-
International Journal of InnovativeComputing, Information and
Control ICIC International c2011 ISSN 1349-4198Volume 7, Number 4,
April 2011 pp. 1971{1982
IMPROVED INTEGRAL SLIDING MODE CONTROL METHODSFOR SPEED CONTROL
OF PMSM SYSTEM
Cunjian Xia, Xiaocui Wang, Shihua Li and Xisong Chen
School of AutomationKey Laboratory of Measurement and Control of
CSE, Ministry of Education
Southeast UniversityNo. 2, Sipailou, Nanjing 210096, P. R.
China
[email protected]
Received December 2009; revised April 2010
Abstract. To improve the disturbance rejection property of
permanent magnet syn-chronous motor (PMSM) speed control system,
the integral sliding mode control (ISMC)method is introduced in the
control design of speed loop. However, the simulation
andimplementation results show that it is dicult to balance the
chattering and the anti-disturbance capacity. To this end, three
kinds of improved ISMC control methods aredeveloped. First, ISMC
using linear varying gain is developed. Using this method,
theswitching gain of ISMC controller can be smaller while still
ensuring that the speed statereaches its steady state and the
steady state uctuations can thus be reduced. Moreover,the
anti-disturbance capacity of the PMSM system can also be assured.
Second, an in-tegral sliding mode control based on extended state
observer (ESO) is developed. ESOcan estimate both of the states and
the disturbances simultaneously. By using ESO, anestimate of the
lumped disturbances is obtained, which is employed for the
feedforwardcompensation design of the composite ISMC control law.
In this case, the controller maytake a smaller value for the
switching gain without sacricing disturbance rejection
per-formance, which helps to reduce large chattering caused by high
control gains. Third, anadaptive composite control method combining
linear varying gain and ESO is developedto take advantages of both
improved methods. These improved methods show advantagesin reducing
the chattering while ensuring the dynamic and disturbance rejection
perfor-mance. Both of simulation and experiment results are
provided.Keywords: PMSM, Integral sliding mode control, Extended
state observer, Linearvarying gain, Composite control,
Speed-regulation
1. Introduction. Permanent magnet synchronous motor has gained
widespread accep-tance in numerical control machine tools, robots,
aviation and so on, due to its excellentfeatures such as high power
density, torque-to-current ratio and eciency [1]. Linear con-trol
schemes such as proportional-integral (PI) control scheme have been
widely used inPMSM servo system because of simple implementation
[2]. However, it is very dicult toachieve a satisfactory
performance in the entire operating rage by only using linear
controlmethods. The reason is that the PMSM servo system is a
nonlinear system with unavoid-able and unmeasured disturbances as
well as parameters variations [3, 4, 5]. Thus, variousmethods of
nonlinear control methods have been developed for PMSM system, such
asadaptive control [6, 7], robust control [8], sliding mode control
[9], input-output lineariza-tion control [4], backstepping control
[10], neural network control [5], fuzzy control [11]and nite-time
control [12], etc.Sliding mode control (SMC) is a very useful
nonlinear control method [13, 14, 15] and
it has been introduced in AC servo drive systems [16, 17] due to
its good robustnessfor external disturbances and variations of
system parameters, fast response and easy
1971
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1972 C. XIA, X. WANG, S. LI AND X. CHEN
implementation. Under the framework of vector control and
cascade structure, in thecontrol design for speed loop, usually a
rst-order model is used to approximately describethe relationship
between the reference quadrature axis current and the speed output.
Toconstruct a common sliding surface, the sliding-mode speed
controller needs both thespeed and the acceleration signals [16,
18]. However, due to noise and uncertainties ofparameters,
acceleration signals are dicult to be measured or to be estimated
accurately[19]. These degrades the closed loop performance of
system. To this end, integral slidingmode control (ISMC) method is
proposed [20] and applied in asynchronous motor [21].The
acceleration information is not required anymore.In this paper, a
standard ISMC method is developed for the speed control of PMSM
system. However, the simulation and implementation results of
the standard ISMCmethod on PMSM system show that it is dicult to
balance the chattering and theanti-disturbance capacity. The
anti-disturbance performance of the system mainly de-pends on the
switching gain. While it is tuned to a small value, the motor speed
can notrecover to its reference value when the load disturbance is
added. When the switchinggain is increased, the anti-disturbance
capacity of closed loop system becomes better.However, in this
case, the closed loop system produces a greater
chattering.According to the SMC theory, if the switching control
gain is selected to be bigger
than the upper bound of disturbances, the disturbances can be
well rejected. However,for practical applications, the bound of
disturbances is dicult to obtain which oftenresults in an
inadequate selection of switching gain. Usually, an overlarge
switching gaincauses a large chattering. To this end, three
improved ISMC methods are proposed toimprove the performance of
servo system. The main idea is to nd some ways to reducethe
conservativeness of selecting switching control gain.First, an
improved ISMC method based on linear varying gain is used. The
switching
gain linearly varies at dierent stages. It takes a larger value
when the states are far awayfrom the sliding surface (e.g., at the
starting stage), so the dynamic response time of statecan be
ensured. When the state approaches the sliding surface including
its steady state,the switching gain takes a smaller value which
helps to reduce the steady state chatteringof system. Meanwhile,
the anti-disturbance capacity can still be assured.Second, an
improved method employs a disturbance estimation technique to have
an
adequate estimate for the lumped disturbances of PMSM system.
Then, after distur-bance compensation based on ESO, the switching
gain only need to be taken biggerthan the bound of the disturbance
compensation error, which is usually much smaller.Thus, an ISMC
method based on extended state observer (ESO), named
ISMC+ESOmethod, is developed. An ESO in the feedback path provides
estimates of both thespeed and the lumped disturbances. The
estimate of the lumped disturbances is em-ployed for feedforward
compensation design of the control law. In [9], a total slidingmode
controller is proposed for the position control problem of PMSM
system, where arecurrent-fuzzy-neural-network is adopted as a bound
observer to facilitate adaptive con-trol gain adjustment. Compared
with that technique, the ESO technique is very simplefor
implementation.Third, to further enhance the performance, an
adaptive ISMC method based on ESO,
named adaptive ISMC+ESO method, is developed. The eectiveness of
the proposedthree improved schemes is veried and compared by
simulation and experiment results.
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INTEGRAL SLIDING MODE CONTROL METHODS FOR PMSM SYSTEM 1973
Figure 1. The diagram of PMSM system
2. The Mathematical Model of PMSM. The model of surface mounted
PMSM isexpressed in d q coordinates as follows [27]:0@ _id_iq
_!
1A =0B@
RsLd
np! 0
np! RsLq np fLq
03np f2J
BJ
1CA 0@ idiq
!
1A+0@ udLduq
Lq
TLJ
1A (1)where Rs the stator resistance, ud, uq the d- and q- axes
stator voltages, id, iq the d- andq- axes stator currents, Ld, Lq
the d- and q- axes stator inductances Ld = Lq = L, npthe number of
pole pairs, ! the rotor angular velocity, f the ux linkage, TL the
loadtorque, B the viscous friction coecient, J the rotor
inertia.The general structure of the PMSM servo system is shown in
Figure 1. The overall
system consists of a PMSM with load, space vector pulse width
modulation (SVPWM),voltage-source inverter (VSI), eld-orientation
mechanism and three controllers. Thecontrollers employ a structure
of cascade control loop including a speed loop and twocurrent
loops. Here two PI controllers, which are used to stabilize the d q
axes currenterrors of the vector controlled drive, are adopted in
the two current loops respectively.As it can be seen from Figure 1,
the rotor angular velocity ! can be obtained from theposition and
speed sensor. The currents id and iq can be calculated from ia and
ib (whichcan be obtained from measurements) by Clarke and Park
transforms.
3. Control Strategy of PMSM Speed-Regulation System.
3.1. The standard integral sliding mode controller. The torque
equation of PMSMsystem can be written as
_! =1:5np f iq B! TL
J=
3np f2J
iq + a(t) (2)
where a(t) =1:5np f
iq iq
B! TL =J is the lumped disturbances of system.Dene the speed
error e = ! ! where ! is reference speed. Taking the derivative
of e and substituting (2) into it, yields:
_e = _! 3np f2J
iq a(t): (3)The sliding surface is designed as
s = e+3np f2J
m
Z t0
ed: (4)
And the speed controller can be designed as
iq = me+ f sgn(s) + (2J=(3np f )) _! (5)where m, f > 0, and
sgn() is the standard signum function.
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1974 C. XIA, X. WANG, S. LI AND X. CHEN
Figure 2. Speed response ofsystem under ISMC scheme
Figure 3. Speed response ofsystem with disturbance underISMC
scheme
Assumption 3.1. The lumped disturbances of system a(t) satises 0
ja(t)j < l.Theorem 3.1. Assume that system (1) satises
Assumption 3.1. Under the control law(5), the speed error of system
(1) converges to zero if f > 2Jl
3np f.
Proof: Choosing Lyapunov function V = 12s2, and taking the
derivative of it along
system (3), yields:
_V = s _s = s
_e+
3np f2J
me
: (6)
Substituting (3) into (6) yields:
_V = s _s = 3np f2J
s
iq +
2J
3np fa(t)me 2J
3np f_!: (7)
Substituting (5) into (7) yields:
_V = s _s = 3np f2J
f jsj+ 2J
3np fa(t)s
: (8)
From Assumption 3.1, if f > 2Jl3np f
, one obtains
_V 3np f2J
jsjf 2J
3np fl
< 0 (s 6= 0) (9)
So the control law (5) makes (4) converge to zero in nite time.
When the speed
error reaches its sliding surface, i.e., s = e +3np f2J
mR t0ed = 0, which implies that
_e = (3np f=(2J))me. Thus, the speed error will converge to
zero. The theorem isproved.
3.2. Simulation results and experiment results. Here, the
parameters of the PMSMare: the resistance of stator Rs = 0:8, the
inductances of d and q axes Ld = Lq =2:9 103H, the ux of rotor f =
0:2wb, the rotor inertia J = 6:5 104kg m2, theviscous friction
coecient B = 1:28 104n m s, the number of poles np = 4.3.2.1.
Simulation results. The speed regulation system of the PMSM is
simulated byMATLAB. The parameters of the integral sliding mode
controller are m = 0:8; f = 10.The reference speed is 1000rpm, and
the load torque TL = 4Nm is added at t = 1s.The speed response of
the PMSM speed-regulation system is shown in Figure 2. Figure2
shows that when the ISMC scheme is used, a speed response with no
overshoot isobtained. Figure 3 shows the anti-disturbance property
of system when a load disturbance
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INTEGRAL SLIDING MODE CONTROL METHODS FOR PMSM SYSTEM 1975
Figure 4. Speed response of system under ISMC method
Figure 5. Speed response ofsystem with disturbance underISMC
method with a smallergain
Figure 6. Speed response ofsystem with disturbance underISMC
method with a largergain
is added. It shows that the speed response of the system used
ISMC scheme has an obviouschattering and the anti-disturbance
capacity is not satisfactory.
3.2.2. Experiment results. As for the experimental test setup,
the whole speed controlalgorithms is implemented by the program of
the DSP TMS320F2808 with a clock fre-quency of 100MHZ. The PMSM is
driven by a three-phase PWM inverter with an IPMwith a switching
frequency of 10kHz. The phase currents are measured by the
Hall-eectdevices and are converted through two 12-bit A/D
converters. An incremental positionencoder of 2500 lines is used to
measure the rotor speed and absolute rotor position. Theparameters
of the ISMC are m = 1200, f = 18. The reference speed is
1000rpm.The speed response of the closed loop system under ISMC
scheme is shown in Figure 4.
The speed responses under dierent switching gains are shown in
Figures 5 and 6 whena step load disturbance of TL = 2Nm is added.
The anti-disturbance performance of thesystem mainly depends on the
switching gain f and m, here m is selected to be m = 1200.When f is
tuned to a small value, e.g., f = 18, the motor speed can not
recover to itsreference value when the load disturbance is added
and the performance of disturbancerejection is worse, as shown in
Figure 5. Usually, it is expected that when the switchinggain is
increased, the anti-disturbance capacity of closed loop system may
become better.As shown in Figure 6, when f is tuned to a larger
value, e.g., f = 80, the speed responsecan recover to its reference
value. However, in this case, the system produces a
greaterchattering. Through parameters tuning of control gain f for
many times, we nd that it isvery dicult to have a good balance
between disturbance rejection and chattering underthe standard ISMC
method. In fact, this phenomena of the ISMC method has alreadybeen
mentioned in [28] for synchronous reluctance motor drive
system.
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1976 C. XIA, X. WANG, S. LI AND X. CHEN
4. The Integral Sliding Mode Controller with Linear Varying
Gain.
4.1. Design of controller. The torque equation of PMSM system is
written as (2). Andthe sliding surface is designed as (4). The
speed controller can be designed as:
iq = me+ f^ sgn(s) +2J
3np f_! (10)
where the state feedback gain m > 0 and f^ is the linear
varying switching gain. Here,
f^ = c1 +jsjc2
and c1, c2 can be changed.
Theorem 4.1. Assume that system (1) satises Assumption 3.1.
Under the control law(10), the speed error of system (1) converges
to zero if c1 >
2Jl3np f
.
Proof: Choosing Lyapunov function V = 12s2, and taking the
derivative of it along
system (3), yields (6). Substituting (3) into (6) yields:
_V = s _s = 3np f2J
s
iq +
2J
3np fa(t)me 2J
3np f_!: (11)
Substituting (10) into (11) yields:
_V = s _s = 3np f2J
c1jsj+ jsj
2
c2+
2J
3np fa(t)s
: (12)
From Assumption 3.1, one obtains
_V 3np f2J
c1 2J
3np fl
jsj+ jsj
2
c2
: (13)
If c1 >2J
3np fl, _V = s _s 0. So the control law (10) can make the
sliding surface (4)
converge to zero in nite time. When the speed error reaches its
sliding surface, i.e.,
_e = 3np f2J
me, it converges to zero. The theorem is proved.
4.2. Simulation and experiment results.
4.2.1. Simulation results. The parameters of the integral
sliding mode controller are m =0:8, f = 10 and the parameters of
the integral sliding mode controller with linear varyinggain are m
= 0:65, c1 = 5, c2 = 40. The reference speed is 1000rpm, and the
loadtorque TL = 4Nm is added at t = 1s. The speed responses of
system are shown inFigure 7. Figure 7 shows that the rising time of
the ISMC method is almost the same asthat of ISMC method with
linear varying gain. Figure 8 is the anti-disturbance
capacitycomparison of the system used two dierent controllers. And
the result shows that thespeed response of ISMC with linear varying
gain has less chattering when the system isat its steady state and
has better anti-disturbance capacity when the same disturbanceload
is added.
4.2.2. Experiment results. The parameters of the ISMC are m =
1200, f = 18 and theparameters of the ISMC with linear varying gain
are m = 850, c1 = 18, c2 = 100. Thereference speed is 1000rpm. The
step responses of the ISMC and ISMC with linear varyinggain are
shown in Figure 9. It shows that the rising time of the ISMC with
linear varyinggain is a little longer than the rising time of the
ISMC and its overshoot is a little bigger.From Figure 10, when load
disturbance is added, the speed can return to the referencespeed in
short time and the maximum speed uctuations of the closed loop
system underISMC with linear varying gain control method is only
about 30rpm, much smaller thanthat of standard ISMC method. As the
switching gain is smaller than that of standard
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INTEGRAL SLIDING MODE CONTROL METHODS FOR PMSM SYSTEM 1977
Figure 7. Speed responses ofsystem under ISMC and ISMCwith
linear varying gain
Figure 8. Speed responses ofsystem with disturbance underISMC
and ISMC with linearvarying gain
Figure 9. Speed responses ofsystem under ISMC and ISMCwith
linear varying gain
Figure 10. Speed responsesof system with disturbance un-der the
ISMC with linear vary-ing gain
ISMC method, so the steady uctuations of the servo system is
also reduced. And theanti-disturbance capacity and the performance
of the system are improved.
5. ESO-based Composite Control Strategy.
5.1. Design of controller. Here, a kind of disturbance observer
techniques is employedto have an adequate estimate for the lumped
disturbances of PMSM system. This tech-nique is extended state
observer [22, 23, 24]. A control frame based on ESO, called
activedisturbance rejection control (ADRC) is also developed. This
method has also been ap-plied in many areas, such as robotic
systems [25], machining processes [26], PMSM systems[2, 6], and so
on. The ESO regards the internal and external disturbances of the
systemas the lumped disturbances, and the lumped disturbances can
be considered as a newextended state. ESO can observe and estimate
the state and the lumped disturbances ofsystem respectively. The
estimate of the lumped disturbances is employed to compensatethe
disturbances through a feedforward design in the control law.A
linear ESO can be constructed as follows [29]:
_z1 = z2 2p(z1 !) + 3np f2J
iq; _z2 = p2(z1 !) (14)where z1 is an estimate of speed !, z2 is
an estimate of the lumped disturbances, and p(p > 0) is the
desired double-pole of ESO. The sliding surface is designed as (4).
And the
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1978 C. XIA, X. WANG, S. LI AND X. CHEN
Figure 11. Speed responsesof system under ISMC andISMC+ESO
methods
Figure 12. Speed responsesof system with disturbance un-der ISMC
and ISMC+ESOmethods
control law can be designed as:
iq = me+ f sgn(s)2J
3np fz2 +
2J
3np f_! (15)
where the state feedback gain m > 0 and f is the switching
gain, 2J3np f
z2 is the feedfor-
ward component of the control law.Here, a(t) represents
disturbances of the system, z2 is the estimate of a(t) from
ESO.
Assumption 5.1. Assume that a(t) z2 is bounded, which satises 0
ja(t) z2j < l0.Theorem 5.1. Assume that system (1) satises
Assumption 5.1. For PMSM speed-regulation system (1), if the
switching gain f > 2Jl
03np f
, the speed error of system (2)
converges to zero under the control law (15).
Proof: Let Lyapunov function V = 12s2, and the derivative of it
with respect to time
is (6). Substituting (3) into (6), yields (7). And substituting
(15) into (7), yields:
_V = s _s = 3np f2J
f jsj+ 2J
3np f[a(t) z2]s
: (16)
From Assumption 5.1, one obtains
_V 3np f2J
f 2J
3np fl0jsj: (17)
If f > 2Jl0
3np f, _V = s _s < 0 (s 6= 0). The control law (15) makes the
state of the speed error
of system (2) converge to s(x) = 0 in nite time. When the speed
error reaches its slidingmode, i.e., s = 0, it will converge to
zero. The theorem is proved.
5.2. Simulation and experiment results.
5.2.1. Simulation results. The parameters of the ISMC are m =
0:8, f = 10 and theparameters of the ISMC+ESO are m = 2, f = 0:1, p
= 3000. The reference speedis 1000rpm, and the load torque TL = 4Nm
is added at t = 1s. Speed responses areshown in Figure 11. Figure
11 shows that rise time of the ISMC+ESO is a little
longer.Simulation results of anti-load disturbance of the two
controllers are shown in Figure12. Figure 12 shows that speed
response of the ISMC+ESO has a less chattering. Andwhen the same
disturbance load is added, the maximum uctuation of the ISMC+ESOis
smaller , so the PMSM speed-regulation system using the ISMC+ESO
controller has abetter anti-disturbance capacity.
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INTEGRAL SLIDING MODE CONTROL METHODS FOR PMSM SYSTEM 1979
Figure 13. Speed responsesof system under ISMC andISMC+ESO
methods
Figure 14. Speed responseof system with disturbance un-der
ISMC+ESO method
Figure 15. The diagram of adaptive ISMC+ESO scheme for PMSM
system
5.2.2. Experiment results. The parameters of the ISMC controller
are m = 1200, f = 18and the parameters of the ISMC+ESO controller
are m = 1050, f = 20, p = 500. Andthe reference speed is 1000rpm.
Comparisons of speed responses under ISMC law (5) andISMC+ESO law
(15) respectively are shown in Figure 13. The rising time of
ISMC+ESOis a little longer and its overshoot is a little bigger.
The speed responses of ISMC+ESOwith disturbance load is shown in
Figure 14. It can be seen that the maximum speed
uctuation of the closed loop system under ISMC+ESO control
method is only about25rpm, much smaller than that of standard ISMC
method. Since the estimate of lumpeddisturbances is employed for
feedforward compensation design of the control law, theperformance
degradation caused by disturbances is suppressed, and the closed
loop systemunder ISMC+ESO control method has a less chattering, and
a better anti-disturbancecapacity.
6. Adaptive ESO-based Composite Control Strategy.
6.1. Design of controller. In this section, an adaptive
ESO-based composite controlleris proposed. Here, the switching gain
is changed with the variations of the speed error.As same as
Section 5.1, ESO is used to estimate the lumped disturbances of
PMSMsystem. Then, the estimate value of lumped disturbances is
employed for feedforwardcompensation design of the control law.
Here, the linear ESO is designed as (14), thesliding surface is
designed as (4). And the speed controller can be designed as
iq = me+c3 +
jsjc4
sgn(s) 2J
3np fz2 +
2J
3np f_! (18)
where c3, c4 > 0 are constants, m is the feedback gain. The
principle diagram of theadaptive ESO-based composite control
(Adaptive ISMC+ESO) is shown in Figure 15.Note that the generalized
plant in Figure 15 represents the two current loops whichinclude
PMSM and other components the same as that of Figure 1.
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1980 C. XIA, X. WANG, S. LI AND X. CHEN
Figure 16. Speed responseof system under ISMC+ESOand adaptive
ISMC+ESOmethods
Figure 17. Speed responseof system with disturbance un-der
ISMC+ESO and adaptiveISMC+ESO methods
Theorem 6.1. Assume that system (1) satises Assumption 5.1. For
PMSM speed-regulation system (1), if switching gain c3 >
2Jl03np f
, the speed error of system (1) converges
to zero under the control law (18).
Proof: The proof process is a combination of that of Theorems
4.1 and 5.1, which isomitted here.
6.2. Simulation results and experimental results.
6.2.1. Simulation results. Here, the parameters of ISMC+ESO
controller are m = 2,f = 0:1, p = 3000, and the parameters of
adaptive ISMC+ESO controller are m = 2,c3 = 0:3, c4 = 80, p = 2000.
The reference speed is 1000rpm, and the load torqueTL = 4Nm is
added at t = 0:2s. Speed responses are shown in Figure 16. Figure
16shows that, compared with that of ISMC+ESO method, the rising
time of the systemunder adaptive ISMC+ESO method is shorter
although the overshoot is a little bigger.When the same load
disturbance is added, as shown in Figure 17, the maximum
uctuationof the system using adaptive ISMC+ESO method is decreased.
It shows that the anti-disturbance capacity is further improved by
using the adaptive ISMC+ESO method.
6.2.2. Experimental results. The parameters of ISMC+ESO
controller are m = 1050,f = 20, p = 500, and the parameters of
adaptive ISMC+ESO controller are m = 1020,c1 = 10, c2 = 100, p =
480. The reference speed is 1000rpm. From Figure 18, comparedwith
that of ISMC+ESO method, we can see that, the system under adaptive
ISMC+ESOmethod has shorter rising time and settling time although
its overshoot is a little bigger.Figure 19 shows that when the same
load disturbance is added, the maximum uctuationof speed under the
ISMC+ESO and the adaptive ISMC+ESO methods are 25rpm and17rpm
respectively. So the system using adaptive ISMC+ESO method has a
better anti-disturbance capacity.
7. Conclusion. In this paper, ISMC technique has been studied
for the speed control ofPMSM system. Since the standard ISMC method
is dicult to balance the chattering andthe anti-disturbance
capacity, three kinds of improved ISMC control methods have
beendeveloped from dierent considerations. An improved method based
on linear varyinggain and an improved method based on extended
state observer have been developed,respectively. To further improve
system performance and take advantages of the bothimproved methods,
an adaptive ISMC method based on combination of linear varying
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INTEGRAL SLIDING MODE CONTROL METHODS FOR PMSM SYSTEM 1981
Figure 18. Speed responseof system under ISMC+ESOand adaptive
ISMC+ESOmethods
Figure 19. Speed responseof system with disturbance un-der
ISMC+ESO and adaptiveISMC+ESO methods
gain and extended state observer has been developed. Simulation
and experimental resultshave shown that the three improved methods
can reduce the steady state chattering whileensuring the system
performance.
Acknowledgement. This work was supported by Natural Science
Foundation of JiangsuProvince (BK2008295) and National 863 Project
(2009AA04Z140,2009AA01Z314).
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