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2086 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS,
VOL. 26, NO. 9, SEPTEMBER 2015
Dynamic Surface Control Using Neural Networksfor a Class of
Uncertain Nonlinear Systems
With Input SaturationMou Chen, Member, IEEE, Gang Tao, Fellow,
IEEE, and Bin Jiang, Senior Member, IEEE
Abstract— In this paper, a dynamic surface control (DSC)scheme
is proposed for a class of uncertain strict-feedbacknonlinear
systems in the presence of input saturation andunknown external
disturbance. The radial basis function neuralnetwork (RBFNN) is
employed to approximate the unknownsystem function. To efficiently
tackle the unknown external dis-turbance, a nonlinear disturbance
observer (NDO) is developed.The developed NDO can relax the known
boundary requirementof the unknown disturbance and can guarantee
the disturbanceestimation error converge to a bounded compact set.
UsingNDO and RBFNN, the DSC scheme is developed for
uncertainnonlinear systems based on a backstepping method. Using
aDSC technique, the problem of explosion of complexity inherentin
the conventional backstepping method is avoided, which isspecially
important for designs using neural network approxima-tions. Under
the proposed DSC scheme, the ultimately boundedconvergence of all
closed-loop signals is guaranteed via Lyapunovanalysis. Simulation
results are given to show the effectivenessof the proposed DSC
design using NDO and RBFNN.
Index Terms— Backstepping control, dynamic surface control(DSC),
nonlinear disturbance observer (NDO), robust control,uncertain
nonlinear system.
I. INTRODUCTION
IN PRACTICAL engineering, lots of plants possess nonlin-ear and
uncertain characteristics. On the other hand, themagnitude of
control signal is always limited due to actuatorphysical
constraints. Thus, it is very important to developeffective robust
control techniques for uncertain nonlinear sys-tems with input
saturation. Saturation as one of the commonnonsmooth nonlinear
constraint of control input should beexplicitly considered in the
control design to enhance robustcontrol performance. If the input
saturation is ignored inthe control design, the closed-loop control
performance willbe severely degraded, and instability may occur. In
recent
Manuscript received July 31, 2013; revised August 21, 2014;
acceptedSeptember 21, 2014. Date of publication December 4, 2014;
date of currentversion August 17, 2015. This work was supported in
part by the JiangsuNatural Science Foundation of China under Grant
SBK20130033, in part bythe National Natural Science Foundation of
China under Grant 61374130 andGrant 61174102, in part by the
Program for New Century Excellent Talentsin the University of China
under Grant NCET-11-0830, and in part by theJiangsu Province Blue
Project through the Innovative Research Team.
M. Chen and B. Jiang are with the College of Automation
Engineering,Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China(e-mail: [email protected];
[email protected]).
G. Tao is with the Department of Electrical and
ComputerEngineering, University of Virginia, Charlottesville, VA
22904 USA(e-mail: [email protected]).
Color versions of one or more of the figures in this paper are
availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNNLS.2014.2360933
years, there have been extensive studies on various systemswith
input saturation in [1]–[3]. Neural network (NN)-basednear-optimal
control was developed for a class of discrete-time affine nonlinear
systems with control constraints in [4].In [5], a robust adaptive
control scheme was proposed foruncertain nonlinear systems in the
presence of input saturationand external disturbance. Robust
adaptive neural networkcontrol was proposed for a class of
uncertain multi-input andmulti-output (MIMO) nonlinear systems with
input nonlin-earities [6]. Backstepping control was studied for
hoveringunmanned aerial vehicle, including input saturations in
[7].In [8], an adaptive tracking control scheme was developedfor
uncertain MIMO nonlinear systems with input saturation.Adaptive
control was studied for minimum phase single-inputand single-output
plants with input saturation [9]. However,there are few existing
research results for the dynamic surfacecontrol (DSC) scheme of
uncertain strict-feedback nonlinearsystems with input saturation
and unknown external distur-bance.
On the other hand, robust adaptive backstepping con-trol as an
efficient control method has been extensivelyused for nonlinear
control system design due to its designflexibility [10]–[13]. At
the same time, NNs and fuzzy logicalsystems as the universal
approximators have been widelyemployed to tackle the system
uncertainty [14]–[19]. In [20],an adaptive sliding-mode control was
proposed for nonlinearactive suspension vehicle systems using
Takagi–Sugeno fuzzyapproach. Robust adaptive tracking control
scheme was pro-posed for nonlinear systems based on the fuzzy
approximatorin [21]. A combined backstepping and small-gain
approachwas developed for the robust adaptive fuzzy output
feedbackcontrol design in [22]. In [23], a globally stable
adaptivebackstepping fuzzy control scheme was studied for
output-feedback systems with unknown high-frequency gain
sign.Adaptive backstepping fuzzy control was proposed for
nonlin-early parameterized systems with periodic disturbance in
[24].In [25], an observer-based adaptive decentralized fuzzy
fault-tolerant control scheme was studied for nonlinear
large-scalesystems with actuator failures. Furthermore,
backstepping con-trol has been extensively used in many practical
systems.Nonlinear adaptive flight control was proposed using
back-stepping method and NNs in [26]. In [27], a fuzzy
adaptivecontrol design was studied for hypersonic vehicles via
back-stepping method. Robust attitude control was developed
forhelicopters with actuator dynamics using NNs in [28]. In
[29],
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CHEN et al.: DSC USING NNs FOR A CLASS OF UNCERTAIN NONLINEAR
SYSTEMS 2087
an observer-based adaptive fuzzy backstepping control schemewas
proposed for a class of stochastic nonlinear
strict-feedbacksystems. However, there are few backstepping control
resultsfor uncertain nonlinear systems using disturbance
observers.To tackle the unknown time-varying disturbance for
effectivebackstepping control design, the robust adaptive
backsteppingcontrol based on disturbance observation should be
furtherdeveloped.
With conventional backstepping, a possible issue is theproblem
of explosion of complexity. That is, the complexityof the
controller grows drastically as the order n of the systemincreases.
This explosion of complexity is caused by therepeated
differentiations of certain nonlinear functions. To effi-ciently
handle the system uncertainty in each subsystem, radialbasis
function NN (RBFNN) with the universal approximationcapability is
employed in [30] and [31]. Since RBFNN is used,we need to take
derivatives of those radial basis functions,which further lead to
heavier calculation burden in each stepdesign. Recently, the DSC
method was employed to solve thisproblem and many research results
were presented [32]. In[33], an adaptive DSC design was proposed
using adaptivebackstepping for nonlinear systems. DSC was presented
for aclass of nonlinear systems in [34]. In [35], NN-based
adaptiveDSC was developed for nonlinear systems in
strict-feedbackform. A robust adaptive NN tracking control design
wasproposed for strict-feedback nonlinear systems using DSCapproach
in [36]. In [37], a NN-based adaptive DSC schemewas studied for
uncertain nonlinear pure-feedback systems.Simultaneous quadratic
stabilization was studied for a class ofnonlinear systems with
input saturation using DSC in [38]. In[39], an output feedback
adaptive DSC scheme was developedfor a class of nonlinear systems
with input saturation. Recently,L∞-type criteria are used in the
DSC design to enhancethe control performance [40]–[42]. However,
DSC shouldbe further investigated for uncertain strict-feedback
nonlinearsystems in the presence of input saturation and
unknownexternal disturbance.
In recent years, disturbance observer design and applicationhave
attracted considerable interest for robust control ofuncertain
nonlinear systems. Thus, different disturbanceobservers have been
developed [43]–[47] and robust controlschemes were proposed using
disturbance observers. Ageneral framework was given for nonlinear
systems usingdisturbance observer based control (DOBC) techniques
in[48]. In [49], composite DOBC and terminal sliding modecontrol
were investigated for uncertain structural systems.The disturbance
attenuation and rejection problem wasinvestigated for a class of
MIMO nonlinear systems usinga DOBC framework in [50]. In [51],
composite DOBC andH∞ control designs were proposed for complex
continuousmodels. Adding robustness to nominal output
feedbackcontrollers was studied for uncertain nonlinear systems
usinga disturbance observer in [52]. Although significant
progresshas been made for the disturbance observer design, thereare
still some open problems that need to be solved. Inalmost all
approaches reported in the literature, the unknowndisturbance is
assumed as a slowly changeable disturbancefor the disturbance
observer design that implies the derivative
of the disturbance approaching to zero. It is apparent thatthis
assumption is restrictive for a practical system. TheNDO can
provide the estimation of the bounded unknowndisturbance and can be
employed in the robust control designto compensate for the unknown
disturbance. At the sametime, the NDO does not rely on complete
knowledge of thedisturbance mathematical model, as an efficient
disturbanceobserver. In this paper, the NDO is proposed for the
uncertainnonlinear systems for which the known upper
boundaryassumption of the unknown disturbance is canceled and
theconvergence of the disturbance estimation error is proved.
This paper develops a new NDO-based DSC design foruncertain
nonlinear systems with unknown external distur-bance and input
saturation. The control objective is that theproposed DSC can track
a desired trajectory in the presenceof unknown time-varying
external disturbance and input satu-ration. The main contributions
of this paper are as follows.
1) An NDO is developed to estimate the unknown distur-bance.
Especially, the known upper boundary requirementof the unknown
disturbance is eliminated for the designof NDO.
2) DSC is implemented using the output of the developedNDO for
uncertain nonlinear systems with input satura-tion and unknown
external disturbances to enhance therobust control performance of
the closed-loop system.
3) Closed-loop system stability is guaranteed using Lya-punov
method, which shows that all closed-loop systemsignals are
semiglobal uniformly ultimately bounded.
The organization of this paper is as follows. Section IIdetails
the problem formulation. Section III presents theDSC scheme with
NDO. Simulation studies are presented inSection IV to demonstrate
the effectiveness of the developednonlinear disturbance
observer-based DSC, followed by someconcluding remarks in Section
V.
Throughout this paper, (·̃) = (·̂) − (·)∗, || · || denotes the
l2norm, and λmin(·) and λmax(·) denote the smallest and
largesteigenvalues of a square matrix ·, respectively.
II. PROBLEM STATEMENT AND PRELIMINARIESA. Problem Statement
Consider a class of uncertain strict-feedback nonlinear sys-tems
with input saturation and unknown disturbance which aredescribed
by
ẋi = fi (x̄i ) + gi(x̄i )xi+1, i = 1, . . . , n − 1ẋn =
fn(x̄n) + gn0(x̄n)u(v(t)) + d(t)y = x1 (1)
where x̄i = [x1, x2, . . . , xi ]T ∈ Ri , i = 1, 2, . . . , n,
arestate vectors which are assumed to be measurable; y ∈ R isthe
output of the uncertain nonlinear system; function termsfi (x̄i) :
Ri → R, i = 1, 2, . . . , n, gi(x̄i ) : Ri → R, i =1, 2, . . . , n
− 1, and gn0(x̄n) : Rn → R are unknown andcontinuous; d ∈ R is an
unknown and bounded disturbance;v(t) ∈ R is the control input and
u(·) denotes the plant inputwhich is subject to saturation
nonlinearity described by [5]
u(v(t)) = sat(v(t)) ={
sign(v(t))uM , |v(t)| ≥ uMv(t), |v(t)| < uM (2)
where uM is a bound of u(t).
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2088 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS,
VOL. 26, NO. 9, SEPTEMBER 2015
To efficiently tackle the saturation u(v(t)) in the DSC, it
isapproximated by the following smooth function [5]:
h(v) = uM tanh(
v
uM
)= uM e
v/uM − e−v/uMev/uM + e−v/uM . (3)
It is apparent that there exists a difference �(v)
betweensat(v(t)) and h(v). Then, we have
�(v) = sat(v(t)) − h(v). (4)Since the bounded property of the
tanh function and the
sat function, we can see that the difference �(v) is
boundedwhich satisfies the following condition [5]:
|�(v)| = |sat(v(t)) − h(v)| ≤ uM (1 − tanh(1)) = d̄. (5)Consider
the saturation characteristic and the corresponding
approximation error, the uncertain nonlinear system (1) can
bewritten as
ẋi = fi (x̄i) + gi (x̄i)xi+1, i = 1, . . . , n − 1ẋn = fn(x̄n)
+ gn0(x̄n)h(v(t)) + gn0(x̄n)�(v) + d(t)y = x1. (6)
To facilitate the DSC design for the uncertain nonlinearsystem
(6), invoking the mean-value theorem [53], we canexpress h(v(t)) in
(6) as follows:
h(v(t)) = h(v0) + ∂h(v)∂v
∣∣∣∣v=vμ
(v − v0) (7)
where vμ = μv + (1 − μ)v0 with 0 < μ < 1.By choosing v0 =
0, we obtain
h(v(t)) = h(0) + ∂h(v)∂v
∣∣∣∣v=vμ
v. (8)
Considering h(0) = 0, we have
h(v(t)) = ∂h(v)∂v
∣∣∣∣v=vμ
v. (9)
Define gn(x̄n) = gn0(x̄n)(∂h(v)/∂v)|v=vμ and D(t) =
d(t)+gn0(x̄n)�(v). Then, the uncertain nonlinear system (6) can
berewritten as
ẋi = fi (x̄i ) + gi(x̄i )xi+1, i = 1, . . . , n − 1ẋn =
fn(x̄n) + gn(x̄n)v + D(t)y = x1. (10)
B. Neural Networks
In many references of robust adaptive control for
uncertainnonlinear systems, RBFNNs are usually employed
asapproximation models for the unknown nonlinear and con-tinuous
function terms using their inherent approximationcapabilities [54].
As a class of linearly parameterized NNs,RBFNNs are adopted to
approximate the unknown and con-tinuous function f (Z) : Rq → R can
be written as follows:
f (Z) = Ŵ T S(Z) + ε(Z) (11)where Z = [z1, z2, . . . , zq ]T ∈
Rq is an input vector ofNN, Ŵ ∈ R p is a weight vector of the NN,
S(Z) =[s1(Z), s2(Z), . . . , sp(Z)]T ∈ R p is a basis function, ε
is the
approximation error which satisfies |ε| ≤ |ε̄|, and ε̄ is a
boundunknown parameter.
In general, RBFNN can smoothly approximate any contin-uous
function f (Z) over the compact set �Z ∈ Rq to anyarbitrary
accuracy as [55]
f (Z) = W∗T S(Z) + ε∗(Z) ∀Z ∈ �Z ⊂ Rq (12)where W∗ is the
optimal weight value and ε∗(Z) is thesmallest approximation error.
The Gaussian function is writtenin the form of
si (Z)=exp[−(Z − ci )T (Z − ci )/b2i ], i =1, 2, . . . , p
(13)where ci and bi are the center and width of the neural cell
ofthe i th hidden layer.
The optimal weight value of RBFNN is given by [55]
W∗ = arg minŴ∈� f
[ supz∈SZ
| f̂ (Z |Ŵ ) − f (Z)|] (14)
where � f = {Ŵ : ‖Ŵ‖ ≤ M} is a valid field of the parameterand
M is a design parameter. SZ ⊂ Rn is an allowable set ofthe state
vector.
Using the optimal weight value yields
| f (Z) − W∗T S(Z)| = |ε∗(Z)| ≤ |ε̄|. (15)In this paper, the NDO
is employed to estimate the unknown
compounded disturbance D(t) which consists of d(t)
andgn0(x̄n)�(v). The RBFNNs are used to approximate theunknown
continuous functions. Based on estimated outputsof the developed
NDO and the RBFNN, the DSC scheme isproposed for uncertain
nonlinear systems. The control objec-tive is that the developed DSC
scheme can make the systemoutput follow a given desired system
output yd of the nonlinearsystem in the presence of the unknown
external disturbanceand the input saturation for all initial
conditions satisfying�i := {∑ij=1(z2j + (W̃ Tj � j W̃ j )) + ∑ij=2
η2j < 2 p}, i =1, . . . , n with p > 0, z1 = x1 − yd , zi =
xi −λi , i = 2, . . . , n,W̃ j = Ŵ j −W∗j , j = 1, . . . , n, λi
and ηi will be given. For thedesired system output yd , the
proposed nonlinear disturbanceobserver-based DSC should ensure that
all closed-loop signalsare convergent.
To proceed with the design of the nonlinear
disturbanceobserver-based DSC for the uncertain nonlinear system
(1),the following assumptions are required.
Assumption 1 [56]: For all t > 0, the reference signalyd(t)
is a sufficiently smooth function of t , and yd , ẏd , andÿd are
bounded, that is, there exists a positive constant B0such that �0
:= {(yd , ẏd , ÿd ) : (yd)2 + (ẏd)2 + (ÿd)2 ≤ B0}.
Assumption 2 [57]: The signs of gi , i = 1, . . . , n − 1 andgn0
are known. Furthermore, there exist positive constants giand ḡi ,
such that gi ≤ |gi | ≤ ḡi . At the same time, there existtwo
positive constants g
0and ḡ0 to render g0 ≤ |gn0| ≤ ḡ0
valid. Without losing generality, we shall assume that gi andgn0
are positive in the DSC design.
Assumption 3: There exist the unknown positive constantsβ0 and
β1 such that the external disturbance satisfy |d| ≤ β0and |ḋ| ≤
β1.
Assumption 4 [57]: There exist constants gdi > 0, i =1, 2, .
. . , n such that |ġi(.)| ≤ gdi in the compact set � j .
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CHEN et al.: DSC USING NNs FOR A CLASS OF UNCERTAIN NONLINEAR
SYSTEMS 2089
At the same time, there exists a positive constant gdn0 suchthat
|ġn0(.)| ≤ gdn0.
Assumption 5: For a practical system described by theuncertain
strict-feedback nonlinear system (1) subject to theinput saturation
(2) and the desired reference signal yd , thereshould exist a
feasible actual control input v which can achievethe given tracking
control objective.
Remark 1: Due to the control input saturation u(v(t)) andthe
unknown external disturbance d(t), the control design ofthe
uncertain nonlinear system (1) becomes more complicated.In
accordance with the characteristic of the DSC, the referencesignal
yd(t) and its time derivatives ẏd(t), ÿd (t) are assumedto be
bounded in Assumption 1. Assumption 2 implies thatsmooth functions
are strictly either positive or negative. ToAssumption 3, the
external disturbance is assumed as boundedand the boundary is
unknown. Since the time-dependent distur-bance d(t) can be largely
attributed to the exogenous effects, ithas finite energy. Hence, it
is bounded and the time derivationis also bounded. On the other
hand, the approximation error�(v) of the control input saturation
is bounded which equalsto �(v) = sat(v(t)) − h(v). For a practical
system, the timederivation of sat(v(t)) is bounded when the
actuator is deter-mined. Furthermore, the time derivation of tanh
function h(v)is also bounded. Thus, the time derivation of �̇(v) is
bounded.At the same time, Ḋ(t) =
ḋ(t)+(ġn0(x̄n)�(v)+gn0(x̄n)�̇(v)).According to Assumptions 2 and
4, we know that gn0 andġn0 are bounded. From above analysis, we
know that thecompounded disturbance D(t) satisfies |D| ≤ θ0 and
|Ḋ| ≤ θ1with the unknown constants θ0 > 0 and θ1 > 0.
Remark 2: For a given practical system, the input satura-tion
should meet the physical requirement of system control.In other
words, there should exist a DSC that can track thegiven desired
output of the nonlinear system in the presenceof the unknown
external disturbance and the input saturationfor all given initial
conditions. Many practical systems arecontrollable under the
control input saturation, such as aflight control system. For an
aircraft, the deflexion anglesof control surfaces are limited,
which lead to the boundedcontrol forces and control moments.
However, there usuallyexists a possible control to meet the flight
control requirementunder the limited control forces and control
moments. Thus,for a given practical system, the input saturation
should meetthe physical requirement of system control. Namely,
thereshould exist a DSC that can track the given desired outputof
the nonlinear system in the presence of the unknownexternal
disturbance and the input saturation for all giveninitial
conditions.
Remark 3: To tackle the control input saturation of theuncertain
strict-feedback nonlinear system, the saturation func-tion is
approximated by the tanh function in the DSC design.To facilitate
the DSC design for the uncertain nonlinearsystem (6), we introduce
gn0(x̄n)(∂h(v)/∂v) |v=vμ to be asa control gain function by
invoking the mean-value theorem.In general, ∂h(v)/∂v goes to zero
as v → ∞ which may leadto gn(x̄n) = gn0(x̄n)(∂h(v)/∂v)|v=vμ going
to zero. However,from Assumption 5, we know that the difference
between thedesigned control input v and the actual control input u
shouldbe bounded to meet the controllable requirement. Due to
the
bounded actual control input u, the designed control input vdoes
not go to infinite which means gn without going to zeroin our DSC
design.
III. DSC USING NONLINEAR DISTURBANCE OBSERVERAND BACKSTEPPING
TECHNIQUE
In this section, the NN-based DSC scheme will bedeveloped for
the uncertain strict-feedback nonlinearsystem (1) using the NDO.
The detailed design process isdescribed as follows.
Step 1: Consider the first equation in (10) when n = 1 anddefine
the error variable as
z1 = x1 − yd . (16)Invoking (10) and differentiating z1 with
respect to time
yields
ż1 = ẋ1 − ẏd = f1(x1) + g1(x1)x2 − ẏd . (17)Assuming x2 as a
virtual control input, the desired feedback
control α∗2 can be designed as
α∗2 = −k1z1 −1
g1( f1 − ẏd) (18)
where k1 is a positive design constant. f1 and g1 are
unknownsmooth functions of x1.
Define ρ1(Z1) = (1/g1(x1))( f1(x1) − ẏd) with Z1 =[x1, ẏd ]T .
By employing the RBFNN to approximate ρ1(Z1)and considering (12),
α∗2 can be expressed as
α∗2 = −k1z1 − W∗T1 S1(Z1) − ε∗1 . (19)Since W∗1 and ε∗1 are
unknown, the virtual control law α2
is proposed as
α2 = −k1z1 − Ŵ T1 S1(Z1) (20)where Ŵ1 is the estimation of W∗1
which is updated by
˙̂W1 = �1(S1(Z1)z1 − σ1Ŵ1) (21)where �1 = �T1 > 0 and σ1
> 0 are the design parameters.
To avoid repeatedly differentiating α2, which leads to
theso-called explosion of complexity in the sequel steps, the
DSCtechnique can be employed to solve it. Introducing a
first-orderfilter λ2, and letting α2 pass through it with time
constant τ2yields
τ2λ̇2 + λ2 = α2, λ2(0) = α2(0). (22)Defining z2 = x2 − λ2 and η2
= λ2 − α2, we have
λ̇2 = −η2/τ2 and x2 = z2+η2+α2. Considering (17) and (20),we
obtain
ż1 = f1 + g1(z2 + η2 + α2) − ẏd= g1ρ1 + g1(z2 + η2 + α2)=
g1
(W∗T1 S1(Z1) + ε∗1
)+ g1
(z2 + η2 − k1z1 − Ŵ T1 S1(Z1)
)= g1
(z2 + η2 − k1z1 − W̃ T1 S1(Z1) + ε∗1
)(23)
where W̃1 = Ŵ1 − W∗1 .
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2090 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS,
VOL. 26, NO. 9, SEPTEMBER 2015
For η2, we have
η̇2 = λ̇2 − α̇2= −η2
τ2+
(− ∂α2
∂x1ẋ1 − ∂α2
∂z1ż1 − ∂α2
∂Ŵ1− ∂α2
∂ydẏd
)
= −η2τ2
+ M2(z1, z2, η2, Ŵ1, yd , ẏd , ÿd) (24)
where M2(z1, z2, η2, Ŵ1, yd , ẏd , ÿd) = −(∂α2/∂x1)ẋ1
−(∂α2/∂z1)ż1 − ∂α2/∂Ŵ1 − (∂α2/∂yd)ẏd is a continuous func-tion.
For any B0 and p, the sets �0 := {(yd , ẏd , ÿd) : (yd)2 +(ẏd)2
+ (ÿd)2 ≤ B0} and �2 := {∑2j=1 z2j + W̃ T1 �1W̃1 +η22 <2 p} are
compact in R3 and RN1+3, respectively, where N1is the dimension of
W̃1. Thus, �0 × �2 is also compact.Considering the continuous
property, the function M2(.) hasa maximum value B2 for the given
initial conditions in thecompact set �0 × �2 [35].
Consider the Lyapunov function candidate
V1 = 12g1
z21 +1
2η22 +
1
2W̃ T1 �
−11 W̃1. (25)
Invoking (21), (23), and (24), the time derivative of V1 isgiven
by
V̇1 = 1g1
z1 ż1 − ġ12g21
z21 + η2η̇2 + W̃ T1 �−11 ˙̃W1
≤ z1(z2 + η2 − k1z1 − W̃ T1 S1(Z1) + ε∗1
) + gd12g2
1
z21
+ η2(
− η2τ2
+ M2)
+ W̃ T1 �−11 ˙̃W1
= − k1z21 + z1z2 + z1η2 + z1ε∗1 +gd1
2g21
z21
−η22
τ2+ η2 M2 − σ1W̃ T1 Ŵ1 (26)
where M2 denotes M2(z1, z2, η2, Ŵ1, yd , ẏd , ÿd).Considering
the following fact:
2W̃ T1 Ŵ1 = ‖W̃1‖2 + ‖Ŵ1‖2 − ‖W∗1 ‖2≥ ‖W̃1‖2 − ‖W∗1 ‖2
(27)
we have
V̇1 ≤ −(
k1 − 1.5 − gd1
2g21
)z21 −
(1
τ2− 1
)η22 −
σ1
2‖W̃1‖2
+ 0.5z22 + 0.5ε∗21 + 0.5B22 +σ1
2‖W∗1 ‖2. (28)
Step i (2 ≤ i ≤ n − 1): In the i th step, we define the
errorvariable as
zi = xi − λi (29)where λi is obtained from the (i − 1)th
step.
Considering (10) and differentiating zi with respect to
timeyields
żi = ẋi − λ̇i = fi (x̄i ) + gi(x̄i )xi+1 − λ̇i . (30)
Assuming xi+1 as a virtual control input, the desired feed-back
control α∗i+1 can be designed as
α∗i+1 = −ki zi −1
gi( fi − λ̇i ) (31)
where ki is a positive design constant. fi and gi are
unknownsmooth functions of x̄i .
Define ρi (Zi ) = (1/gi (x̄i))( fi (x̄i )−λ̇i ) with Zi = [x̄i ,
˙̄λi ]T.By employing the RBFNN to approximate ρi (Zi )
andconsidering (12), α∗i+1 can be expressed as
α∗i+1 = −ki zi − W∗Ti Si (Zi ) − ε∗i . (32)Since W∗i and ε∗i are
unknown, the virtual control law αi+1
is proposed as
αi+1 = −ki zi − Ŵ Ti Si (Zi ) (33)where Ŵi is the estimation
of W∗i which is updated by
˙̂Wi = �i (Si (Zi )zi − σi Ŵi ) (34)where �i = �Ti > 0 and
σi > 0 are the design parameters.
To avoid repeatedly differentiating αi+1, which leads to
theso-called explosion of complexity in the sequel steps, the
DSCtechnique can be employed to solve it. Introducing a
first-orderfilter λi+1, and letting αi+1 pass through it with time
constantτi+1 yields
τi+1λ̇i+1 + λi+1 = αi+1, λi+1(0) = αi+1(0). (35)Defining zi+1 =
xi+1 − λi+1 and ηi+1 = λi+1 − αi+1, we
have λ̇i+1 = −ηi+1/τi+1 and xi+1 = zi+1 + ηi+1 +
αi+1.Considering (30) and (33), we obtain
żi = fi + gi(zi+1 + ηi+1 + αi+1) − λ̇i= giρi + gi (zi+1 + ηi+1
+ αi+1)= gi
(W∗Ti Si (Zi ) + ε∗i
)+ gi
(zi+1 + ηi+1 − ki zi − Ŵ Ti Si (Zi )
)= gi
(zi+1 + ηi+1 − ki zi − W̃ Ti Si (Zi ) + ε∗i
)(36)
where W̃i = Ŵi − W∗i .For ηi+1, we have
η̇i+1 = λ̇i+1 − α̇i+1 = −ηi+1τi+1
+(
− ∂α∂xi
ẋi − ∂αi+1∂zi
żi − ∂αi+1∂Ŵi
− ∂αi+1∂λi
λ̇i
)
= −ηi+1τi+1
+ Mi+1 (37)
where Mi+1 denotes Mi+1(z1, . . . , zi+1, η1, . . . , ηi , Ŵ1,
. . . ,Ŵi , yd , ẏd , ÿd) and Mi+1 = −(∂α/∂xi )ẋi − (∂αi+1/∂zi
)żi −∂αi+1/∂Ŵi − (∂αi+1/∂λi )λ̇i is a continuous function. For
anyB0 and p, the sets �0 := {(yd , ẏd , ÿd) : (yd)2+(ẏd)2+(ÿd)2
≤B0} and �i := {∑ij=1(z2j + (W̃ Tj � j W̃ j )) + ∑ij=2 η2j <2
p}, i = 1, . . . , n − 1 are compact in R3 and R
∑ij=1 Ni +2i−1,
respectively, where Ni is the dimension of W̃i . Thus, �0 ×�iis
also compact. Considering the continuous property, thefunction Mi+1
has a maximum value Bi+1 for the given initialconditions in the
compact set �0 × �i [35].
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SYSTEMS 2091
Consider the Lyapunov function candidate
Vi = 12gi
z2i +1
2η2i+1 +
1
2W̃ Ti �
−1i W̃i . (38)
Invoking (34), (36), and (37), the time derivative of V1 isgiven
by
V̇i = 1gi
zi żi − ġi2g2i
z2i + ηi+1η̇i+1 + W̃ Ti �−1i ˙̃Wi
≤ zi(zi+1 + ηi+1 − ki zi − W̃ Ti Si (Zi ) + ε∗i
) + gdi2g2
i
z2i
+ ηi+1(
− ηi+1τi+1
+ Mi+1)
+ W̃ Ti �−1i ˙̃Wi= − ki z2i + zi zi+1 + ziηi+1 + ziε∗i
+ gdi
2g2i
z2i −η2i+1τi+1
− σi W̃ Ti Ŵi + ηi+1 Mi+1. (39)
Considering the following fact:2W̃ Ti Ŵi = ‖W̃i‖2 + ‖Ŵi ‖2 −
‖W∗i ‖2
≥ ‖W̃i‖2 − ‖W∗i ‖2 (40)we have
V̇i ≤ −(
ki − 1.5 − gdi
2g2i
)z2i −
(1
τi+1− 1
)η2i+1 −
σi
2‖W̃i‖2
+ 0.5z2i+1 + 0.5ε∗2i + 0.5B2i+1 +σi
2‖W∗i ‖2. (41)
Step n: In this step, the error variable is defined as
zn = xn − λn (42)where λn is obtained from the (n − 1)th
step.
Considering (10) and differentiating zn with respect to
timeyields
żn = ẋn − λ̇n = fn(x̄n) + gn(x̄n)v + D − λ̇n . (43)The desired
feedback control v∗ can be designed as
v∗ = −knzn − 1gn
( fn − λ̇n) − D (44)where kn is a positive design constant. fn
and gn are unknownsmooth functions of x̄n .
Define ρn(Zn) = 1/gn(x̄n)( fn(x̄n) − λ̇n) with Zn =[x̄n, ˙̄λn]T.
By employing the RBFNN to approximate ρn(Zn)and considering (12),
v∗ can be expressed as
v∗ = −knzn − W∗Tn Sn(Zn) − ε∗n − D. (45)Since W∗n , ε∗n , and D
are unknown, the control law v is
proposed as
v = −knzn − Ŵ Tn Sn(Zn) − D̂ (46)where D̂ is the estimation of
D and Ŵn is the estimation ofW∗n which is updated by
˙̂Wn = �n(Sn(Zn)zn − σn Ŵn) (47)where �n = �Tn > 0 and σn
> 0 are the design parameters.
Considering (43) and (46), we obtain
żn = fn + gnv + D(t) − λ̇n = gn(W∗Tn Sn(x̄n) + ε∗n
)+ gn
( − knzn − Ŵ Tn Sn(Zn) − D̂) + D(t)= gn
( − knzn − W̃ Tn Sn(Zn) − D̂ + ε∗n) + D(t) (48)where W̃n = Ŵn −
W∗n .
To facilitate the design of the NDO, (43) can be also
writtenas
żn = l−1ρ(x̄n, v) + D − λ̇n= l−1W∗Tρ Sρ(x̄n) + l−1ερ + D − λ̇n
(49)
where ρ(x̄n, v) = l( fn(x̄n) + gn(x̄n)v), W∗ρ is optimal
weightvalue of the RBFNN, ερ is the approximation error of
theRBFNN, and l > 0 is a design parameter of the
developedNDO.
Invoking (49), an auxiliary variable is given by
s = zn − ξ (50)and the intermedial variable ξ is proposed as
ξ̇ = cs + l−1Ŵ Tρ Sρ(x̄n) − λ̇n (51)where c > 0 is a
designed parameter and Ŵρ is the estimateof the optimal weight
value W∗ρ .
Differentiating (50) and considering (49) and (51), we have
ṡ = żn − ξ̇ = l−1W∗Tρ Sρ(x̄n) + l−1ερ + D − λ̇n− (cs + l−1Ŵ
Tρ Sρ(x̄n) − λ̇n)
= − cs − l−1W̃ Tρ Sρ(x̄n) + l−1ερ + D (52)where W̃ρ = Ŵρ − W∗ρ
.
Considering (52) yields
sṡ = −cs2 − l−1sW̃ Tρ Sρ(x̄n) + l−1sερ + s D≤ − (c −
1.0)s2−l−1sW̃ Tρ Sρ(x̄n)+0.5l−2ε2ρ +0.5θ20 .
(53)
On the basis of the auxiliary variable s, the NDO isdesigned
as
D̂ = l(s − φ) (54)and the intermedial variable φ is given by
φ̇ = −cs + D̂. (55)Define D̃ = D − D̂. Differentiating (54), and
considering
(52) and (55) yield
˙̂D = l(ṡ−φ̇)= l((−cs−l−1W̃ Tρ Sρ(x̄n)+l−1ερ +D)−(−cs+ D̂))=
−W̃ Tρ Sρ(x̄n) + ερ + l(D − D̂)= −W̃ Tρ Sρ(x̄n) + ερ + l D̃.
(56)Considering (56), we have
˙̃D = Ḋ − ˙̂D = Ḋ − l D̃ + W̃ Tρ Sρ(x̄n) − ερ. (57)Invoking
(57), we obtain
D̃ ˙̃D = D̃ Ḋ − l D̃2 + D̃W̃ Tρ Sρ(x̄n) − D̃ερ. (58)
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2092 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS,
VOL. 26, NO. 9, SEPTEMBER 2015
Considering the following fact:2D̃W̃ Tρ Sρ(x̄n) ≤ 2|D̃|||W̃ρ
||||Sρ(x̄n)||
≤ γ0ϑ2 D̃2 + 1γ0
||W̃ρ ||2 (59)yields
D̃ ˙̃D ≤ D̃2 + 0.5Ḋ2 − l D̃2 + γϑ2 D̃2 + 1γ
||W̃ρ ||2 + 0.5ε2ρ≤ −(l − (1.0 + γϑ2))D̃2 + 1
γ||W̃ρ ||2 + 0.5θ21 + 0.5ε2ρ
(60)
where ||Sρ(x̄n)|| ≤ ϑ , γ = 0.5γ0 and γ0 > 0 is a
designparameter.
The parameter updated law Ŵρ is designed as
˙̂Wρ = �ρ(l−1STρ (x̄n)s − σρ Ŵρ
)(61)
where �ρ = �Tρ > 0 and σρ > 0 are the design
parameters.Consider the Lyapunov function candidate
Vn = 12gn
z2n +1
2W̃ Tn �
−1n W̃n +
1
2W̃ Tρ �
−1ρ W̃ρ +
1
2s2 + 1
2D̃2.
(62)
Invoking (48), (53), and (60), the time derivative of Vn is
V̇n = 1gn
zn żn − ġn2g2n
z2n + W̃ Tn �−1n ˙̃Wn+W̃ Tρ �−1ρ ˙̃Wρ + sṡ + D̃ ˙̃D
≤ zn( − knzn − W̃ Tn Sn(Zn) − D̂ + ε∗n) + g
dn
2g2n
z2n
+ zn Dgn
+ W̃ Tn �−1n ˙̃Wn + W̃ Tρ �−1ρ ˙̃Wρ− (c − 1.0)s2 − l−1sW̃ Tρ
Sρ(x̄n) + 0.5l−2ε2ρ + 0.5θ20− (l − (1.0 + γϑ2))D̃2 + 1
γ||W̃ρ ||2
+ 0.5θ21 + 0.5ε2ρ. (63)Considering D̃ = D − D̂ and Assumption 3,
we have
V̇n ≤ zn( − knzn − W̃ Tn Sn(Zn) + ε∗n) + g
dn
2g2n
z2n
+ zn D̃ + zn D(
1
gn− 1
)+ W̃ Tn �−1n ˙̃Wn + W̃ Tρ �−1ρ ˙̃Wρ
− (c − 1.0)s2 − l−1sW̃ Tρ Sρ(x̄n) + 0.5l−2ε2ρ + 0.5θ20− (l −
(1.0 + γϑ2))D̃2 + 1
γ||W̃ρ ||2 + 0.5θ21 + 0.5ε2ρ
≤ −(
kn − 1.5 − gdn
2g2n
)z2n − zn W̃ Tn Sn(Zn)
+ W̃ Tn �−1n ˙̃Wn + W̃ Tρ �−1ρ ˙̃Wρ − l−1sW̃ Tρ Sρ(x̄n)− (c −
1.0)s2 − (l − (1.5 + γϑ2))D̃2
+ 1γ
||W̃ρ ||2 +(
0.5 + 0.5| 1g
n
− 1|2)
θ20
+ 0.5θ21 + 0.5ε∗2n + (0.5 + 0.5l−2)ε2ρ. (64)
Substituting (47) and (61) into (64), we obtain
V̇n ≤ −(
kn − 1.5 − gdn
2g2n
)z2n − (c − 1.0)s2
−(l − (1.5 + γϑ2))D̃2 − σnW̃ Tn Ŵn − σρ W̃ Tρ Ŵρ+ 1
γ||W̃ρ ||2 +
(0.5 + 0.5| 1
gn
− 1|2)
θ20
+ 0.5θ21 + 0.5ε∗2n + (0.5 + 0.5l−2)ε2ρ. (65)Considering the
following facts:
2W̃ Tn Ŵn = ‖W̃n‖2 + ‖Ŵn‖2 − ‖W∗n ‖2≥ ‖W̃n‖2 − ‖W∗n ‖2
(66)
and
2W̃ Tρ Ŵρ = ‖W̃ρ‖2 + ‖Ŵρ‖2 − ‖W∗ρ ‖2
≥ ‖W̃ρ‖2 − ‖W∗ρ ‖2 (67)we have
V̇n ≤ −(
kn − 1.5 − gdn
2g2n
)z2n − (c − 1.0)s2
−(l − (1.5 + γϑ2))D̃2 − σn2
‖W̃n‖2
−(
σρ
2− 1
γ
)‖W̃ρ‖2 +
(0.5 + 0.5| 1
gn
− 1|2)
θ20
+ 0.5θ21 + 0.5ε∗2n + (0.5 + 0.5l−2)ε2ρ+ σn
2‖W∗n ‖2 +
σρ
2‖W∗ρ ‖2. (68)
The above DSC design procedure and stability analysis canbe
summarized in the following theorem, which contains theresults for
the uncertain strict-feedback nonlinear system (1)using the NDO and
backstepping technique.
Theorem 1: Consider the uncertain strict-feedback nonlin-ear
system (1) with the input saturation and the unknownexternal
disturbance and suppose that full state information isavailable.
The NDO is designed as (50), (51), (54), and (55).The updated laws
of the NN weight values are chosen as (21),(34), (47), and (61).
The nonlinear disturbance observer-basedDSC is proposed in (46).
Given any positive number p, for allinitial conditions satisfying
�n := {∑nj=1(z2j +(W̃ Tj � j W̃ j ))+∑n
j=2 η2j < 2 p}, the appropriate design parameters ki , l, c,
τi ,�i , �ρ , σi , σρ , and γ can be chosen according to (72)
suchthat all closed-loop signals are uniformly bounded
convergenceunder the proposed dynamic surface control based on
thenonlinear disturbance observer. Furthermore, the tracking
errorz1 = x1−yd can be made small by proper choice of the
designparameters ki , l, c, τi , �i , �ρ , σi , σρ , and γ.
Proof: For considering the convergence of disturbanceestimate
error and closed-loop states, the Lyapunov func-tion candidate of
the whole closed-loop control system isconsidered as
V =n∑
i=1Vi . (69)
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CHEN et al.: DSC USING NNs FOR A CLASS OF UNCERTAIN NONLINEAR
SYSTEMS 2093
Differentiating V and considering (28), (41), and (68),
weobtain
V̇ ≤ −(
k1 − 1.5 − gd1
2g21
)z21 −
n∑i=2
(ki − 2.0 − g
di
2g2i
)z2i
−n∑
i=2
(1
τi− 1
)η2i −
n∑i=1
σi
2‖W̃i‖2
− (c − 1.0)s2 − (l − (1.5 + γϑ2))D̃2
−(
σρ
2− 1
γ
)‖W̃ρ‖2 + 0.5
n∑i=2
B2i
+(
0.5 + 0.5| 1g
n
− 1|2)
θ20 + 0.5θ21 + 0.5n∑
i=1ε∗2i
+ (0.5 + 0.5l−2)ε2ρ +n∑
i=1
σi
2‖W∗i ‖2 +
σρ
2‖W∗ρ ‖2
≤ −�V + C (70)where � and C are given by
� : = min
⎛⎜⎜⎜⎜⎜⎜⎝
(k1 − 1.5 − g
d1
2g21
),(
ki − 2.0 − gdi
2g2i
)
(1
τi− 1), (c − 1.0), (l − (1.5 + γϑ2))σi
λmax(�−1i )
,2( σρ2 − 1γ )λmax(�
−1ρ )
⎞⎟⎟⎟⎟⎟⎟⎠
C : = 0.5n∑
i=2B2i +
n∑i=1
σi
2‖W∗i ‖2 +
σρ
2‖W∗ρ ‖2
+(
0.5 + 0.5∣∣∣∣ 1g
n
− 1∣∣∣∣ 2
)θ20
+ 0.5θ21 + 0.5n∑
i=1ε∗2i + (0.5 + 0.5l−2)ε2ρ. (71)
To ensure the closed-loop system stability, the correspond-ing
design parameters ki , l, c, τi , σρ , and γ should be chosento
make the following inequalities hold:
k1 − 1.5 − gd1
2g21
> 0
ki − 2.0 − gdi
2g2i
> 0, i = 2, . . . , n
1
τi− 1 > 0, i = 2, . . . , n − 1
c − 1.0 > 0l − (1.5 + γϑ2) > 0σρ
2− 1
γ> 0. (72)
According to (70), we have
0 ≤ V ≤ C�
+[
V (0) − C�]e−�t. (73)
From (73), we can know that V is convergent, i.e.,limt−→∞ V =
C/�. According to (73), it may directly show
that the signals e, zi , W̃i , W̃ρ , and D̃ are semiglobally
uni-formly bounded when t → 0. Hence, the tracking error e,
theapproximation errors W̃i , W̃ρ , and the disturbance
estimationerror D̃ of the closed-loop system are bounded. This
concludesthe proof. ♦
Remark 4: To enhance the closed-loop system robustnessof the
uncertain nonlinear system, the nonlinear disturbanceobserver-based
DSC scheme has been developed. For fullyutilizing the dynamic
information of the external disturbance,the NDO is proposed to
estimate the unknown disturbance ofthe uncertain nonlinear system
and the output of the NDO isused to design the DSC law, as shown in
(46). Due to theintroduction of the output of NDO, the control gain
can beadjusted according to the variation of unknown disturbanceand
the disturbance rejection ability of the closed-loop systemhas been
improved.
Remark 5: In this paper, the NDO is developed to estimatethe
unknown disturbance of the uncertain strict-feedback non-linear
system. In the developed NDO, the known boundaryrequirement of the
disturbance is canceled and the boundeddisturbance estimation error
is guaranteed. At the same time,the slowly changeable assumption of
the external disturbanceis eliminated.
Remark 6: In the developed DSC, the design parameters ki ,l, c,
τi , �i , �ρ , σi , σρ , and γ need to be tuned to obtain a
goodtransient performance and the closed-loop stable performance.If
the tracking error is desired to be lower, we should increasek1. �i
, �ρ , σi , and σρ are design parameters in adaptationlaw of NN
weight value. Decreases in σi and σρ or increasesin the adaptive
gain �i and �ρ will result in a better trackingperformance.
Furthermore, the L∞ performance of systemtracking error can be
guaranteed by choosing the properinitial conditions for all
closed-loop system signals accordingto (73) [13], [32], [40],
[42].
IV. SIMULATION STUDY
In this section, simulation results are presented to illus-trate
the effectiveness of the developed nonlinear
disturbanceobserver-based DSC and an example system is used in
thesimulation study. In the simulation, the NDO is designedas (50),
(51), (54), and (55). The updated laws of theNN weight values are
chosen as (21), (34), (47), and (61).The nonlinear disturbance
observer-based DSC is proposedin (46).
Let us consider the one-link manipulator with the inclusionof
motor dynamics. The model of the one-link manipulator isgiven by
[36]
D̄q̈ + Bq̇ + N sin(q) = τM τ̇ + H τ = u − Kmq̇ (74)
where q , q̇, and q̈ denote the link angular position,
velocity,and acceleration, respectively. τ is the motor current. u
is theinput control voltage. The parameter values with
appropriateunits are given by D̄ = 1, M = 0.05, B = 1, Km = 10,H =
10, and N = 10.
Defining x1 = q , x2 = q̇, and x3 = τ , and considering theinput
saturation, the above one-link manipulator model can be
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2094 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS,
VOL. 26, NO. 9, SEPTEMBER 2015
Fig. 1. Output x1 (solid line) follows desired trajectory yd
(dashed line) ofthe single-link robot system for case 1.
written as
ẋ1 = x2ẋ2 = x3
D̄− Bx2
D̄− N
D̄sin(x1)
ẋ3 = 1M
u(v(t)) − KmM
x2 − HM
x3y = x1.
When uncertainty and disturbance are involved, the dynam-ics of
one-link manipulator can be expressed as the followingform:
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
ẋ1 = x2ẋ2 = x22 e−x
21 − Bx2
D̄− N
D̄sin(x1) + x3D̄
ẋ3 = − KmM x2 − HM x3 + 1M u(v(t)) + d(t)y = x1.
(75)
Define f1(x1) = 0, g1(x1) = 1, f2(x̄2) = x22e−x21 −
Bx2/D̄ − (N/D̄) sin(x1), g2(x̄2) = 1/D̄, f3(x̄3) =−(Km/M)x2 −
(H/M)x3, and g3(x̄3) = 1/M . It is apparentthat the numerical
example (75) is suitable for the case ofsystem (1). In the
simulation, the external disturbance ischosen as d(t) = 0.4
cos(2t).
To proceed with the design of nonlinear
disturbanceobserver-based DSC scheme, all design parameters are
chosenas l = 200, c = 13, σi = 0.02, k1 = 20, k2 = 20, k3 = 10,and
γ = 20. The initial state conditions are chosen as x0 = 0,x2 = 0,
and x3 = 0. The input saturation value is given asuM = 80.
A. Case 1: For Constant Desired Trajectory
To illustrate the effectiveness of the developed
nonlineardisturbance observer-based DSC design, the desired
trajectoryis taken as yd = 1. Under the proposed nonlinear
disturbanceobserver-based DSC scheme (46), the tracking control
resultsare shown in Figs. 1–4. From Figs. 1 and 2, we note thatthe
tracking performance is satisfactory and the tracking error
Fig. 2. Tracking error of the single-link robot system for case
1.
Fig. 3. Control input of the single-link robot system for case
1.
quickly converge to zero for the uncertain one-link manipula-tor
system (75) in the presence of the time-varying externaldisturbance
and input saturation. Although the better trackingerror is obtain
without considering the input saturation, theaccepted tracking
performance is still maintained for theuncertain one-link
manipulator system (75) in the presenceof the time-varying external
disturbance and input saturationunder our developed DSC scheme.
Using the output of theNDO, the control input is bounded and
convergent, as shownin Fig. 3. The plots of the NN weight values
are shown inFig. 4, which are convergent.
B. Case 2: For Time-Varying Desired Trajectory
Here, the desired trajectory is taken as yd = sin(t) +cos(0.5t)
to illustrate the effectiveness of the developednonlinear
disturbance observer-based DSC design. Usingthe proposed nonlinear
disturbance observer to design DSCscheme (46), all tracking control
results are given inFigs. 5–8. According to Figs. 5 and 6, the
satisfactory trackingperformance is obtained and the tracking error
maintains in
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CHEN et al.: DSC USING NNs FOR A CLASS OF UNCERTAIN NONLINEAR
SYSTEMS 2095
Fig. 4. Norms of NN weight values for case 1.
Fig. 5. Output x1 (solid line) follows desired trajectory yd
(dashed line) ofthe single-link robot system for case 2.
Fig. 6. Tracking error of the single-link robot system for case
2.
a small compact set for the uncertain one-link manipulatorsystem
(75) under the integrated effects of the time-varyingexternal
disturbance and input saturation. On the basis of the
Fig. 7. Control input of the single-link robot system for case
2.
Fig. 8. Norms of NN weight values for case 2.
output of the NDO, the control input command is bounded
andconvergent, as shown in Fig. 7. From Fig. 8, the convergentplots
of the NN weight values are noted.
Based on above simulation results, we can obtain that
theproposed disturbance observer-based DSC scheme is valid forthe
uncertain the one-link manipulator system with the time-varying
unknown external disturbance and input saturation.
V. CONCLUSION
In this paper, the nonlinear disturbance observer-based NNDSC
scheme has been developed for a class of uncertainnonlinear systems
with input saturation. To improve the abil-ity of the disturbance
attenuation and system performancerobustness, the NDO has been used
to monitor the unknowncompounded disturbance, and its output signal
is utilizedin the construction of nonlinear disturbance
observer-basedNN DSC scheme. Closed-loop system stability and
trackingperformance have been proved and analyzed using a
rigorousLyapunov analysis. Finally, simulation results of a
one-linkmanipulator control system have been presented to
illus-trate the effectiveness of the proposed disturbance
observer-based NN DSC scheme. In the future, the application of
the
-
2096 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS,
VOL. 26, NO. 9, SEPTEMBER 2015
developed nonlinear disturbance observer-based DSC schemeshould
be further studied. Furthermore, the DSC scheme canbe developed
using L∞-type criteria to enhance the controlperformance when the
control input saturation appear for thestudied uncertain nonlinear
systems.
REFERENCES
[1] Y.-Y. Cao and Z. Lin, “Robust stability analysis and
fuzzy-schedulingcontrol for nonlinear systems subject to actuator
saturation,” IEEE Trans.Fuzzy Syst., vol. 11, no. 1, pp. 57–67,
Feb. 2003.
[2] Q. Hu, G. Ma, and L. Xie, “Robust and adaptive variable
structureoutput feedback control of uncertain systems with input
nonlinearity,”Automatica, vol. 44, no. 2, pp. 552–559, 2008.
[3] Y. Luo and H. Zhang, “Approximate optimal control for a
class ofnonlinear discrete-time systems with saturating actuators,”
Prog. NaturalSci., vol. 18, no. 8, pp. 1023–1029, 2008.
[4] H. Zhang, Y. Luo, and D. Liu, “Neural-network-based
near-optimalcontrol for a class of discrete-time affine nonlinear
systems with controlconstraints,” IEEE Trans. Neural Netw., vol.
20, no. 9, pp. 1490–1503,Sep. 2009.
[5] C. Wen, J. Zhou, Z. Liu, and H. Su, “Robust adaptive control
ofuncertain nonlinear systems in the presence of input saturation
andexternal disturbance,” IEEE Trans. Autom. Control, vol. 56, no.
7,pp. 1672–1678, Jul. 2011.
[6] M. Chen, S. S. Ge, and B. Voon Ee How, “Robust adaptive
neuralnetwork control for a class of uncertain MIMO nonlinear
systemswith input nonlinearities,” IEEE Trans. Neural Netw., vol.
21, no. 5,pp. 796–812, May 2010.
[7] J. R. Azinheira and A. Moutinho, “Hover control of an UAV
withbackstepping design including input saturations,” IEEE Trans.
ControlSyst. Technol., vol. 16, no. 3, pp. 517–526, May 2008.
[8] M. Chen, S. S. Ge, and B. Ren, “Adaptive tracking control of
uncertainMIMO nonlinear systems with input constraints,”
Automatica, vol. 47,no. 3, pp. 452–465, 2011.
[9] Y.-S. Zhong, “Globally stable adaptive system design for
minimumphase SISO plants with input saturation,” Automatica, vol.
41, no. 9,pp. 1539–1547, 2005.
[10] M. Chen, C.-S. Jiang, and Q.-X. Wu, “Robust adaptive
control ofuncertain time delay systems with FLS,” Int. J.
Innovative Comput.,Inf., Control, vol. 4, no. 8, pp. 1551–1561,
2008.
[11] D. Chwa, “Tracking control of differential-drive wheeled
mobile robotsusing a backstepping-like feedback linearization,”
IEEE Trans. Syst.,Man, Cybern. A, Syst., Humans, vol. 40, no. 6,
pp. 1285–1295,Nov. 2010.
[12] S. C. Tong, Y. Li, and H.-G. Zhang, “Adaptive neural
network decen-tralized backstepping output-feedback control for
nonlinear large-scalesystems with time delays,” IEEE Trans. Neural
Netw., vol. 22, no. 7,pp. 1073–1086, Jul. 2011.
[13] C. Wang and Y. Lin, “Multivariable adaptive backstepping
control:A norm estimation approach,” IEEE Trans. Autom. Control,
vol. 57,no. 4, pp. 989–995, Apr. 2012.
[14] H. Li, H. Liu, H. Gao, and P. Shi, “Reliable fuzzy control
for activesuspension systems with actuator delay and fault,” IEEE
Trans. FuzzySyst., vol. 20, no. 2, pp. 342–357, Apr. 2012.
[15] Z. Li and C. Yang, “Neural-adaptive output feedback control
of a classof transportation vehicles based on wheeled inverted
pendulum models,”IEEE Trans. Control Syst. Technol., vol. 20, no.
6, pp. 1583–1591,Nov. 2012.
[16] Q. Zhou, P. Shi, and S. Xu, “Neural-network-based
decentralizedadaptive output-feedback control for large-scale
stochastic nonlinearsystems,” IEEE Trans. Syst., Man, Cybern. B,
Cybern., vol. 42, no. 6,pp. 1608–1619, Dec. 2012.
[17] Z. Li and C.-Y. Su, “Neural-adaptive control of
single-master–multiple-slaves teleoperation for coordinated
multiple mobile manipulators withtime-varying communication delays
and input uncertainties,” IEEETrans. Neural Netw. Learn. Syst.,
vol. 24, no. 9, pp. 1400–1413,Sep. 2013.
[18] Q. Zhou, P. Shi, S. Xu, and H. Li, “Observer-based adaptive
neuralnetwork control for nonlinear stochastic systems with time
delay,” IEEETrans. Neural Netw. Learn. Syst., vol. 24, no. 1, pp.
71–80, Jan. 2013.
[19] Z. Li, S. S. Ge, and S. Liu, “Contact-force distribution
optimizationand control for quadruped robots using both gradient
and adaptiveneural networks,” IEEE Trans. Neural Netw. Learn.
Syst., vol. 25, no. 8,pp. 1460–1473, Aug. 2014.
[20] H. Li, J. Yu, C. Hilton, and H. Liu, “Adaptive sliding-mode
control fornonlinear active suspension vehicle systems using T–S
fuzzy approach,”IEEE Trans. Ind. Electron., vol. 60, no. 8, pp.
3328–3338, Aug. 2013.
[21] Y.-J. Liu, W. Wang, S.-C. Tong, and Y.-S. Liu, “Robust
adaptive trackingcontrol for nonlinear systems based on bounds of
fuzzy approximationparameters,” IEEE Trans. Syst., Man, Cybern. A,
Syst., Humans, vol. 40,no. 1, pp. 170–184, Jan. 2010.
[22] S.-C. Tong, X.-L. He, and H.-G. Zhang, “A combined
backstepping andsmall-gain approach to robust adaptive fuzzy output
feedback control,”IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp.
1059–1069, Oct. 2009.
[23] W. Chen and Z. Zhang, “Globally stable adaptive
backstepping fuzzycontrol for output-feedback systems with unknown
high-frequency gainsign,” Fuzzy Sets Syst., vol. 161, no. 6, pp.
821–836, 2010.
[24] W. Chen, L. Jiao, R. Li, and J. Li, “Adaptive backstepping
fuzzy controlfor nonlinearly parameterized systems with periodic
disturbances,” IEEETrans. Fuzzy Syst., vol. 18, no. 4, pp. 674–685,
Aug. 2010.
[25] S. Tong, B. Huo, and Y. Li, “Observer-based adaptive
decentralizedfuzzy fault-tolerant control of nonlinear large-scale
systems with actuatorfailures,” IEEE Trans. Fuzzy Syst., vol. 22,
no. 1, pp. 1–15, Feb. 2014.
[26] T. Lee and Y. Kim, “Nonlinear adaptive flight control using
backsteppingand neural networks controller,” J. Guid., Control,
Dyn., vol. 24, no. 4,pp. 675–686, 2001.
[27] D. Gao, Z. Sun, and X. Luo, “Fuzzy adaptive control for
hypersonicvehicle via backstepping method,” Control Theory Appl.,
vol. 25, no. 5,pp. 805–810, 2008.
[28] M. Chen, S. S. Ge, and B. Ren, “Robust attitude control of
helicopterswith actuator dynamics using neural networks,” IET
Control TheoryAppl., vol. 44, no. 12, pp. 2837–2854, Dec. 2010.
[29] S. Tong, Y. Li, Y. Li, and Y. Liu, “Observer-based adaptive
fuzzybackstepping control for a class of stochastic nonlinear
strict-feedbacksystems,” IEEE Trans. Syst., Man, Cybern. B,
Cybern., vol. 41, no. 6,pp. 1693–1704, Dec. 2011.
[30] Y.-J. Liu, C. L. P. Chen, G.-X. Wen, and S. Tong, “Adaptive
neuraloutput feedback tracking control for a class of uncertain
discrete-time nonlinear systems,” IEEE Trans. Neural Netw., vol.
22, no. 7,pp. 1162–1167, Jul. 2011.
[31] R. Cui, B. Ren, and S. S. Ge, “Synchronised tracking
control of multi-agent system with high order dynamics,” IET
Control Theory Appl.,vol. 6, no. 5, pp. 603–614, Mar. 2012.
[32] W. Chenliang and L. Yan, “Adaptive dynamic surface control
for linearmultivariable systems,” Automatica, vol. 46, no. 10, pp.
1703–1711,2010.
[33] P. P. Yip and J. K. Hedrick, “Adaptive dynamic surface
control:A simplified algorithm for adaptive backstepping control of
nonlinearsystems,” Int. J. Control, vol. 71, no. 5, pp. 959–979,
1998.
[34] D. Swaroop, J. K. Hedrick, P. P. Yip, and J. C. Gerdes,
“Dynamic surfacecontrol for a class of nonlinear systems,” IEEE
Trans. Autom. Control,vol. 45, no. 10, pp. 1893–1899, Oct.
2000.
[35] D. Wang and J. Huang, “Neural network-based adaptive
dynamic surfacecontrol for a class of uncertain nonlinear systems
in strict-feedbackform,” IEEE Trans. Neural Netw., vol. 16, no. 1,
pp. 195–202, Jan. 2005.
[36] T.-S. Li, D. Wang, G. Feng, and S.-C. Tong, “A DSC approach
to robustadaptive NN tracking control for strict-feedback nonlinear
systems,”IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no.
3, pp. 915–926,Jun. 2010.
[37] D. Wang, “Neural network-based adaptive dynamic surface
control ofuncertain nonlinear pure-feedback systems,” Int. J.
Robust NonlinearControl, vol. 21, no. 5, pp. 527–541, 2011.
[38] B. Song and J. K. Hedrick, “Simultaneous quadratic
stabilizationfor a class of non-linear systems with input
saturation using dynamicsurface control,” Int. J. Control, vol. 77,
no. 1, pp. 19–26, 2004.
[39] L. Dong, W. Jie, and Y. Xiuduan, “Output feedback adaptive
dynamicsurface control with input saturation,” in Proc. 31st Chin.
Control Conf.,Jul. 2012, pp. 3125–3130.
[40] Q. Zhao and Y. Lin, “Adaptive dynamic surface control for a
classof output-feedback nonlinear systems with guaranteed L∞
track-ing performance,” Int. J. Syst. Sci., vol. 42, no. 8, p.
1351–1362,2011.
[41] X. Yu, X. Sun, Y. Lin, and W. Dong, “Output feedback
adaptiveDSC for nonlinear systems with guaranteed L∞ tracking
performance,”J. Control Theory Appl., vol. 10, no. 1, pp. 124–131,
2012.
[42] C. Wang and Y. Lin, “Output-feedback robust adaptive
backsteppingcontrol for a class of multivariable nonlinear systems
with guaranteedL∞ tracking performance,” Int. J. Robust Nonlinear
Control, vol. 23,no. 18, pp. 2082–2096, 2013.
-
CHEN et al.: DSC USING NNs FOR A CLASS OF UNCERTAIN NONLINEAR
SYSTEMS 2097
[43] W.-H. Chen, D. J. Ballance, P. J. Gawthrop, J. J. Gribble,
and J. O’Reilly,“Nonlinear PID predictive controller,” IEE Proc.
Control Theory Appl.,vol. 146, no. 6, pp. 603–611, Nov. 1999.
[44] Z. G. Sun, N. C. Cheung, S. W. Zhao, and W. C. Gan, “The
application ofdisturbance observer-based sliding mode control for
magnetic levitationsystems,” Proc. Inst. Mech. Eng. C, J. Mech.
Eng. Sci., vol. 224, no. 8,pp. 1635–1644, 2010.
[45] M. Chen and W.-H. Chen, “Disturbance-observer-based robust
controlfor time delay uncertain systems,” Int. J. Control, Autom.,
Syst., vol. 8,no. 2, pp. 445–453, 2010.
[46] M. Chen and W.-H. Chen, “Sliding mode controller design for
a classof uncertain nonlinear system based disturbance observer,”
Int. J. Adapt.Control Signal Process., vol. 24, no. 1, pp. 51–64,
2010.
[47] J. Yang, S. Li, and X. Yu, “Sliding-mode control for
systems withmismatched uncertainties via a disturbance observer,”
IEEE Trans. Ind.Electron., vol. 60, no. 1, pp. 160–169, Jan.
2013.
[48] W.-H. Chen, “Disturbance observer based control for
nonlinear systems,”IEEE/ASME Trans. Mechatronics, vol. 9, no. 4,
pp. 706–710, Dec. 2004.
[49] X. Wei, H.-F. Zhang, and L. Guo, “Composite
disturbance-observer-based control and terminal sliding mode
control for uncertain structuralsystems,” Int. J. Syst. Sci., vol.
40, no. 10, pp. 1009–1017, 2009.
[50] L. Guo and W.-H. Chen, “Disturbance attenuation and
rejection forsystems with nonlinearity via DOBC approach,” Int. J.
Robust NonlinearControl, vol. 15, no. 3, pp. 109–125, 2005.
[51] X. Wei and L. Guo, “Composite disturbance-observer-based
control andH∞ control for complex continuous models,” Int. J.
Robust NonlinearControl, vol. 20, no. 1, pp. 106–118, 2009.
[52] J. Back and H. Shim, “Adding robustness to nominal
output-feedbackcontrollers for uncertain nonlinear systems: A
nonlinear version ofdisturbance observer,” Automatica, vol. 44, no.
10, pp. 2528–2537, 2008.
[53] M. Chen and B. Jiang, “Robust bounded control for uncertain
flightdynamics using disturbance observer,” J. Syst. Eng.
Electron., vol. 25,no. 4, pp. 640–647, Aug. 2014.
[54] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable
Adaptive NeuralNetwork Control. Norwell, MA, USA: Kluwer, 2001.
[55] M. Chen and S. S. Ge, “Direct adaptive neural control for a
class ofuncertain nonaffine nonlinear systems based on disturbance
observer,”IEEE Trans. Cybern., vol. 43, no. 4, pp. 1213–1225, Aug.
2013.
[56] K. P. Tee and S. S. Ge, “Control of fully actuated ocean
surface vesselsusing a class of feedforward approximators,” IEEE
Trans. Control Syst.Technol., vol. 14, no. 4, pp. 750–756, Jul.
2006.
[57] J.-X. Xu, Y.-J. Pan, and T.-H. Lee, “Sliding mode control
with closed-loop filtering architecture for a class of nonlinear
systems,” IEEE Trans.Circuits Syst. II, Exp. Briefs, vol. 51, no.
4, pp. 168–173, Apr. 2004.
Mou Chen (M’10) received the B.Sc. degree inmaterial science and
engineering, and the M.Sc.degree and the Ph.D. degree in automatic
controlengineering from the Nanjing University of Aero-nautics and
Astronautics, Nanjing, China, in 1998and 2004, respectively.
He was an Academic Visitor with the Depart-ment of Aeronautical
and Automotive Engineer-ing, Loughborough University, Leicester,
U.K., from2007 to 2008, and a Research Fellow with theDepartment of
Electrical and Computer Engineering,
National University of Singapore, Singapore, from 2008 to 2009.
He iscurrently a Professor with the College of Automation
Engineering, NanjingUniversity of Aeronautics and Astronautics. His
current research interestsinclude nonlinear system control,
intelligent control, and flight control.
Gang Tao (S’84–M’89–SM’96–F’07) received theB.S. degree from the
University of Science andTechnology of China, Hefei, China, in
1982, andthe M.S. and Ph.D. degrees from the Universityof Southern
California, Los Angeles, CA, USA,in 1984 and 1989 respectively, all
in electricalengineering.
He is currently a Professor with the Universityof Virginia,
Charlottesville, VA, USA. His researchhas been mainly in adaptive
control, with particularinterests in adaptive control of systems
with multiple
inputs and multiple outputs, with nonsmooth nonlinearities or
uncertain faults,stability and robustness of adaptive control
systems, and system passivitycharacterizations. He has authored or
co-authored, and co-edited six books andover 350 papers on some
related topics. His current research interests includeadaptive
control of systems with actuator failures and structural
damage,adaptive approximation-based control, and resilient aircraft
and spacecraftcontrol applications.
Prof. Tao served as an Associate Editor of Automatica, the
InternationalJournal of Adaptive Control and Signal Processing, and
the IEEE TRANS-ACTIONS ON AUTOMATIC CONTROL, an Editorial Board
Member of theInternational Journal of Control, Automation and
Systems, and a Guest Editorof the Journal of Systems Engineering
and Electronics.
Bin Jiang (SM’05) received the Ph.D. degreein automatic control
from Northeastern University,Shenyang, China, in 1995.
He was a Post-Doctoral Fellow or Research Fellowin Singapore,
France, and the U.S., respectively.He is currently a Professor and
the Dean of theCollege of Automation Engineering with the
NanjingUniversity of Aeronautics and Astronautics, Nan-jing, China.
His current research interests includefault diagnosis and fault
tolerant control, and theirapplications.
Dr. Jiang is a member of the IFAC Technical Committee on Fault
Detection,Supervision, and Safety of Technical Processes. He
currently serves as anAssociate Editor or Editorial Board Member
for a number of journals,such as the IEEE TRANSACTIONS ON CONTROL
SYSTEM TECHNOLOGY,the International Journal of Systems Science, the
International Journal ofControl, Automation, and Systems, the
International Journal of InnovativeComputing, Information and
Control, the International Journal of AppliedMathematics and
Computer Science, Acta Automatica Sinica, and the Journalof
Astronautics.
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