = [1199091 119909
119899] isin 119877
119899 is the state vector available formeasurement u = [119906
1 119906
119898] isin 119877
119898 is the control vectorwith |119906
119894| le 119906119894max where 119906
119894max denote the actuator amplitudey = [119910
1 119910
119898] isin 119877
119898 is the output vector ℎ1(x) ℎ
119898(x)
are smooth functions defined on the open set of R119899 f(x)g1(x) g
119898(x) are continuous unknown but smooth vector
fields d1015840 = [1198891015840
1 119889
1015840
119899]119879
isin R119899 is external disturbance vectorwhere 119889
1015840
1is unknown but bounded
Define that 119871 fℎ119894 is the Lie derivative of ℎ119894along vector
field f(x) and 119871119896
fℎ119894 is recursively as 119871119896
fℎ119894 = 119871 f(119871119896minus1
f ℎ119894) A
multivariable uncertain nonlinear system of the form (1) hasa vector relative degree [119903
1 1199032 119903
119898] at a point x
0if
119871g119895119871119896
fℎ119894 (x) = 0 forall1 le 119895 le 119898 119896 le 119903119894minus 1 (2)
Using feedback linearization the nonlinear system (1) canbe transformed into the following form [43]
y(r) = F (x) + G (x) u + d (4)
where
y(r) = [119910(1199031)
1 119910(1199032)
2 119910
(119903119898)
119898]119879
(5)
F (x) =
[[[[[[[
[
1198711199031
f ℎ1 (x)
1198711199032
f ℎ2 (x)
119871119903119898
f ℎ119898
(x)
]]]]]]]
]
=
[[[[[[
[
1198911(x)
1198912(x)
119891119898
(x)
]]]]]]
]
(6)
d =
[[[[[[[[[[[[[
[
1199031
sum
119894=1
119871119894minus1
x 11987111988910158401198711199031minus119894
f ℎ1(x)
1199032
sum
119894=1
119871119894minus1
x 11987111988910158401198711199032minus119894
f ℎ2(x)
119903119898
sum
119894=1
119871119894minus1
x 1198711198891015840119871119903119898minus119894
f ℎ119898
(x)
]]]]]]]]]]]]]
]
=
[[[[[[
[
1198891
1198892
119889119898
]]]]]]
]
(7)
x = [1199101 119910
(1199031)
1 1199102 119910
(1199032)
2 119910
119898 119910
(119903119898)
119898] (8)
In system (4) d still represents unknown boundedexternal disturbance vector The relative degree of the systemis assumed to be equal to the order of the system and isexpressed as sum
119898
119894=1119903119894= 119899 which implies that the system does
not have any zero dynamicsLet the desired output trajectory be given by
y119889= [1199101198891
1199101198892
119910119889119898
]119879
(9)
This paper aims to develop a robust adaptive fuzzy track-ing control scheme such that all closed-loop system signalsasymptotically converge to a compact set in the presence ofinput saturation system uncertainties and external distur-bances and ensure the system outputs y track the desiredtrajectory y
119889 For system (4) to be controllable the following
assumptions are made
Assumption 1 The desired trajectory 119910119889119894
and its 119899th orderderivatives are known and bounded
Assumption 2 For x in certain controllability regionU119888isin 119877119899
120590(G(x)) = 0 where 120590(G(x)) denotes the minimum singularvalue of the matrix G(x)
22 Fuzzy Logic Systems In this paper fuzzy logic systemsare used to approximate unknown nonlinear functions F(x)and G(x) Following the approximation presentation offuzzy logic systems is given Without loss of generality theunknown uncertainty is assumed as 119891(x) A fuzzy logic sys-tem consists of four parts the knowledge base the fuzzifier
the fuzzy inference engine working on fuzzy rules and thedefuzzifier The fuzzy inference engine uses fuzzy IF-THENrules to perform a mapping from an input linguistic vectorx = [119909
1 1199092 119909
119899]119879
isin 119877119899 to an output variable 119910 isin 119877 Then
the 119894th fuzzy rule can be represented as [3]
119877119897
If 1199091is 119865119897
1and sdot sdot sdot and 119909
119899is 119865119897
119899 then 119910 is 119866
119897
(119897 = 1 2 119903)
(10)
where 119865119897
119894and 119866
119897 are fuzzy sets characterized with the fuzzymembership functions 120583
119865119897119894(119909119894) and 120583
119866119897(119910) respectively and 119903
is the number of fuzzy rulesThrough singleton function center average defuzzifier
and product inference the fuzzy logic system can beexpressed as follows
119910 (x) =
sum119903
119897=1119910119897
(prod119899
119894=1120583119865119897119894(119909119894))
sum119903
119897=1prod119899
119894=1120583119865119897119894(119909119894)
(11)
where 119910119897
= max119910isin119877
120583119866119897(119910)
By introducing the concept of fuzzy basis function vectorthe final output of the fuzzy logic system can be expressed as
119910 (x) = 120579119879
120585 (x) (12)
where 120579 = [1199101
119910119903
]119879 is the adjustable parameter vector
and 120585(x) = [1205851(x) 120585
119903(x)]119879 is the fuzzy basis function
vector The fuzzy basis function is defined as follows
120585119897(x) =
sum119903
119897=1(prod119899
119894=1120583119865119897119894(119909119894))
sum119903
119897=1prod119899
119894=1120583119865119897119894(119909119894)
(13)
Lemma 3 Let 119891(x) be a continuous function defined on acompact set U
119888 Then for any constants 120576 gt 0 there exists a
fuzzy logic system such as [2]
supxisinU119888
10038161003816100381610038161003816119891 (x) minus 120579
119879
120585 (x)10038161003816100381610038161003816 le 120576 (14)
Define the optimal parameter vector 120579lowast as
120579lowast
= arg min120579isinΩ119891
[supxisinU119888
10038161003816100381610038161003816119891 (x) minus 120579
where ΩF ΩG and U119888denote the sets of suitable bounds on
120579119891119894 120579119892119894119895 and x respectively 120579lowast
119891119894(119894 = 1 119898) and 120579lowast
119892119894119895(119894 119895 =
1 119898) are constants vectorsThe unknown function 119891
119894and 119892
119894119895can be expressed as
119891119894(x) = 120579
lowast119879
119891119894120585119891119894
(x) + 120576119891119894
119894 = 1 119898
119892119894119895(x) = 120579
lowast119879
119892119894119895120585119892119894119895
(x) + 120576119892119894119895
119894 119895 = 1 119898
(20)
where 120576119891119894and 120576119892119894119895
are the smallest approximation errors of thefuzzy logic systems
3 Adaptive Fuzzy Robust Control Designs
In this section we first design the adaptive fuzzy approx-imation based control problem without control saturationand then consider the case where actuators have physicallimitations such as magnitude and rate constraints
31 Adaptive Fuzzy Robust Control The tracking errors aredefined as
119890119894= 119910119889119894
minus 119910119894
119894 = 1 2 119898 (21)
Define 119906119897119894
(119894 = 1 119898) as follows
119906119897119894= 119910(119903119894)
119889119894+
119903119894
sum
119895=1
119896119894119895119890(119895minus1)
119894119894 = 1 119898 (22)
where 1198961198941 119896
119894119903119894are parameters to be chosen such that the
roots of the equation 119904119903119894 + 119896119894119903119894119904119903119894minus1 + sdot sdot sdot + 119896
1198941= 0 in the open
left-half complex planeIf F(x) and G(x) are known external disturbances are
ignored then according to nonlinear dynamic inversioncontrol techniques the control law is given by
u = Gminus1 (x) [minusF (x) + u119897] (23)
where u119897= [1199061198971 119906
119897119898]119879
Because F(x) and G(x) are unknown 119889119894
= 0 thecontrol law (23) cannot be implemented in practice Usingthe approximation ofF(x) andG(x) and considering externaldisturbances the controller is modified as follows
u = Gminus1 (x | 120579119866) [minusF (x | 120579
119865) + u119897+ u119889] (24)
where u119889is the robust compensation term which is used to
attenuate the effect of external disturbances and approxima-tion errors
Substituting (24) into system (4) yields
y(119903) = F (x) minus F (x | 120579119865) + [G (x) minus G (x | 120579
119866)] u + u
119897
+ u119889+ d
(25)
Considering (22) the 119894th subsystem of (25) can be rewrit-ten as
e119894= A119894e119894+ B119894
[119891119894(x | 120579
119891119894) minus 119891119894(x)]
+
119898
sum
119895=1
[119892119894119895(x | 120579
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus B119894119906119889119894
minus B119894119889119894
(26)
where 119906119889119894is the 119894th element of u
119889and
A119894=
[[[[[[[[[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1198961198941
minus1198961198942
minus1198961198943
sdot sdot sdot minus119896119894119903119894
]]]]]]]]]
]
B119894=
[[[[[[[[[
[
0
0
0
1
]]]]]]]]]
]
e119894=
[[[[[[[[[[
[
119890119894
119890119894
119890(119903119894minus2)
119894
119890(119903119894minus1)
119894
]]]]]]]]]]
]
(27)
Define the minimum approximation error as
1205961015840
119894= [119891119894(x | 120579
lowast
119891119894) minus 119891119894(x)] +
119898
sum
119895=1
[119892119894119895(x | 120579
lowast
119892119894119895) minus 119892119894119895(x)] 119906
119895
(28)
Mathematical Problems in Engineering 5
According to fuzzy system theory the following assump-tion is reasonable
Assumption 4 The minimum approximation error is squareintegrable that is int119879
0
1205961015840119879
1198941205961015840
119894119889119905 lt infin
Using the definition (28) and the optimal approximationfor 119891119894(x) and 119892
119894119895(x) (26) can be rewritten in the following
form
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(29)
where 119891119894
= 120579119891119894
minus 120579lowast
119891119894 119892119894119895
= 120579119892119894119895
minus 120579lowast
119892119894119895 and 120596
119894= 1205961015840
119894minus 119889119894
Theorem 5 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance If the controller ischosen as (24) the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (38)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(39)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
The controller proposed by (24) is able to guarantee theLyapunov stability of the closed-loop system and attenuatethe effect of system uncertainties and external disturbancesHowever no saturation on actuators is taken into accountwhich is rather important for practical applications (includ-ing satellite systems) Assume that the control input u is con-straint by saturation functions it can be readily shown thatthe control law (24) the parameters update laws (30) (31)and the robust compensation term (32) with saturation limitscannot guarantee the stability of the closed-loop system
It is expected that during saturation the magnitude of thetracking error will increase since the control signal is notbeing achieved This tracking error is not the result of func-tion approximation error therefore we need to be careful so
6 Mathematical Problems in Engineering
that the approximator does not cause ldquounlearningrdquo duringthe period when the actuators are saturated Clearly theparameters update laws (30) and (31) depend on the trackingerror e
119894 thus if the tracking error increases due to saturation
the parameters update laws may cause a significant change intheweights in response to the increase in tracking error Nextwe develop a robust fuzzy adaptive control scheme to addressthe input saturation problem
32 Adaptive Fuzzy Control with Input Saturation When theactuators have physical constraints such as themagnitude andrate limitations the above approach may not be able to besuccessfully implemented Considering the magnitude andrate limitations on the actuator controller (24) is modified as
u119888= Gminus1 (x | 120579
119866) [minusF (x | 120579
119865) + u119897+ u119886119908
+ u119889]
u = 119878119872119877
(u119888)
(40)
where u119888is obtained by certainty equivalence principle u
119886119908
is the auxiliary control term which is used to compensate theeffect of input saturation 119878
119872119877(sdot) is a function including the
magnitude and rate constraints which can produce a limitedoutput
The state space representation of each component of u119888is
[
1199031198941
1199031198942
] = [
1199031198942
119878119877(120596119894(119878119872
(119906119888119894) minus 1199031198941))
]
[
119906119894
119894
] = [
1199031198941
1199031198942
]
(41)
where 119906119888119894is the 119894th element of u
119888 and 119906
119894is the 119894th element
of u 120596119894is the natural frequency and 119878
119872(sdot) and 119878
119877(sdot) are the
saturation functions corresponding to magnitude and raterespectively The function 119878
119872(sdot) is defined as
119878119872
(sdot) =
119872 if 119909 ge 119872
119909 if |119909| lt 119872
minus119872 if 119909 le minus119872
(42)
and 119878119877(sdot) has the same definition
To compensate the effects of input limitations an auxil-iary system is introduced as follows [35]
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus 119906119889119894
minus 119889119894
(48)
Define e119894
= [119890119894 119890119894 119890
(119903119894minus1)
119894]119879 and using the optimal
approximation for 119891119894(x) and 119892
119894119895(x) (48) can be rewritten as
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(49)
Theorem 6 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance and input satura-tion if the controller is chosen as (40) the auxiliary control
Mathematical Problems in Engineering 7
term u119886119908 the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (60)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(61)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
From the above analysis it is concluded that in the caseof no control saturations the signals 120585
119894119895(119894 = 1 119898 119895 =
1 119903119894) remain zeros and the control law becomes the same
as the standard robust adaptive control law described in theprevious section In the presence of control saturations 120585
119894119895
is nonzero thus giving rise to a modified tracking error119890119894= 119910119889119894
minus 119910119894minus 1205851198941 which is used in fuzzy parameter update
laws The auxiliary control term u119886119908
in (40) will be usedto compensate the effects of control saturations Note thatthe fuzzy parameter update laws (52) and (53) are similar tothe corresponding update laws (30) and (31) derived in thestandard fuzzy approximation based control problem withtracking error 119890
119894being replaced by the modified tracking
error 119890119894 The use of the modified tracking error in the fuzzy
update laws is crucial in preventing actuator constraints inonline approximation schemes
8 Mathematical Problems in Engineering
Corollary 7 For the 119894th subsystem of (4) it is assumed thatint119879
0
1198892
119894119889119905 lt infin If the control law (40) the auxiliary control term
(51) and the parameter update laws (52)ndash(54) are adoptedthen the following statements hold
(i) The closed loop system is stable and the signals e119894 120579119891119894
120579119892119894119895 and 119906
119894are bounded
(ii) Thesteadymodified tracking error satisfies lim119905rarrinfin
119890119894=
0 and the bound of the transient modified trackingerror will be given as follows
According to the definition of 119881119894 (62) is obtained
Assumption 4 implies int119879
0
12059610158402
119894119889119905 lt infin then int
119879
0
1205962
119894119889119905 =
int119879
0
(1205961015840
119894minus 119889119894)2
119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can
conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that
lim119905rarrinfin
119890119894(119905) = 0 This completes the proof
Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867
infin tracking performance with a prescribeddisturbance attenuation level 120588
119894 that is the 119871
2gain from 120596
119894
to the modified tracking error 119890119894is equal or less than 120588
119894
Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890
119894will converge to zero The bound of the
transient modified tracking error 119890119894is an explicit function
of the design parameters and the external disturbance andfuzzy approximation error 120596
119894 The bound can be decreased
by choosing the initial estimates 120579119891119894(0) 120579
119892119894119895(0) closing to
the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial
estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574
119891119894 120574119892119894119895
and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596
119894on the transient tracking
performance can be reduced by decreasing 120588119894and increasing
120582min(Q119894) Small 120588119894implies high disturbance attenuation level
4 Simulation Examples
The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]
y = F (x) + G (x) u + d (67)
where x = [q120596]119879 is the state vector where q =
[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed
reference frame relative to the inertial frame satisfying 1199022
0+
1199022
1+ 1199022
2+ 1199022
3= 1 here 119902
0is chosen as 119902
0= radic1 minus 119902
2
1minus 1199022
2minus 1199022
3
120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed
reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the
output vector u = [1199061 1199062 1199063]119879 is the control torque vector
d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows
1198911(x) = minus
1
41199021(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199022120596119909120596119910
+119868119909minus 119868119911
2119868119910
1199023120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199020120596119910120596119911
1198912(x) = minus
1
41199022(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119910minus 119868119909
2119868119911
1199021120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199020120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199023120596119910120596119911
Mathematical Problems in Engineering 9
1198913(x) = minus
1
41199023(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199020120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199021120596119909120596119911+
119868119911minus 119868119910
2119868119909
1199022120596119910120596119911
G (x) =
[[[[[[[
[
1199020
2119868119909
minus1199023
2119868119910
1199022
2119868119911
1199023
2119868119909
1199020
2119868119910
minus1199021
2119868119911
minus1199022
2119868119909
1199021
2119868119910
1199020
2119868119911
]]]]]]]
]
(68)
where 119868119909 119868119910 and 119868
119911are the principal central moments of
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
The design of tracking controllers for uncertain MIMOnonlinear systems with actuator constraints is a challengingproblem During the past decades there have been extensiveresearches on the control of nonlinear systems with variousconstraints Analysis and design of control systems with con-trol saturation have been widely studied in [24ndash42] Farrell etal [24ndash26] have presented an adaptive backstepping approachfor unknown nonlinear systems with known magnituderate and bandwidth constraints on intermediate states oractuators without disturbance To tackle with the physical sat-uration an auxiliary systemwith the same order as that of theplant was constructed to compensate the effect of saturationThe control input saturation is investigated through onlineapproximation based control for uncertain nonlinear systemsin [26] An adaptive control and the constrained adaptivecontrol in combination with the backstepping technique areproposed in [29] A direct adaptive fuzzy control approachfor uncertain nonlinear systems with input saturation ispresented in [30] in which a Nussbaum function is usedto compensate for the nonlinear term arising from theinput saturation In [31] an adaptive fuzzy output feedbackcontrol algorithm for a class of output constrained uncertainnonlinear systems with input saturation is developed byemploying a barrier Lyapunov function and an auxiliarysystem Adaptive backstepping tracking control based onfuzzy neural networks is investigated for unknown chaoticsystems in [32] and with control input constraints in [33]Neural network based adaptive control schemeswith externaldisturbances and actuator saturations are presented in [35]in which auxiliary systems are added to attenuate the effectsof input saturation It is apparent that the presence of inputsaturation constraint substantially increases the complexityof control system design for uncertain MIMO nonlinearsystems In the constrained adaptive control the key problemis how to handle the constraint effect of the actuatorrsquos physicalconstraints To this end we introduce an auxiliary designsystem to handle the constraint effect in this paper Basedon the states of the auxiliary design system constrainedadaptive control is investigated for a class of uncertainMIMOnonlinear systems with input constraints using robust fuzzycontrol technique
In this paper a robust adaptive fuzzy tracking controlscheme is presented to handle the external disturbances andactuator physical constraints for uncertain MIMO nonlinearsystems In control design fuzzy logic systems are used toapproximate unknown nonlinear systems Note that inputsaturations are nonsmooth functions but the adaptive fuzzycontrol technique requires all functions differentiable [6]To compensate the effect of input saturations an auxiliarysystem is constructed and the actuator saturations then canbe augmented into the controller The modified trackingerror is introduced and used in fuzzy parameter update lawsBesides in order to deal with fuzzy approximation errorsfor unknown nonlinear systems and external disturbancesa robust compensation control is designed It is proved thatthe proposed control approach can guarantee that all thesignals of the resulting closed-loop system are bounded andthe closed-loop system obtains 119867
infintracking performance
through Lyapunov analysis The transient modified tracking
errors performance is derived to be explicit functions ofdesign parameters and thus bounds of modified trackingerrors can be adjusted by tuning design parameters
The rest of this paper is organized as followsThe descrip-tion of the uncertain MIMO nonlinear system under consid-eration and necessary preliminaries are given in Section 2 InSection 3 the robust adaptive fuzzy control without controlsaturation is firstly designed When the actuators have phys-ical limitations this approach may not be able to be success-fully implemented In order to solve this problem a robustadaptive fuzzy control scheme with input saturation is inves-tigated The simulation results of satellite attitude controlare presented to demonstrate the effectiveness of proposedcontroller in Section 4 Section 5 contains the conclusion
2 Problem Formulation and Preliminary
21 Problem Formulation Consider a class of uncertainMIMO affine nonlinear system
x = f (x) +
119898
sum
119894=1
g119894(x) 119906119894+ d1015840
1199101= ℎ1(x)
119910119898
= ℎ119898
(x)
(1)
where x = [1199091 119909
119899] isin 119877
119899 is the state vector available formeasurement u = [119906
1 119906
119898] isin 119877
119898 is the control vectorwith |119906
119894| le 119906119894max where 119906
119894max denote the actuator amplitudey = [119910
1 119910
119898] isin 119877
119898 is the output vector ℎ1(x) ℎ
119898(x)
are smooth functions defined on the open set of R119899 f(x)g1(x) g
119898(x) are continuous unknown but smooth vector
fields d1015840 = [1198891015840
1 119889
1015840
119899]119879
isin R119899 is external disturbance vectorwhere 119889
1015840
1is unknown but bounded
Define that 119871 fℎ119894 is the Lie derivative of ℎ119894along vector
field f(x) and 119871119896
fℎ119894 is recursively as 119871119896
fℎ119894 = 119871 f(119871119896minus1
f ℎ119894) A
multivariable uncertain nonlinear system of the form (1) hasa vector relative degree [119903
1 1199032 119903
119898] at a point x
0if
119871g119895119871119896
fℎ119894 (x) = 0 forall1 le 119895 le 119898 119896 le 119903119894minus 1 (2)
Using feedback linearization the nonlinear system (1) canbe transformed into the following form [43]
y(r) = F (x) + G (x) u + d (4)
where
y(r) = [119910(1199031)
1 119910(1199032)
2 119910
(119903119898)
119898]119879
(5)
F (x) =
[[[[[[[
[
1198711199031
f ℎ1 (x)
1198711199032
f ℎ2 (x)
119871119903119898
f ℎ119898
(x)
]]]]]]]
]
=
[[[[[[
[
1198911(x)
1198912(x)
119891119898
(x)
]]]]]]
]
(6)
d =
[[[[[[[[[[[[[
[
1199031
sum
119894=1
119871119894minus1
x 11987111988910158401198711199031minus119894
f ℎ1(x)
1199032
sum
119894=1
119871119894minus1
x 11987111988910158401198711199032minus119894
f ℎ2(x)
119903119898
sum
119894=1
119871119894minus1
x 1198711198891015840119871119903119898minus119894
f ℎ119898
(x)
]]]]]]]]]]]]]
]
=
[[[[[[
[
1198891
1198892
119889119898
]]]]]]
]
(7)
x = [1199101 119910
(1199031)
1 1199102 119910
(1199032)
2 119910
119898 119910
(119903119898)
119898] (8)
In system (4) d still represents unknown boundedexternal disturbance vector The relative degree of the systemis assumed to be equal to the order of the system and isexpressed as sum
119898
119894=1119903119894= 119899 which implies that the system does
not have any zero dynamicsLet the desired output trajectory be given by
y119889= [1199101198891
1199101198892
119910119889119898
]119879
(9)
This paper aims to develop a robust adaptive fuzzy track-ing control scheme such that all closed-loop system signalsasymptotically converge to a compact set in the presence ofinput saturation system uncertainties and external distur-bances and ensure the system outputs y track the desiredtrajectory y
119889 For system (4) to be controllable the following
assumptions are made
Assumption 1 The desired trajectory 119910119889119894
and its 119899th orderderivatives are known and bounded
Assumption 2 For x in certain controllability regionU119888isin 119877119899
120590(G(x)) = 0 where 120590(G(x)) denotes the minimum singularvalue of the matrix G(x)
22 Fuzzy Logic Systems In this paper fuzzy logic systemsare used to approximate unknown nonlinear functions F(x)and G(x) Following the approximation presentation offuzzy logic systems is given Without loss of generality theunknown uncertainty is assumed as 119891(x) A fuzzy logic sys-tem consists of four parts the knowledge base the fuzzifier
the fuzzy inference engine working on fuzzy rules and thedefuzzifier The fuzzy inference engine uses fuzzy IF-THENrules to perform a mapping from an input linguistic vectorx = [119909
1 1199092 119909
119899]119879
isin 119877119899 to an output variable 119910 isin 119877 Then
the 119894th fuzzy rule can be represented as [3]
119877119897
If 1199091is 119865119897
1and sdot sdot sdot and 119909
119899is 119865119897
119899 then 119910 is 119866
119897
(119897 = 1 2 119903)
(10)
where 119865119897
119894and 119866
119897 are fuzzy sets characterized with the fuzzymembership functions 120583
119865119897119894(119909119894) and 120583
119866119897(119910) respectively and 119903
is the number of fuzzy rulesThrough singleton function center average defuzzifier
and product inference the fuzzy logic system can beexpressed as follows
119910 (x) =
sum119903
119897=1119910119897
(prod119899
119894=1120583119865119897119894(119909119894))
sum119903
119897=1prod119899
119894=1120583119865119897119894(119909119894)
(11)
where 119910119897
= max119910isin119877
120583119866119897(119910)
By introducing the concept of fuzzy basis function vectorthe final output of the fuzzy logic system can be expressed as
119910 (x) = 120579119879
120585 (x) (12)
where 120579 = [1199101
119910119903
]119879 is the adjustable parameter vector
and 120585(x) = [1205851(x) 120585
119903(x)]119879 is the fuzzy basis function
vector The fuzzy basis function is defined as follows
120585119897(x) =
sum119903
119897=1(prod119899
119894=1120583119865119897119894(119909119894))
sum119903
119897=1prod119899
119894=1120583119865119897119894(119909119894)
(13)
Lemma 3 Let 119891(x) be a continuous function defined on acompact set U
119888 Then for any constants 120576 gt 0 there exists a
fuzzy logic system such as [2]
supxisinU119888
10038161003816100381610038161003816119891 (x) minus 120579
119879
120585 (x)10038161003816100381610038161003816 le 120576 (14)
Define the optimal parameter vector 120579lowast as
120579lowast
= arg min120579isinΩ119891
[supxisinU119888
10038161003816100381610038161003816119891 (x) minus 120579
where ΩF ΩG and U119888denote the sets of suitable bounds on
120579119891119894 120579119892119894119895 and x respectively 120579lowast
119891119894(119894 = 1 119898) and 120579lowast
119892119894119895(119894 119895 =
1 119898) are constants vectorsThe unknown function 119891
119894and 119892
119894119895can be expressed as
119891119894(x) = 120579
lowast119879
119891119894120585119891119894
(x) + 120576119891119894
119894 = 1 119898
119892119894119895(x) = 120579
lowast119879
119892119894119895120585119892119894119895
(x) + 120576119892119894119895
119894 119895 = 1 119898
(20)
where 120576119891119894and 120576119892119894119895
are the smallest approximation errors of thefuzzy logic systems
3 Adaptive Fuzzy Robust Control Designs
In this section we first design the adaptive fuzzy approx-imation based control problem without control saturationand then consider the case where actuators have physicallimitations such as magnitude and rate constraints
31 Adaptive Fuzzy Robust Control The tracking errors aredefined as
119890119894= 119910119889119894
minus 119910119894
119894 = 1 2 119898 (21)
Define 119906119897119894
(119894 = 1 119898) as follows
119906119897119894= 119910(119903119894)
119889119894+
119903119894
sum
119895=1
119896119894119895119890(119895minus1)
119894119894 = 1 119898 (22)
where 1198961198941 119896
119894119903119894are parameters to be chosen such that the
roots of the equation 119904119903119894 + 119896119894119903119894119904119903119894minus1 + sdot sdot sdot + 119896
1198941= 0 in the open
left-half complex planeIf F(x) and G(x) are known external disturbances are
ignored then according to nonlinear dynamic inversioncontrol techniques the control law is given by
u = Gminus1 (x) [minusF (x) + u119897] (23)
where u119897= [1199061198971 119906
119897119898]119879
Because F(x) and G(x) are unknown 119889119894
= 0 thecontrol law (23) cannot be implemented in practice Usingthe approximation ofF(x) andG(x) and considering externaldisturbances the controller is modified as follows
u = Gminus1 (x | 120579119866) [minusF (x | 120579
119865) + u119897+ u119889] (24)
where u119889is the robust compensation term which is used to
attenuate the effect of external disturbances and approxima-tion errors
Substituting (24) into system (4) yields
y(119903) = F (x) minus F (x | 120579119865) + [G (x) minus G (x | 120579
119866)] u + u
119897
+ u119889+ d
(25)
Considering (22) the 119894th subsystem of (25) can be rewrit-ten as
e119894= A119894e119894+ B119894
[119891119894(x | 120579
119891119894) minus 119891119894(x)]
+
119898
sum
119895=1
[119892119894119895(x | 120579
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus B119894119906119889119894
minus B119894119889119894
(26)
where 119906119889119894is the 119894th element of u
119889and
A119894=
[[[[[[[[[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1198961198941
minus1198961198942
minus1198961198943
sdot sdot sdot minus119896119894119903119894
]]]]]]]]]
]
B119894=
[[[[[[[[[
[
0
0
0
1
]]]]]]]]]
]
e119894=
[[[[[[[[[[
[
119890119894
119890119894
119890(119903119894minus2)
119894
119890(119903119894minus1)
119894
]]]]]]]]]]
]
(27)
Define the minimum approximation error as
1205961015840
119894= [119891119894(x | 120579
lowast
119891119894) minus 119891119894(x)] +
119898
sum
119895=1
[119892119894119895(x | 120579
lowast
119892119894119895) minus 119892119894119895(x)] 119906
119895
(28)
Mathematical Problems in Engineering 5
According to fuzzy system theory the following assump-tion is reasonable
Assumption 4 The minimum approximation error is squareintegrable that is int119879
0
1205961015840119879
1198941205961015840
119894119889119905 lt infin
Using the definition (28) and the optimal approximationfor 119891119894(x) and 119892
119894119895(x) (26) can be rewritten in the following
form
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(29)
where 119891119894
= 120579119891119894
minus 120579lowast
119891119894 119892119894119895
= 120579119892119894119895
minus 120579lowast
119892119894119895 and 120596
119894= 1205961015840
119894minus 119889119894
Theorem 5 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance If the controller ischosen as (24) the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (38)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(39)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
The controller proposed by (24) is able to guarantee theLyapunov stability of the closed-loop system and attenuatethe effect of system uncertainties and external disturbancesHowever no saturation on actuators is taken into accountwhich is rather important for practical applications (includ-ing satellite systems) Assume that the control input u is con-straint by saturation functions it can be readily shown thatthe control law (24) the parameters update laws (30) (31)and the robust compensation term (32) with saturation limitscannot guarantee the stability of the closed-loop system
It is expected that during saturation the magnitude of thetracking error will increase since the control signal is notbeing achieved This tracking error is not the result of func-tion approximation error therefore we need to be careful so
6 Mathematical Problems in Engineering
that the approximator does not cause ldquounlearningrdquo duringthe period when the actuators are saturated Clearly theparameters update laws (30) and (31) depend on the trackingerror e
119894 thus if the tracking error increases due to saturation
the parameters update laws may cause a significant change intheweights in response to the increase in tracking error Nextwe develop a robust fuzzy adaptive control scheme to addressthe input saturation problem
32 Adaptive Fuzzy Control with Input Saturation When theactuators have physical constraints such as themagnitude andrate limitations the above approach may not be able to besuccessfully implemented Considering the magnitude andrate limitations on the actuator controller (24) is modified as
u119888= Gminus1 (x | 120579
119866) [minusF (x | 120579
119865) + u119897+ u119886119908
+ u119889]
u = 119878119872119877
(u119888)
(40)
where u119888is obtained by certainty equivalence principle u
119886119908
is the auxiliary control term which is used to compensate theeffect of input saturation 119878
119872119877(sdot) is a function including the
magnitude and rate constraints which can produce a limitedoutput
The state space representation of each component of u119888is
[
1199031198941
1199031198942
] = [
1199031198942
119878119877(120596119894(119878119872
(119906119888119894) minus 1199031198941))
]
[
119906119894
119894
] = [
1199031198941
1199031198942
]
(41)
where 119906119888119894is the 119894th element of u
119888 and 119906
119894is the 119894th element
of u 120596119894is the natural frequency and 119878
119872(sdot) and 119878
119877(sdot) are the
saturation functions corresponding to magnitude and raterespectively The function 119878
119872(sdot) is defined as
119878119872
(sdot) =
119872 if 119909 ge 119872
119909 if |119909| lt 119872
minus119872 if 119909 le minus119872
(42)
and 119878119877(sdot) has the same definition
To compensate the effects of input limitations an auxil-iary system is introduced as follows [35]
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus 119906119889119894
minus 119889119894
(48)
Define e119894
= [119890119894 119890119894 119890
(119903119894minus1)
119894]119879 and using the optimal
approximation for 119891119894(x) and 119892
119894119895(x) (48) can be rewritten as
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(49)
Theorem 6 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance and input satura-tion if the controller is chosen as (40) the auxiliary control
Mathematical Problems in Engineering 7
term u119886119908 the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (60)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(61)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
From the above analysis it is concluded that in the caseof no control saturations the signals 120585
119894119895(119894 = 1 119898 119895 =
1 119903119894) remain zeros and the control law becomes the same
as the standard robust adaptive control law described in theprevious section In the presence of control saturations 120585
119894119895
is nonzero thus giving rise to a modified tracking error119890119894= 119910119889119894
minus 119910119894minus 1205851198941 which is used in fuzzy parameter update
laws The auxiliary control term u119886119908
in (40) will be usedto compensate the effects of control saturations Note thatthe fuzzy parameter update laws (52) and (53) are similar tothe corresponding update laws (30) and (31) derived in thestandard fuzzy approximation based control problem withtracking error 119890
119894being replaced by the modified tracking
error 119890119894 The use of the modified tracking error in the fuzzy
update laws is crucial in preventing actuator constraints inonline approximation schemes
8 Mathematical Problems in Engineering
Corollary 7 For the 119894th subsystem of (4) it is assumed thatint119879
0
1198892
119894119889119905 lt infin If the control law (40) the auxiliary control term
(51) and the parameter update laws (52)ndash(54) are adoptedthen the following statements hold
(i) The closed loop system is stable and the signals e119894 120579119891119894
120579119892119894119895 and 119906
119894are bounded
(ii) Thesteadymodified tracking error satisfies lim119905rarrinfin
119890119894=
0 and the bound of the transient modified trackingerror will be given as follows
According to the definition of 119881119894 (62) is obtained
Assumption 4 implies int119879
0
12059610158402
119894119889119905 lt infin then int
119879
0
1205962
119894119889119905 =
int119879
0
(1205961015840
119894minus 119889119894)2
119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can
conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that
lim119905rarrinfin
119890119894(119905) = 0 This completes the proof
Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867
infin tracking performance with a prescribeddisturbance attenuation level 120588
119894 that is the 119871
2gain from 120596
119894
to the modified tracking error 119890119894is equal or less than 120588
119894
Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890
119894will converge to zero The bound of the
transient modified tracking error 119890119894is an explicit function
of the design parameters and the external disturbance andfuzzy approximation error 120596
119894 The bound can be decreased
by choosing the initial estimates 120579119891119894(0) 120579
119892119894119895(0) closing to
the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial
estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574
119891119894 120574119892119894119895
and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596
119894on the transient tracking
performance can be reduced by decreasing 120588119894and increasing
120582min(Q119894) Small 120588119894implies high disturbance attenuation level
4 Simulation Examples
The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]
y = F (x) + G (x) u + d (67)
where x = [q120596]119879 is the state vector where q =
[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed
reference frame relative to the inertial frame satisfying 1199022
0+
1199022
1+ 1199022
2+ 1199022
3= 1 here 119902
0is chosen as 119902
0= radic1 minus 119902
2
1minus 1199022
2minus 1199022
3
120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed
reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the
output vector u = [1199061 1199062 1199063]119879 is the control torque vector
d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows
1198911(x) = minus
1
41199021(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199022120596119909120596119910
+119868119909minus 119868119911
2119868119910
1199023120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199020120596119910120596119911
1198912(x) = minus
1
41199022(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119910minus 119868119909
2119868119911
1199021120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199020120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199023120596119910120596119911
Mathematical Problems in Engineering 9
1198913(x) = minus
1
41199023(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199020120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199021120596119909120596119911+
119868119911minus 119868119910
2119868119909
1199022120596119910120596119911
G (x) =
[[[[[[[
[
1199020
2119868119909
minus1199023
2119868119910
1199022
2119868119911
1199023
2119868119909
1199020
2119868119910
minus1199021
2119868119911
minus1199022
2119868119909
1199021
2119868119910
1199020
2119868119911
]]]]]]]
]
(68)
where 119868119909 119868119910 and 119868
119911are the principal central moments of
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
Using feedback linearization the nonlinear system (1) canbe transformed into the following form [43]
y(r) = F (x) + G (x) u + d (4)
where
y(r) = [119910(1199031)
1 119910(1199032)
2 119910
(119903119898)
119898]119879
(5)
F (x) =
[[[[[[[
[
1198711199031
f ℎ1 (x)
1198711199032
f ℎ2 (x)
119871119903119898
f ℎ119898
(x)
]]]]]]]
]
=
[[[[[[
[
1198911(x)
1198912(x)
119891119898
(x)
]]]]]]
]
(6)
d =
[[[[[[[[[[[[[
[
1199031
sum
119894=1
119871119894minus1
x 11987111988910158401198711199031minus119894
f ℎ1(x)
1199032
sum
119894=1
119871119894minus1
x 11987111988910158401198711199032minus119894
f ℎ2(x)
119903119898
sum
119894=1
119871119894minus1
x 1198711198891015840119871119903119898minus119894
f ℎ119898
(x)
]]]]]]]]]]]]]
]
=
[[[[[[
[
1198891
1198892
119889119898
]]]]]]
]
(7)
x = [1199101 119910
(1199031)
1 1199102 119910
(1199032)
2 119910
119898 119910
(119903119898)
119898] (8)
In system (4) d still represents unknown boundedexternal disturbance vector The relative degree of the systemis assumed to be equal to the order of the system and isexpressed as sum
119898
119894=1119903119894= 119899 which implies that the system does
not have any zero dynamicsLet the desired output trajectory be given by
y119889= [1199101198891
1199101198892
119910119889119898
]119879
(9)
This paper aims to develop a robust adaptive fuzzy track-ing control scheme such that all closed-loop system signalsasymptotically converge to a compact set in the presence ofinput saturation system uncertainties and external distur-bances and ensure the system outputs y track the desiredtrajectory y
119889 For system (4) to be controllable the following
assumptions are made
Assumption 1 The desired trajectory 119910119889119894
and its 119899th orderderivatives are known and bounded
Assumption 2 For x in certain controllability regionU119888isin 119877119899
120590(G(x)) = 0 where 120590(G(x)) denotes the minimum singularvalue of the matrix G(x)
22 Fuzzy Logic Systems In this paper fuzzy logic systemsare used to approximate unknown nonlinear functions F(x)and G(x) Following the approximation presentation offuzzy logic systems is given Without loss of generality theunknown uncertainty is assumed as 119891(x) A fuzzy logic sys-tem consists of four parts the knowledge base the fuzzifier
the fuzzy inference engine working on fuzzy rules and thedefuzzifier The fuzzy inference engine uses fuzzy IF-THENrules to perform a mapping from an input linguistic vectorx = [119909
1 1199092 119909
119899]119879
isin 119877119899 to an output variable 119910 isin 119877 Then
the 119894th fuzzy rule can be represented as [3]
119877119897
If 1199091is 119865119897
1and sdot sdot sdot and 119909
119899is 119865119897
119899 then 119910 is 119866
119897
(119897 = 1 2 119903)
(10)
where 119865119897
119894and 119866
119897 are fuzzy sets characterized with the fuzzymembership functions 120583
119865119897119894(119909119894) and 120583
119866119897(119910) respectively and 119903
is the number of fuzzy rulesThrough singleton function center average defuzzifier
and product inference the fuzzy logic system can beexpressed as follows
119910 (x) =
sum119903
119897=1119910119897
(prod119899
119894=1120583119865119897119894(119909119894))
sum119903
119897=1prod119899
119894=1120583119865119897119894(119909119894)
(11)
where 119910119897
= max119910isin119877
120583119866119897(119910)
By introducing the concept of fuzzy basis function vectorthe final output of the fuzzy logic system can be expressed as
119910 (x) = 120579119879
120585 (x) (12)
where 120579 = [1199101
119910119903
]119879 is the adjustable parameter vector
and 120585(x) = [1205851(x) 120585
119903(x)]119879 is the fuzzy basis function
vector The fuzzy basis function is defined as follows
120585119897(x) =
sum119903
119897=1(prod119899
119894=1120583119865119897119894(119909119894))
sum119903
119897=1prod119899
119894=1120583119865119897119894(119909119894)
(13)
Lemma 3 Let 119891(x) be a continuous function defined on acompact set U
119888 Then for any constants 120576 gt 0 there exists a
fuzzy logic system such as [2]
supxisinU119888
10038161003816100381610038161003816119891 (x) minus 120579
119879
120585 (x)10038161003816100381610038161003816 le 120576 (14)
Define the optimal parameter vector 120579lowast as
120579lowast
= arg min120579isinΩ119891
[supxisinU119888
10038161003816100381610038161003816119891 (x) minus 120579
where ΩF ΩG and U119888denote the sets of suitable bounds on
120579119891119894 120579119892119894119895 and x respectively 120579lowast
119891119894(119894 = 1 119898) and 120579lowast
119892119894119895(119894 119895 =
1 119898) are constants vectorsThe unknown function 119891
119894and 119892
119894119895can be expressed as
119891119894(x) = 120579
lowast119879
119891119894120585119891119894
(x) + 120576119891119894
119894 = 1 119898
119892119894119895(x) = 120579
lowast119879
119892119894119895120585119892119894119895
(x) + 120576119892119894119895
119894 119895 = 1 119898
(20)
where 120576119891119894and 120576119892119894119895
are the smallest approximation errors of thefuzzy logic systems
3 Adaptive Fuzzy Robust Control Designs
In this section we first design the adaptive fuzzy approx-imation based control problem without control saturationand then consider the case where actuators have physicallimitations such as magnitude and rate constraints
31 Adaptive Fuzzy Robust Control The tracking errors aredefined as
119890119894= 119910119889119894
minus 119910119894
119894 = 1 2 119898 (21)
Define 119906119897119894
(119894 = 1 119898) as follows
119906119897119894= 119910(119903119894)
119889119894+
119903119894
sum
119895=1
119896119894119895119890(119895minus1)
119894119894 = 1 119898 (22)
where 1198961198941 119896
119894119903119894are parameters to be chosen such that the
roots of the equation 119904119903119894 + 119896119894119903119894119904119903119894minus1 + sdot sdot sdot + 119896
1198941= 0 in the open
left-half complex planeIf F(x) and G(x) are known external disturbances are
ignored then according to nonlinear dynamic inversioncontrol techniques the control law is given by
u = Gminus1 (x) [minusF (x) + u119897] (23)
where u119897= [1199061198971 119906
119897119898]119879
Because F(x) and G(x) are unknown 119889119894
= 0 thecontrol law (23) cannot be implemented in practice Usingthe approximation ofF(x) andG(x) and considering externaldisturbances the controller is modified as follows
u = Gminus1 (x | 120579119866) [minusF (x | 120579
119865) + u119897+ u119889] (24)
where u119889is the robust compensation term which is used to
attenuate the effect of external disturbances and approxima-tion errors
Substituting (24) into system (4) yields
y(119903) = F (x) minus F (x | 120579119865) + [G (x) minus G (x | 120579
119866)] u + u
119897
+ u119889+ d
(25)
Considering (22) the 119894th subsystem of (25) can be rewrit-ten as
e119894= A119894e119894+ B119894
[119891119894(x | 120579
119891119894) minus 119891119894(x)]
+
119898
sum
119895=1
[119892119894119895(x | 120579
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus B119894119906119889119894
minus B119894119889119894
(26)
where 119906119889119894is the 119894th element of u
119889and
A119894=
[[[[[[[[[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1198961198941
minus1198961198942
minus1198961198943
sdot sdot sdot minus119896119894119903119894
]]]]]]]]]
]
B119894=
[[[[[[[[[
[
0
0
0
1
]]]]]]]]]
]
e119894=
[[[[[[[[[[
[
119890119894
119890119894
119890(119903119894minus2)
119894
119890(119903119894minus1)
119894
]]]]]]]]]]
]
(27)
Define the minimum approximation error as
1205961015840
119894= [119891119894(x | 120579
lowast
119891119894) minus 119891119894(x)] +
119898
sum
119895=1
[119892119894119895(x | 120579
lowast
119892119894119895) minus 119892119894119895(x)] 119906
119895
(28)
Mathematical Problems in Engineering 5
According to fuzzy system theory the following assump-tion is reasonable
Assumption 4 The minimum approximation error is squareintegrable that is int119879
0
1205961015840119879
1198941205961015840
119894119889119905 lt infin
Using the definition (28) and the optimal approximationfor 119891119894(x) and 119892
119894119895(x) (26) can be rewritten in the following
form
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(29)
where 119891119894
= 120579119891119894
minus 120579lowast
119891119894 119892119894119895
= 120579119892119894119895
minus 120579lowast
119892119894119895 and 120596
119894= 1205961015840
119894minus 119889119894
Theorem 5 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance If the controller ischosen as (24) the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (38)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(39)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
The controller proposed by (24) is able to guarantee theLyapunov stability of the closed-loop system and attenuatethe effect of system uncertainties and external disturbancesHowever no saturation on actuators is taken into accountwhich is rather important for practical applications (includ-ing satellite systems) Assume that the control input u is con-straint by saturation functions it can be readily shown thatthe control law (24) the parameters update laws (30) (31)and the robust compensation term (32) with saturation limitscannot guarantee the stability of the closed-loop system
It is expected that during saturation the magnitude of thetracking error will increase since the control signal is notbeing achieved This tracking error is not the result of func-tion approximation error therefore we need to be careful so
6 Mathematical Problems in Engineering
that the approximator does not cause ldquounlearningrdquo duringthe period when the actuators are saturated Clearly theparameters update laws (30) and (31) depend on the trackingerror e
119894 thus if the tracking error increases due to saturation
the parameters update laws may cause a significant change intheweights in response to the increase in tracking error Nextwe develop a robust fuzzy adaptive control scheme to addressthe input saturation problem
32 Adaptive Fuzzy Control with Input Saturation When theactuators have physical constraints such as themagnitude andrate limitations the above approach may not be able to besuccessfully implemented Considering the magnitude andrate limitations on the actuator controller (24) is modified as
u119888= Gminus1 (x | 120579
119866) [minusF (x | 120579
119865) + u119897+ u119886119908
+ u119889]
u = 119878119872119877
(u119888)
(40)
where u119888is obtained by certainty equivalence principle u
119886119908
is the auxiliary control term which is used to compensate theeffect of input saturation 119878
119872119877(sdot) is a function including the
magnitude and rate constraints which can produce a limitedoutput
The state space representation of each component of u119888is
[
1199031198941
1199031198942
] = [
1199031198942
119878119877(120596119894(119878119872
(119906119888119894) minus 1199031198941))
]
[
119906119894
119894
] = [
1199031198941
1199031198942
]
(41)
where 119906119888119894is the 119894th element of u
119888 and 119906
119894is the 119894th element
of u 120596119894is the natural frequency and 119878
119872(sdot) and 119878
119877(sdot) are the
saturation functions corresponding to magnitude and raterespectively The function 119878
119872(sdot) is defined as
119878119872
(sdot) =
119872 if 119909 ge 119872
119909 if |119909| lt 119872
minus119872 if 119909 le minus119872
(42)
and 119878119877(sdot) has the same definition
To compensate the effects of input limitations an auxil-iary system is introduced as follows [35]
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus 119906119889119894
minus 119889119894
(48)
Define e119894
= [119890119894 119890119894 119890
(119903119894minus1)
119894]119879 and using the optimal
approximation for 119891119894(x) and 119892
119894119895(x) (48) can be rewritten as
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(49)
Theorem 6 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance and input satura-tion if the controller is chosen as (40) the auxiliary control
Mathematical Problems in Engineering 7
term u119886119908 the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (60)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(61)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
From the above analysis it is concluded that in the caseof no control saturations the signals 120585
119894119895(119894 = 1 119898 119895 =
1 119903119894) remain zeros and the control law becomes the same
as the standard robust adaptive control law described in theprevious section In the presence of control saturations 120585
119894119895
is nonzero thus giving rise to a modified tracking error119890119894= 119910119889119894
minus 119910119894minus 1205851198941 which is used in fuzzy parameter update
laws The auxiliary control term u119886119908
in (40) will be usedto compensate the effects of control saturations Note thatthe fuzzy parameter update laws (52) and (53) are similar tothe corresponding update laws (30) and (31) derived in thestandard fuzzy approximation based control problem withtracking error 119890
119894being replaced by the modified tracking
error 119890119894 The use of the modified tracking error in the fuzzy
update laws is crucial in preventing actuator constraints inonline approximation schemes
8 Mathematical Problems in Engineering
Corollary 7 For the 119894th subsystem of (4) it is assumed thatint119879
0
1198892
119894119889119905 lt infin If the control law (40) the auxiliary control term
(51) and the parameter update laws (52)ndash(54) are adoptedthen the following statements hold
(i) The closed loop system is stable and the signals e119894 120579119891119894
120579119892119894119895 and 119906
119894are bounded
(ii) Thesteadymodified tracking error satisfies lim119905rarrinfin
119890119894=
0 and the bound of the transient modified trackingerror will be given as follows
According to the definition of 119881119894 (62) is obtained
Assumption 4 implies int119879
0
12059610158402
119894119889119905 lt infin then int
119879
0
1205962
119894119889119905 =
int119879
0
(1205961015840
119894minus 119889119894)2
119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can
conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that
lim119905rarrinfin
119890119894(119905) = 0 This completes the proof
Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867
infin tracking performance with a prescribeddisturbance attenuation level 120588
119894 that is the 119871
2gain from 120596
119894
to the modified tracking error 119890119894is equal or less than 120588
119894
Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890
119894will converge to zero The bound of the
transient modified tracking error 119890119894is an explicit function
of the design parameters and the external disturbance andfuzzy approximation error 120596
119894 The bound can be decreased
by choosing the initial estimates 120579119891119894(0) 120579
119892119894119895(0) closing to
the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial
estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574
119891119894 120574119892119894119895
and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596
119894on the transient tracking
performance can be reduced by decreasing 120588119894and increasing
120582min(Q119894) Small 120588119894implies high disturbance attenuation level
4 Simulation Examples
The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]
y = F (x) + G (x) u + d (67)
where x = [q120596]119879 is the state vector where q =
[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed
reference frame relative to the inertial frame satisfying 1199022
0+
1199022
1+ 1199022
2+ 1199022
3= 1 here 119902
0is chosen as 119902
0= radic1 minus 119902
2
1minus 1199022
2minus 1199022
3
120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed
reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the
output vector u = [1199061 1199062 1199063]119879 is the control torque vector
d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows
1198911(x) = minus
1
41199021(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199022120596119909120596119910
+119868119909minus 119868119911
2119868119910
1199023120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199020120596119910120596119911
1198912(x) = minus
1
41199022(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119910minus 119868119909
2119868119911
1199021120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199020120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199023120596119910120596119911
Mathematical Problems in Engineering 9
1198913(x) = minus
1
41199023(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199020120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199021120596119909120596119911+
119868119911minus 119868119910
2119868119909
1199022120596119910120596119911
G (x) =
[[[[[[[
[
1199020
2119868119909
minus1199023
2119868119910
1199022
2119868119911
1199023
2119868119909
1199020
2119868119910
minus1199021
2119868119911
minus1199022
2119868119909
1199021
2119868119910
1199020
2119868119911
]]]]]]]
]
(68)
where 119868119909 119868119910 and 119868
119911are the principal central moments of
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
where ΩF ΩG and U119888denote the sets of suitable bounds on
120579119891119894 120579119892119894119895 and x respectively 120579lowast
119891119894(119894 = 1 119898) and 120579lowast
119892119894119895(119894 119895 =
1 119898) are constants vectorsThe unknown function 119891
119894and 119892
119894119895can be expressed as
119891119894(x) = 120579
lowast119879
119891119894120585119891119894
(x) + 120576119891119894
119894 = 1 119898
119892119894119895(x) = 120579
lowast119879
119892119894119895120585119892119894119895
(x) + 120576119892119894119895
119894 119895 = 1 119898
(20)
where 120576119891119894and 120576119892119894119895
are the smallest approximation errors of thefuzzy logic systems
3 Adaptive Fuzzy Robust Control Designs
In this section we first design the adaptive fuzzy approx-imation based control problem without control saturationand then consider the case where actuators have physicallimitations such as magnitude and rate constraints
31 Adaptive Fuzzy Robust Control The tracking errors aredefined as
119890119894= 119910119889119894
minus 119910119894
119894 = 1 2 119898 (21)
Define 119906119897119894
(119894 = 1 119898) as follows
119906119897119894= 119910(119903119894)
119889119894+
119903119894
sum
119895=1
119896119894119895119890(119895minus1)
119894119894 = 1 119898 (22)
where 1198961198941 119896
119894119903119894are parameters to be chosen such that the
roots of the equation 119904119903119894 + 119896119894119903119894119904119903119894minus1 + sdot sdot sdot + 119896
1198941= 0 in the open
left-half complex planeIf F(x) and G(x) are known external disturbances are
ignored then according to nonlinear dynamic inversioncontrol techniques the control law is given by
u = Gminus1 (x) [minusF (x) + u119897] (23)
where u119897= [1199061198971 119906
119897119898]119879
Because F(x) and G(x) are unknown 119889119894
= 0 thecontrol law (23) cannot be implemented in practice Usingthe approximation ofF(x) andG(x) and considering externaldisturbances the controller is modified as follows
u = Gminus1 (x | 120579119866) [minusF (x | 120579
119865) + u119897+ u119889] (24)
where u119889is the robust compensation term which is used to
attenuate the effect of external disturbances and approxima-tion errors
Substituting (24) into system (4) yields
y(119903) = F (x) minus F (x | 120579119865) + [G (x) minus G (x | 120579
119866)] u + u
119897
+ u119889+ d
(25)
Considering (22) the 119894th subsystem of (25) can be rewrit-ten as
e119894= A119894e119894+ B119894
[119891119894(x | 120579
119891119894) minus 119891119894(x)]
+
119898
sum
119895=1
[119892119894119895(x | 120579
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus B119894119906119889119894
minus B119894119889119894
(26)
where 119906119889119894is the 119894th element of u
119889and
A119894=
[[[[[[[[[
[
0 1 0 sdot sdot sdot 0
0 0 1 sdot sdot sdot 0
d
0 0 0 sdot sdot sdot 1
minus1198961198941
minus1198961198942
minus1198961198943
sdot sdot sdot minus119896119894119903119894
]]]]]]]]]
]
B119894=
[[[[[[[[[
[
0
0
0
1
]]]]]]]]]
]
e119894=
[[[[[[[[[[
[
119890119894
119890119894
119890(119903119894minus2)
119894
119890(119903119894minus1)
119894
]]]]]]]]]]
]
(27)
Define the minimum approximation error as
1205961015840
119894= [119891119894(x | 120579
lowast
119891119894) minus 119891119894(x)] +
119898
sum
119895=1
[119892119894119895(x | 120579
lowast
119892119894119895) minus 119892119894119895(x)] 119906
119895
(28)
Mathematical Problems in Engineering 5
According to fuzzy system theory the following assump-tion is reasonable
Assumption 4 The minimum approximation error is squareintegrable that is int119879
0
1205961015840119879
1198941205961015840
119894119889119905 lt infin
Using the definition (28) and the optimal approximationfor 119891119894(x) and 119892
119894119895(x) (26) can be rewritten in the following
form
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(29)
where 119891119894
= 120579119891119894
minus 120579lowast
119891119894 119892119894119895
= 120579119892119894119895
minus 120579lowast
119892119894119895 and 120596
119894= 1205961015840
119894minus 119889119894
Theorem 5 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance If the controller ischosen as (24) the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (38)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(39)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
The controller proposed by (24) is able to guarantee theLyapunov stability of the closed-loop system and attenuatethe effect of system uncertainties and external disturbancesHowever no saturation on actuators is taken into accountwhich is rather important for practical applications (includ-ing satellite systems) Assume that the control input u is con-straint by saturation functions it can be readily shown thatthe control law (24) the parameters update laws (30) (31)and the robust compensation term (32) with saturation limitscannot guarantee the stability of the closed-loop system
It is expected that during saturation the magnitude of thetracking error will increase since the control signal is notbeing achieved This tracking error is not the result of func-tion approximation error therefore we need to be careful so
6 Mathematical Problems in Engineering
that the approximator does not cause ldquounlearningrdquo duringthe period when the actuators are saturated Clearly theparameters update laws (30) and (31) depend on the trackingerror e
119894 thus if the tracking error increases due to saturation
the parameters update laws may cause a significant change intheweights in response to the increase in tracking error Nextwe develop a robust fuzzy adaptive control scheme to addressthe input saturation problem
32 Adaptive Fuzzy Control with Input Saturation When theactuators have physical constraints such as themagnitude andrate limitations the above approach may not be able to besuccessfully implemented Considering the magnitude andrate limitations on the actuator controller (24) is modified as
u119888= Gminus1 (x | 120579
119866) [minusF (x | 120579
119865) + u119897+ u119886119908
+ u119889]
u = 119878119872119877
(u119888)
(40)
where u119888is obtained by certainty equivalence principle u
119886119908
is the auxiliary control term which is used to compensate theeffect of input saturation 119878
119872119877(sdot) is a function including the
magnitude and rate constraints which can produce a limitedoutput
The state space representation of each component of u119888is
[
1199031198941
1199031198942
] = [
1199031198942
119878119877(120596119894(119878119872
(119906119888119894) minus 1199031198941))
]
[
119906119894
119894
] = [
1199031198941
1199031198942
]
(41)
where 119906119888119894is the 119894th element of u
119888 and 119906
119894is the 119894th element
of u 120596119894is the natural frequency and 119878
119872(sdot) and 119878
119877(sdot) are the
saturation functions corresponding to magnitude and raterespectively The function 119878
119872(sdot) is defined as
119878119872
(sdot) =
119872 if 119909 ge 119872
119909 if |119909| lt 119872
minus119872 if 119909 le minus119872
(42)
and 119878119877(sdot) has the same definition
To compensate the effects of input limitations an auxil-iary system is introduced as follows [35]
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus 119906119889119894
minus 119889119894
(48)
Define e119894
= [119890119894 119890119894 119890
(119903119894minus1)
119894]119879 and using the optimal
approximation for 119891119894(x) and 119892
119894119895(x) (48) can be rewritten as
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(49)
Theorem 6 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance and input satura-tion if the controller is chosen as (40) the auxiliary control
Mathematical Problems in Engineering 7
term u119886119908 the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (60)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(61)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
From the above analysis it is concluded that in the caseof no control saturations the signals 120585
119894119895(119894 = 1 119898 119895 =
1 119903119894) remain zeros and the control law becomes the same
as the standard robust adaptive control law described in theprevious section In the presence of control saturations 120585
119894119895
is nonzero thus giving rise to a modified tracking error119890119894= 119910119889119894
minus 119910119894minus 1205851198941 which is used in fuzzy parameter update
laws The auxiliary control term u119886119908
in (40) will be usedto compensate the effects of control saturations Note thatthe fuzzy parameter update laws (52) and (53) are similar tothe corresponding update laws (30) and (31) derived in thestandard fuzzy approximation based control problem withtracking error 119890
119894being replaced by the modified tracking
error 119890119894 The use of the modified tracking error in the fuzzy
update laws is crucial in preventing actuator constraints inonline approximation schemes
8 Mathematical Problems in Engineering
Corollary 7 For the 119894th subsystem of (4) it is assumed thatint119879
0
1198892
119894119889119905 lt infin If the control law (40) the auxiliary control term
(51) and the parameter update laws (52)ndash(54) are adoptedthen the following statements hold
(i) The closed loop system is stable and the signals e119894 120579119891119894
120579119892119894119895 and 119906
119894are bounded
(ii) Thesteadymodified tracking error satisfies lim119905rarrinfin
119890119894=
0 and the bound of the transient modified trackingerror will be given as follows
According to the definition of 119881119894 (62) is obtained
Assumption 4 implies int119879
0
12059610158402
119894119889119905 lt infin then int
119879
0
1205962
119894119889119905 =
int119879
0
(1205961015840
119894minus 119889119894)2
119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can
conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that
lim119905rarrinfin
119890119894(119905) = 0 This completes the proof
Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867
infin tracking performance with a prescribeddisturbance attenuation level 120588
119894 that is the 119871
2gain from 120596
119894
to the modified tracking error 119890119894is equal or less than 120588
119894
Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890
119894will converge to zero The bound of the
transient modified tracking error 119890119894is an explicit function
of the design parameters and the external disturbance andfuzzy approximation error 120596
119894 The bound can be decreased
by choosing the initial estimates 120579119891119894(0) 120579
119892119894119895(0) closing to
the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial
estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574
119891119894 120574119892119894119895
and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596
119894on the transient tracking
performance can be reduced by decreasing 120588119894and increasing
120582min(Q119894) Small 120588119894implies high disturbance attenuation level
4 Simulation Examples
The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]
y = F (x) + G (x) u + d (67)
where x = [q120596]119879 is the state vector where q =
[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed
reference frame relative to the inertial frame satisfying 1199022
0+
1199022
1+ 1199022
2+ 1199022
3= 1 here 119902
0is chosen as 119902
0= radic1 minus 119902
2
1minus 1199022
2minus 1199022
3
120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed
reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the
output vector u = [1199061 1199062 1199063]119879 is the control torque vector
d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows
1198911(x) = minus
1
41199021(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199022120596119909120596119910
+119868119909minus 119868119911
2119868119910
1199023120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199020120596119910120596119911
1198912(x) = minus
1
41199022(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119910minus 119868119909
2119868119911
1199021120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199020120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199023120596119910120596119911
Mathematical Problems in Engineering 9
1198913(x) = minus
1
41199023(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199020120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199021120596119909120596119911+
119868119911minus 119868119910
2119868119909
1199022120596119910120596119911
G (x) =
[[[[[[[
[
1199020
2119868119909
minus1199023
2119868119910
1199022
2119868119911
1199023
2119868119909
1199020
2119868119910
minus1199021
2119868119911
minus1199022
2119868119909
1199021
2119868119910
1199020
2119868119911
]]]]]]]
]
(68)
where 119868119909 119868119910 and 119868
119911are the principal central moments of
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
According to fuzzy system theory the following assump-tion is reasonable
Assumption 4 The minimum approximation error is squareintegrable that is int119879
0
1205961015840119879
1198941205961015840
119894119889119905 lt infin
Using the definition (28) and the optimal approximationfor 119891119894(x) and 119892
119894119895(x) (26) can be rewritten in the following
form
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(29)
where 119891119894
= 120579119891119894
minus 120579lowast
119891119894 119892119894119895
= 120579119892119894119895
minus 120579lowast
119892119894119895 and 120596
119894= 1205961015840
119894minus 119889119894
Theorem 5 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance If the controller ischosen as (24) the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (38)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(39)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
The controller proposed by (24) is able to guarantee theLyapunov stability of the closed-loop system and attenuatethe effect of system uncertainties and external disturbancesHowever no saturation on actuators is taken into accountwhich is rather important for practical applications (includ-ing satellite systems) Assume that the control input u is con-straint by saturation functions it can be readily shown thatthe control law (24) the parameters update laws (30) (31)and the robust compensation term (32) with saturation limitscannot guarantee the stability of the closed-loop system
It is expected that during saturation the magnitude of thetracking error will increase since the control signal is notbeing achieved This tracking error is not the result of func-tion approximation error therefore we need to be careful so
6 Mathematical Problems in Engineering
that the approximator does not cause ldquounlearningrdquo duringthe period when the actuators are saturated Clearly theparameters update laws (30) and (31) depend on the trackingerror e
119894 thus if the tracking error increases due to saturation
the parameters update laws may cause a significant change intheweights in response to the increase in tracking error Nextwe develop a robust fuzzy adaptive control scheme to addressthe input saturation problem
32 Adaptive Fuzzy Control with Input Saturation When theactuators have physical constraints such as themagnitude andrate limitations the above approach may not be able to besuccessfully implemented Considering the magnitude andrate limitations on the actuator controller (24) is modified as
u119888= Gminus1 (x | 120579
119866) [minusF (x | 120579
119865) + u119897+ u119886119908
+ u119889]
u = 119878119872119877
(u119888)
(40)
where u119888is obtained by certainty equivalence principle u
119886119908
is the auxiliary control term which is used to compensate theeffect of input saturation 119878
119872119877(sdot) is a function including the
magnitude and rate constraints which can produce a limitedoutput
The state space representation of each component of u119888is
[
1199031198941
1199031198942
] = [
1199031198942
119878119877(120596119894(119878119872
(119906119888119894) minus 1199031198941))
]
[
119906119894
119894
] = [
1199031198941
1199031198942
]
(41)
where 119906119888119894is the 119894th element of u
119888 and 119906
119894is the 119894th element
of u 120596119894is the natural frequency and 119878
119872(sdot) and 119878
119877(sdot) are the
saturation functions corresponding to magnitude and raterespectively The function 119878
119872(sdot) is defined as
119878119872
(sdot) =
119872 if 119909 ge 119872
119909 if |119909| lt 119872
minus119872 if 119909 le minus119872
(42)
and 119878119877(sdot) has the same definition
To compensate the effects of input limitations an auxil-iary system is introduced as follows [35]
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus 119906119889119894
minus 119889119894
(48)
Define e119894
= [119890119894 119890119894 119890
(119903119894minus1)
119894]119879 and using the optimal
approximation for 119891119894(x) and 119892
119894119895(x) (48) can be rewritten as
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(49)
Theorem 6 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance and input satura-tion if the controller is chosen as (40) the auxiliary control
Mathematical Problems in Engineering 7
term u119886119908 the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (60)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(61)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
From the above analysis it is concluded that in the caseof no control saturations the signals 120585
119894119895(119894 = 1 119898 119895 =
1 119903119894) remain zeros and the control law becomes the same
as the standard robust adaptive control law described in theprevious section In the presence of control saturations 120585
119894119895
is nonzero thus giving rise to a modified tracking error119890119894= 119910119889119894
minus 119910119894minus 1205851198941 which is used in fuzzy parameter update
laws The auxiliary control term u119886119908
in (40) will be usedto compensate the effects of control saturations Note thatthe fuzzy parameter update laws (52) and (53) are similar tothe corresponding update laws (30) and (31) derived in thestandard fuzzy approximation based control problem withtracking error 119890
119894being replaced by the modified tracking
error 119890119894 The use of the modified tracking error in the fuzzy
update laws is crucial in preventing actuator constraints inonline approximation schemes
8 Mathematical Problems in Engineering
Corollary 7 For the 119894th subsystem of (4) it is assumed thatint119879
0
1198892
119894119889119905 lt infin If the control law (40) the auxiliary control term
(51) and the parameter update laws (52)ndash(54) are adoptedthen the following statements hold
(i) The closed loop system is stable and the signals e119894 120579119891119894
120579119892119894119895 and 119906
119894are bounded
(ii) Thesteadymodified tracking error satisfies lim119905rarrinfin
119890119894=
0 and the bound of the transient modified trackingerror will be given as follows
According to the definition of 119881119894 (62) is obtained
Assumption 4 implies int119879
0
12059610158402
119894119889119905 lt infin then int
119879
0
1205962
119894119889119905 =
int119879
0
(1205961015840
119894minus 119889119894)2
119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can
conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that
lim119905rarrinfin
119890119894(119905) = 0 This completes the proof
Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867
infin tracking performance with a prescribeddisturbance attenuation level 120588
119894 that is the 119871
2gain from 120596
119894
to the modified tracking error 119890119894is equal or less than 120588
119894
Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890
119894will converge to zero The bound of the
transient modified tracking error 119890119894is an explicit function
of the design parameters and the external disturbance andfuzzy approximation error 120596
119894 The bound can be decreased
by choosing the initial estimates 120579119891119894(0) 120579
119892119894119895(0) closing to
the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial
estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574
119891119894 120574119892119894119895
and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596
119894on the transient tracking
performance can be reduced by decreasing 120588119894and increasing
120582min(Q119894) Small 120588119894implies high disturbance attenuation level
4 Simulation Examples
The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]
y = F (x) + G (x) u + d (67)
where x = [q120596]119879 is the state vector where q =
[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed
reference frame relative to the inertial frame satisfying 1199022
0+
1199022
1+ 1199022
2+ 1199022
3= 1 here 119902
0is chosen as 119902
0= radic1 minus 119902
2
1minus 1199022
2minus 1199022
3
120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed
reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the
output vector u = [1199061 1199062 1199063]119879 is the control torque vector
d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows
1198911(x) = minus
1
41199021(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199022120596119909120596119910
+119868119909minus 119868119911
2119868119910
1199023120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199020120596119910120596119911
1198912(x) = minus
1
41199022(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119910minus 119868119909
2119868119911
1199021120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199020120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199023120596119910120596119911
Mathematical Problems in Engineering 9
1198913(x) = minus
1
41199023(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199020120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199021120596119909120596119911+
119868119911minus 119868119910
2119868119909
1199022120596119910120596119911
G (x) =
[[[[[[[
[
1199020
2119868119909
minus1199023
2119868119910
1199022
2119868119911
1199023
2119868119909
1199020
2119868119910
minus1199021
2119868119911
minus1199022
2119868119909
1199021
2119868119910
1199020
2119868119911
]]]]]]]
]
(68)
where 119868119909 119868119910 and 119868
119911are the principal central moments of
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
that the approximator does not cause ldquounlearningrdquo duringthe period when the actuators are saturated Clearly theparameters update laws (30) and (31) depend on the trackingerror e
119894 thus if the tracking error increases due to saturation
the parameters update laws may cause a significant change intheweights in response to the increase in tracking error Nextwe develop a robust fuzzy adaptive control scheme to addressthe input saturation problem
32 Adaptive Fuzzy Control with Input Saturation When theactuators have physical constraints such as themagnitude andrate limitations the above approach may not be able to besuccessfully implemented Considering the magnitude andrate limitations on the actuator controller (24) is modified as
u119888= Gminus1 (x | 120579
119866) [minusF (x | 120579
119865) + u119897+ u119886119908
+ u119889]
u = 119878119872119877
(u119888)
(40)
where u119888is obtained by certainty equivalence principle u
119886119908
is the auxiliary control term which is used to compensate theeffect of input saturation 119878
119872119877(sdot) is a function including the
magnitude and rate constraints which can produce a limitedoutput
The state space representation of each component of u119888is
[
1199031198941
1199031198942
] = [
1199031198942
119878119877(120596119894(119878119872
(119906119888119894) minus 1199031198941))
]
[
119906119894
119894
] = [
1199031198941
1199031198942
]
(41)
where 119906119888119894is the 119894th element of u
119888 and 119906
119894is the 119894th element
of u 120596119894is the natural frequency and 119878
119872(sdot) and 119878
119877(sdot) are the
saturation functions corresponding to magnitude and raterespectively The function 119878
119872(sdot) is defined as
119878119872
(sdot) =
119872 if 119909 ge 119872
119909 if |119909| lt 119872
minus119872 if 119909 le minus119872
(42)
and 119878119877(sdot) has the same definition
To compensate the effects of input limitations an auxil-iary system is introduced as follows [35]
119892119894119895) minus 119892119894119895(x)] 119906
119895
minus 119906119889119894
minus 119889119894
(48)
Define e119894
= [119890119894 119890119894 119890
(119903119894minus1)
119894]119879 and using the optimal
approximation for 119891119894(x) and 119892
119894119895(x) (48) can be rewritten as
e119894= A119894e119894+ B119894
[
[
119879
119891119894120585119891119894
(x) +
119898
sum
119895=1
119879
119892119894119895120585119892119894119895
(x) 119906119895
]
]
minus B119894119906119889119894
+ B119894120596119894
(49)
Theorem 6 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance and input satura-tion if the controller is chosen as (40) the auxiliary control
Mathematical Problems in Engineering 7
term u119886119908 the parameters update laws and 119906
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (60)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(61)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
From the above analysis it is concluded that in the caseof no control saturations the signals 120585
119894119895(119894 = 1 119898 119895 =
1 119903119894) remain zeros and the control law becomes the same
as the standard robust adaptive control law described in theprevious section In the presence of control saturations 120585
119894119895
is nonzero thus giving rise to a modified tracking error119890119894= 119910119889119894
minus 119910119894minus 1205851198941 which is used in fuzzy parameter update
laws The auxiliary control term u119886119908
in (40) will be usedto compensate the effects of control saturations Note thatthe fuzzy parameter update laws (52) and (53) are similar tothe corresponding update laws (30) and (31) derived in thestandard fuzzy approximation based control problem withtracking error 119890
119894being replaced by the modified tracking
error 119890119894 The use of the modified tracking error in the fuzzy
update laws is crucial in preventing actuator constraints inonline approximation schemes
8 Mathematical Problems in Engineering
Corollary 7 For the 119894th subsystem of (4) it is assumed thatint119879
0
1198892
119894119889119905 lt infin If the control law (40) the auxiliary control term
(51) and the parameter update laws (52)ndash(54) are adoptedthen the following statements hold
(i) The closed loop system is stable and the signals e119894 120579119891119894
120579119892119894119895 and 119906
119894are bounded
(ii) Thesteadymodified tracking error satisfies lim119905rarrinfin
119890119894=
0 and the bound of the transient modified trackingerror will be given as follows
According to the definition of 119881119894 (62) is obtained
Assumption 4 implies int119879
0
12059610158402
119894119889119905 lt infin then int
119879
0
1205962
119894119889119905 =
int119879
0
(1205961015840
119894minus 119889119894)2
119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can
conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that
lim119905rarrinfin
119890119894(119905) = 0 This completes the proof
Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867
infin tracking performance with a prescribeddisturbance attenuation level 120588
119894 that is the 119871
2gain from 120596
119894
to the modified tracking error 119890119894is equal or less than 120588
119894
Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890
119894will converge to zero The bound of the
transient modified tracking error 119890119894is an explicit function
of the design parameters and the external disturbance andfuzzy approximation error 120596
119894 The bound can be decreased
by choosing the initial estimates 120579119891119894(0) 120579
119892119894119895(0) closing to
the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial
estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574
119891119894 120574119892119894119895
and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596
119894on the transient tracking
performance can be reduced by decreasing 120588119894and increasing
120582min(Q119894) Small 120588119894implies high disturbance attenuation level
4 Simulation Examples
The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]
y = F (x) + G (x) u + d (67)
where x = [q120596]119879 is the state vector where q =
[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed
reference frame relative to the inertial frame satisfying 1199022
0+
1199022
1+ 1199022
2+ 1199022
3= 1 here 119902
0is chosen as 119902
0= radic1 minus 119902
2
1minus 1199022
2minus 1199022
3
120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed
reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the
output vector u = [1199061 1199062 1199063]119879 is the control torque vector
d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows
1198911(x) = minus
1
41199021(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199022120596119909120596119910
+119868119909minus 119868119911
2119868119910
1199023120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199020120596119910120596119911
1198912(x) = minus
1
41199022(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119910minus 119868119909
2119868119911
1199021120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199020120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199023120596119910120596119911
Mathematical Problems in Engineering 9
1198913(x) = minus
1
41199023(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199020120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199021120596119909120596119911+
119868119911minus 119868119910
2119868119909
1199022120596119910120596119911
G (x) =
[[[[[[[
[
1199020
2119868119909
minus1199023
2119868119910
1199022
2119868119911
1199023
2119868119909
1199020
2119868119910
minus1199021
2119868119911
minus1199022
2119868119909
1199021
2119868119910
1199020
2119868119911
]]]]]]]
]
(68)
where 119868119909 119868119910 and 119868
119911are the principal central moments of
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
Integrating both sides of the above inequality from 0 to 119879
yields
1
2int
119879
0
e119879119894Q119894e119894119889119905 le 119881
119894(0) minus 119881
119894(119879) +
1205882
119894
2int
119879
0
1205962
119894119889119905 (60)
Since 119881119894(119879) is nonnegative according to the definition of
119881119894 the following inequality is obtained
int
119879
0
e119879119894Q119894e119894119889119905 le e119879
119894(0)P119894e119894(0) +
1
120574119891119894
119879
119891119894(0) 119891119894
(0)
+
119898
sum
119894=1
1
120574119892119894119895
119879
119892119894119895(0) 119892119894119895
(0) + 1205882
119894int
119879
0
1205962
119894119889119905
(61)
Considering the Lyapunov function 119881 = sum119898
119894=1119881119894 it is
easy to obtain the 119867infin
tracking performance index (55) Thiscompletes the proof
From the above analysis it is concluded that in the caseof no control saturations the signals 120585
119894119895(119894 = 1 119898 119895 =
1 119903119894) remain zeros and the control law becomes the same
as the standard robust adaptive control law described in theprevious section In the presence of control saturations 120585
119894119895
is nonzero thus giving rise to a modified tracking error119890119894= 119910119889119894
minus 119910119894minus 1205851198941 which is used in fuzzy parameter update
laws The auxiliary control term u119886119908
in (40) will be usedto compensate the effects of control saturations Note thatthe fuzzy parameter update laws (52) and (53) are similar tothe corresponding update laws (30) and (31) derived in thestandard fuzzy approximation based control problem withtracking error 119890
119894being replaced by the modified tracking
error 119890119894 The use of the modified tracking error in the fuzzy
update laws is crucial in preventing actuator constraints inonline approximation schemes
8 Mathematical Problems in Engineering
Corollary 7 For the 119894th subsystem of (4) it is assumed thatint119879
0
1198892
119894119889119905 lt infin If the control law (40) the auxiliary control term
(51) and the parameter update laws (52)ndash(54) are adoptedthen the following statements hold
(i) The closed loop system is stable and the signals e119894 120579119891119894
120579119892119894119895 and 119906
119894are bounded
(ii) Thesteadymodified tracking error satisfies lim119905rarrinfin
119890119894=
0 and the bound of the transient modified trackingerror will be given as follows
According to the definition of 119881119894 (62) is obtained
Assumption 4 implies int119879
0
12059610158402
119894119889119905 lt infin then int
119879
0
1205962
119894119889119905 =
int119879
0
(1205961015840
119894minus 119889119894)2
119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can
conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that
lim119905rarrinfin
119890119894(119905) = 0 This completes the proof
Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867
infin tracking performance with a prescribeddisturbance attenuation level 120588
119894 that is the 119871
2gain from 120596
119894
to the modified tracking error 119890119894is equal or less than 120588
119894
Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890
119894will converge to zero The bound of the
transient modified tracking error 119890119894is an explicit function
of the design parameters and the external disturbance andfuzzy approximation error 120596
119894 The bound can be decreased
by choosing the initial estimates 120579119891119894(0) 120579
119892119894119895(0) closing to
the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial
estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574
119891119894 120574119892119894119895
and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596
119894on the transient tracking
performance can be reduced by decreasing 120588119894and increasing
120582min(Q119894) Small 120588119894implies high disturbance attenuation level
4 Simulation Examples
The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]
y = F (x) + G (x) u + d (67)
where x = [q120596]119879 is the state vector where q =
[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed
reference frame relative to the inertial frame satisfying 1199022
0+
1199022
1+ 1199022
2+ 1199022
3= 1 here 119902
0is chosen as 119902
0= radic1 minus 119902
2
1minus 1199022
2minus 1199022
3
120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed
reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the
output vector u = [1199061 1199062 1199063]119879 is the control torque vector
d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows
1198911(x) = minus
1
41199021(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199022120596119909120596119910
+119868119909minus 119868119911
2119868119910
1199023120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199020120596119910120596119911
1198912(x) = minus
1
41199022(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119910minus 119868119909
2119868119911
1199021120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199020120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199023120596119910120596119911
Mathematical Problems in Engineering 9
1198913(x) = minus
1
41199023(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199020120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199021120596119909120596119911+
119868119911minus 119868119910
2119868119909
1199022120596119910120596119911
G (x) =
[[[[[[[
[
1199020
2119868119909
minus1199023
2119868119910
1199022
2119868119911
1199023
2119868119909
1199020
2119868119910
minus1199021
2119868119911
minus1199022
2119868119909
1199021
2119868119910
1199020
2119868119911
]]]]]]]
]
(68)
where 119868119909 119868119910 and 119868
119911are the principal central moments of
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
According to the definition of 119881119894 (62) is obtained
Assumption 4 implies int119879
0
12059610158402
119894119889119905 lt infin then int
119879
0
1205962
119894119889119905 =
int119879
0
(1205961015840
119894minus 119889119894)2
119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can
conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that
lim119905rarrinfin
119890119894(119905) = 0 This completes the proof
Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867
infin tracking performance with a prescribeddisturbance attenuation level 120588
119894 that is the 119871
2gain from 120596
119894
to the modified tracking error 119890119894is equal or less than 120588
119894
Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890
119894will converge to zero The bound of the
transient modified tracking error 119890119894is an explicit function
of the design parameters and the external disturbance andfuzzy approximation error 120596
119894 The bound can be decreased
by choosing the initial estimates 120579119891119894(0) 120579
119892119894119895(0) closing to
the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial
estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574
119891119894 120574119892119894119895
and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596
119894on the transient tracking
performance can be reduced by decreasing 120588119894and increasing
120582min(Q119894) Small 120588119894implies high disturbance attenuation level
4 Simulation Examples
The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]
y = F (x) + G (x) u + d (67)
where x = [q120596]119879 is the state vector where q =
[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed
reference frame relative to the inertial frame satisfying 1199022
0+
1199022
1+ 1199022
2+ 1199022
3= 1 here 119902
0is chosen as 119902
0= radic1 minus 119902
2
1minus 1199022
2minus 1199022
3
120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed
reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the
output vector u = [1199061 1199062 1199063]119879 is the control torque vector
d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows
1198911(x) = minus
1
41199021(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199022120596119909120596119910
+119868119909minus 119868119911
2119868119910
1199023120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199020120596119910120596119911
1198912(x) = minus
1
41199022(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119910minus 119868119909
2119868119911
1199021120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199020120596119909120596119911+
119868119910minus 119868119911
2119868119909
1199023120596119910120596119911
Mathematical Problems in Engineering 9
1198913(x) = minus
1
41199023(1205962
119909+ 1205962
119910+ 1205962
119911) +
119868119909minus 119868119910
2119868119911
1199020120596119909120596119910
+119868119911minus 119868119909
2119868119910
1199021120596119909120596119911+
119868119911minus 119868119910
2119868119909
1199022120596119910120596119911
G (x) =
[[[[[[[
[
1199020
2119868119909
minus1199023
2119868119910
1199022
2119868119911
1199023
2119868119909
1199020
2119868119910
minus1199021
2119868119911
minus1199022
2119868119909
1199021
2119868119910
1199020
2119868119911
]]]]]]]
]
(68)
where 119868119909 119868119910 and 119868
119911are the principal central moments of
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
inertia of the satelliteWithin this simulation the nonlinear functions F(x) and
G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose
Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902
1 1199022 1199023 1205961 1205962 and 120596
3as inputs
and the ones used to describe G(x) have 1199021 1199022 and 119902
3as
inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we
025 025]119879 and the initial value of the angular velocity
is 120596(0) = [0005 0005 0005]119879 rads The parameters
of system are given as 119868119909
= 1875 kgsdotm2 119868119910
= 4685 kgsdotm2and 119868
119911= 4685 kgsdotm2 The external disturbance torque
vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579
119892119894119895are set to random values uniformly distributed
between [0 1] The desired output trajectory is chosen asy119889
= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y
119889
41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31
Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally
10 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
Figure 1 Trajectories of outputs without saturation
42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection
Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906
1 1199062 1199063]119879 has amplitude lim-
its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed
1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term
(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4
The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus20
0
20
u1
minus20
0
20
u2
minus20
0
20
u3
Figure 2 Trajectories of control inputs without saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 3 Trajectories of outputs with amplitude saturation
Mathematical Problems in Engineering 11
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
Figure 4 Trajectories of control inputs with amplitude saturation
inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered
43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906
1 1199062 1199063]119879 has the
amplitude and rate limitation |119906119894| le 1 |
119894| le 2 119894 = 1 2 3
The other initial parameters are the same as mentioned inSection 42
According to (41) the dynamics of control inputs areexpressed as follows
119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)
where 120596119894= 205 (119894 = 1 2 3)
Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
ReferenceResponse
minus1
0
1
q1
minus1
0
1
q2
minus1
0
1
q3
Figure 5 Trajectories of outputs with amplitude and rate saturation
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus1
0
1
u1
minus1
0
1
u2
minus1
0
1
u3
Figure 6 Trajectories of control inputs with amplitude and ratesaturation
12 Mathematical Problems in Engineering
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus2
0
2
du1
minus2
0
2
du2
minus2
0
2
du3
Figure 7 Derivatives of control inputs
0 500 1000 1500
0 500 1000 1500
0 500 1000 1500Time (s)
minus50
0
50
uc1
minus50
0
50
uc2
minus50
0
50
uc3
Figure 8 1199061198881 1199061198882 and 119906
1198883
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
rate saturation limits Figure 8 shows the signals 119906119888119894
(119894 =
1 2 3) Obviously they do not satisfy the control inputlimitations
From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation
5 Conclusions
In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903
119894minus 1 order as in (22) which is also
practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106
References
[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999
[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000
Mathematical Problems in Engineering 13
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin
tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996
[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000
[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867
infintracking control of uncertain nonlinear systemsrdquo
Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in
the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004
[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003
[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000
[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011
[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005
[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009
[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010
[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012
[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014
[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014
[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015
[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013
[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear
strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011
[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014
[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006
[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012
[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009
[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003
[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005
[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003
[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013
[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013
[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013
[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014
[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010
[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012
[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006
14 Mathematical Problems in Engineering
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012
[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011
[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014
[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010
[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011
[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014
[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013
[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011
[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011
[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999
[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012