Robust iterative learning control via continuous sliding-mode technique with validation on an SRV02 rotary plant

Mechatronics 22 (2012) 588–593

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Robust iterative learning control via continuous sliding-mode technique with validation on an SRV02 rotary plant Wen Chen a,⇑, Yang-Quan Chen b, Chih-Ping Yeh a a b

Division of Engineering Technology, Wayne State University, Detroit, MI 48202, USA Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical Engineering, Utah State University, 4120 Old Main Hill, Logan, UT 84322-4120, USA

a r t i c l e

i n f o

Article history: Received 18 March 2011 Accepted 23 December 2011 Available online 13 March 2012 Keywords: Iterative learning control Second-order sliding mode Integral Output-tracking control

a b s t r a c t This paper is to present a new design of robust Iterative Learning Control (ILC) for the purpose of output tracking using continuous sliding mode technique. The main feature of the design is that the controller signal is continuous due to the use of integral and employment of second-order sliding mode technique. The proposed ILC is more robust to noises and disturbances than the saturation approximation of the traditional sliding mode control because the control amount required to maintain the region of convergence is less. The robust ILC is suggested and the convergence of output-tracking error is also proven. The experimental results have clearly exhibited the excellent output-tracking performance by the continuous second-order sliding-mode-based robust iterative learning control.  2012 Elsevier Ltd. All rights reserved.

1. Introduction Iterative Learning Control (ILC) is a control strategy for systems that execute the same trajectory, motion or operation repetitively. ILC attempts to improve the transient responses by adjusting control inputs during next system operation based on the errors observed in past operations. Robustness, against uncertainties, time-varying, and/or stochastic noises and disturbances, includes problems such as disturbance rejection, and stochastic affects. Ref. [1] improves the ILC convergence speed for time-varying linear systems with unknown and bounded disturbances using the predicted errors. Time-periodic and non-structured disturbances are compensated for in [2] using a simple recursive technique. For general ideas about robust ILC, refer to Moon et al. [3] for the linear systems and see [4–8] for the nonlinear systems. At present, the design of robust ILC often adopts standard H1 control techniques [9–11]; that is, given a system, an ILC controller gain is calculated using H1 synthesis. According to Norrlof and Gunnarsson [12], the main drawback of this approach is that the obtained ILC controllers are causal and the strength of ILC is related to the non-causality of its controllers. In recent years, more robust ILC schemes have been addressed. Research papers related to this topic can be found in [13–20], just to name a few. Particularly, a robust ILC synthesizing learning control and sliding mode technique with the help of Lyapunov direct method is ⇑ Corresponding author. E-mail address: [email protected] (W. Chen). 0957-4158/$ - see front matter  2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2011.12.005

proposed in [7]. The learning control is applied to the structured uncertainties while the variable structure scheme is to handle the unknown and unstructured uncertainties to ensure the global asymptotic stability. Another similar work is suggested in [21], where a Learning Variable Structure Control (LVSC) is formalized by combining variable structure control, as the robust part, and learning control, as the intelligent part. The proposed LVSC system achieves both uniform convergence of the tracking-error sequences to zero and that of the learning control sequences to the equivalent control. In the aforementioned two research papers, saturation functions are employed to avoid discontinuity and eliminate the undesired chattering caused by the traditional Sliding Mode Control (SMC). This is because the discontinuous control signal will damage actuators or control devices in practice. The problem is that once the error signals excess the designated boundary layer, a signum function is in charge of the control action. Hence, the saturation function itself can reduce the chattering to an extent that when the tracking-error signal is within the boundary. Therefore, the saturation function cannot avoid the discontinuity completely. Higher-order sliding mode technique is able to eliminate the discontinuity with enhanced accuracy and robustness to disturbances [22–26]. In other words, compared with the saturation approximation of the traditional SMC, second-order sliding mode control is continuous and requires less amount of control efforts to maintain the operation within the region of convergence due to noises and disturbances. This paper is to present and validate a robust ILC using continuous second-order sliding mode technique so that control signals

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are continuous and therefore chattering is reduced; thus, the continuous robust ILC can be applied broadly without damaging actuation devices. This paper is organized as follows: in Section 2, the considered nonlinear system is illustrated and the objective of this paper is also addressed. The sliding surface and the controller design are described in Section 3. The convergence of the output-tracking error is also proven using Lyapunov direct method in the same section. An experiment is included to demonstrate the effectiveness of the proposed robust ILC. At last, concluding remarks are made in Section 5. 2. System formulation

Taking derivatives with respect to time t on both sides of (2), it is obtained:

_ þ c2 €eðtÞ þ    þ cm eðmÞ ¼ r_ ðtÞ ¼ c1 eðtÞ

i ¼ 1; . . . ; m  1;

ð1Þ

x_ m ðtÞ ¼ h> ðtÞnðx; tÞ þ bðx; tÞuðtÞ þ dðtÞ;

where the measurable system state x(t) = [x1, x2, . . . , xm]>, u(t) and y(t) are the control input and system output, respectively, b(x, t) is a known non-zero function, h(t) is a p  1 unknown and time-varying function to be learnt, n(x, t) is a known vector-valued function with dimension of p  1. The variable d(t) represents the unknown disturbance. Assumption 1. The desired output trajectory yd(t) is differentiable with respect to time t up to the mth order on a finite time interval [0, T], and all of the higher-order derivatives are available.

r_ ðtÞ ¼ c1 ½y_ d ðtÞ  x2 ðtÞ þ c2 ½y€d ðtÞ  x3 ðtÞ þ   

h i ðmÞ þ yd ðtÞ  h> ðtÞnðx; tÞ  bðx; tÞuðtÞ  dðtÞ m X

ðiÞ

c i yd 

Assumption 3. The initial condition e(0) = e˙(0) = ë(0) =    = e(m)(0) = 0 at any iteration "t 2 [0, T], such that the sliding surface r(0) = 0, where e(t) is the output tracking error that is defined as e(t) = yd(t)  y(t). The control objective is to design a continuous second-order sliding-mode iterative learning controller, u(t), for the uncertain nonlinear system (1) such that system output can follow a desired one with a prescribed accuracy  as follows:

8t 2 ½0; T; jyd ðtÞ  yðtÞj 6 : 3. Main results The underlying robust ILC is to learn and approach the unknown and time-varying function and leave the remaining unknown function to the robust control. The global asymptotic convergence with respect to iterations is established by Lyapunov direct method.

In Levant [24] and Levant [23], the second-order sliding-mode concept is originated. It is further developed in [25]. In reference to these work, an ILC via continuous second-order sliding-mode concept, at kth iteration, is designed as follows: m X

ðiÞ

ci yd ðtÞ 

2

For the considered system (1), a sliding surface dynamics is defined as follows: m X

ci eði1Þ ;

ð2Þ

i¼1

where cm = 1,ci, s(i = 1, . . . , m  1) are coefficients of a Hurwitz polynomial, and e(t) = yd(t)  y(t) = yd(t)  x1(t).

!

ci xiþ1;k  ^h>k ðtÞnðxk ;tÞ  v k ðtÞ þ a1 jrk j3 sgnðrk Þ þ a3 rk ðtÞ ;

i¼1

ð5Þ where k indicates the number of iterations, xk(t) = [x1,k, x2,k, . . . , xm,k]> , a1, a2, and a3 are positive constants, jj is the absolute value, sgn is the signum function, ^ hðtÞ is the recursive control part that is used to learn the unknown function h(t) and generated by the following update law

  ^hk ðtÞ ¼ ^hk1 ðtÞ  qnðxk ; tÞ 4g jrk j13 sgn ðrk Þ þ crk ðtÞ ; 3

ð6Þ

where q, g and c are positive constants. The variable v(t) is an integral term that is defined below:

v_ k ðtÞ ¼ b1 rk ðtÞ  b2 jrk j

1 3

sgn ðrk Þ;

ð7Þ

where b1 and b2 are positive constants. Remark 1. Controller (5), together with (6) and (7), defines the continuous second-order sliding-mode ILC because jrkjs/3 sgn (rk), where s = 1, 2, are two continuous functions. Moreover, jrkj1/3 sgn (rk) is integrated in (7) such that v(t) is absolutely a continuous function. In summary, controller (5) is a continuous signal; it therefore leads to a chattering-free control action. Therefore, the sliding surface dynamics (4) can be simplified by inserting the ILC law (5):

r_ k ðtÞ ¼ a3 rk ðtÞ þ U>k ðtÞnðxk ; tÞ þ v k ðtÞ  dk ðtÞ 2

_ þ    þ cm eðm1Þ ¼ rðtÞ ¼ c1 eðtÞ þ c2 eðtÞ

m 1 X

 a1 jrk j3 sgn ðrk Þ;

3.1. Derivation of sliding surface

ð4Þ

i¼1

i¼1

where bd is a known positive constant.

ci xiþ1  h> ðtÞnðx; tÞ  bðx; tÞuðtÞ  dðtÞ:

3.2. Design of the robust ILC using integral and continuous sliding mode technique

1

jdðtÞj 6 bd ; 8t 2 ½0; T;

m1 X

The above equation can be further interpreted as the sliding variable dynamics. The condition, r(t) = 0, defines the system motion on the sliding surface. The control signal, u(t), is to be designed as an iterative and continuous control input signal. The task of this work is to design such a continuous and iterative control input to steer the sliding surface to be convergent to a region in finite time interval.

uk ðtÞ ¼ b ðxk ;tÞ

Assumption 2. The unknown disturbance variable d(t) is bounded such that

ð3Þ

Considering the fact that e(t) = yd(t)  x1(t), the above equation can be further expanded:

i¼1

x_ i ðtÞ ¼ xiþ1 ðtÞ;

ci eðiÞ :

i¼1

¼

Consider the following higher-order single-input and singleoutput nonlinear dynamical system described by

m X

ð8Þ

where Uk ðtÞ ¼ ^ hk ðtÞ  hðtÞ. The sliding surface dynamics (8) implies that the integral term v(t) is used to attenuate the effect of the unknown disturbance d(t). Remark 2. According to Brown et al. [26], the second-order SMC is more robust to noises and disturbances than the saturation approximation of the traditional SMC because the control amount required to maintain the region of convergence is less.

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W. Chen et al. / Mechatronics 22 (2012) 588–593

sgn ðrk Þ sgn ðrk Þ ¼ 1;

3.3. Convergence of the output-tracking error The following theorem constitutes the convergence of the sliding surface dynamics and output tracking error when the continuous ILC is applied to the system.

(15) can be simplified as: 4

DV 2k ðtÞ 6 gjrk1 ðtÞj3 

Theorem 1. The robust ILC law using second-order continuous sliding surface proposed in (5), (6), and (7) can guarantee that the system output-tracking errors will converge to a neighborhood of the origin.

 

Z Z

t

Proof. To evaluate the convergence property of the output tracking error e(t), we define the following composite energy function at kth iteration:

ð9Þ 4

where V 1k ðtÞ ¼ v k ðtÞ2v k ðtÞ, V 2k ðtÞ ¼ gjrk ðtÞj3 , V 3k ðtÞ ¼ c rk ðtÞ2rk ðtÞ, and

V 4k ðtÞ ¼

Z

1 2q

where Uk ðtÞ ¼ ^hk ðtÞ  hðtÞ:

U>k ðsÞUk ðsÞds;

0

v k ðtÞv k ðtÞ v k1 ðtÞv k1 ðtÞ 

2

2

¼

Z

t

v k ðsÞv_ k ðsÞds 

v k1 ðtÞv k1 ðtÞ 2

0

DV 3k ðtÞ ¼ c

ð10Þ

Substituting the integral term, v(t), proposed in (7), into (10), it is obtained: DV 1k ¼ b1

Z

t

v k ðsÞrk ðsÞds  b2

0

Z

t

1 3

v k ðsÞjrk j

sgnðrk Þds 

v k1 ðtÞv k1 ðtÞ

0

2

4 3

4 3

¼ gjrk ðtÞj  gjrk1 ðtÞj :

ð12Þ

4 3

Take derivative on jrk ðtÞj with respect to time t, we have:

 4 4 1 d jrk ðtÞj3 ¼ jrk ðtÞj3 sgn ðrk Þr_ k ðtÞ: dt 3

ð13Þ

Based on (13) and (12) has an alternative form:

4 DV 2k ðtÞ ¼ g 3

Z

t

1 3

4 3

jrk j sgn ðrk Þr_ k ðsÞds  gjrk1 j :

t

1

jrk ðsÞj3 sgn ðrk Þv k ðsÞds þ

Z

t

Z

t

4 a3 g 4g jrk j sgnðrk Þrk ds þ jrk j sgnðrk ÞU>k ðsÞnðxk ; sÞds 3 3 0 0 Z Z 1 1 4g t 4g t þ jrk j3 sgnðrk Þv k ðsÞds  jrk j3 sgnðrk Þdk ðsÞds 3 0 3 0 Z 1 2 4 4a1 g t  jrk j3 sgnðrk Þjrk j3 sgnðrk Þds  gjrk1 ðtÞj3 ð15Þ 3 0 1 3

1 3

Considering the fact that

sgn ðrk Þrk ¼ jrk j; j sgn ðrk Þj ¼ 1; and

4a1 g 3

Z

4bd g 3

Z

4g 3 t

1

jrk ðsÞj3 ds

0

t

jrk ðsÞj ds:

ð18Þ

0

rk ðtÞrk ðtÞ 2

Z

c

rk1 ðtÞrk1 ðtÞ 2

t

rk ðsÞr_ k ðsÞds  c

rk1 ðtÞrk1 ðtÞ 2

:

ð19Þ

DV 3k ðtÞ 6 c

rk1 ðtÞrk1 ðtÞ

þc

Z 0

þ bd c

2

 ca3

Z

t

rk ðsÞrk ðsÞds

0

t

rk ðsÞU>k ðtÞnðxk ; tÞds þ c Z

t

jrk ðsÞj ds  ca1

Z

t

Z

t

rk ðsÞv k ðsÞds

0 5

jrk ðsÞj3 ds:

ð20Þ

0

At last, the difference of the fourth energy function between kth and (k  1)th iterations is shown below:

Z t  >  1 1 Uk ðsÞUk ðsÞ ds  2q 0 2q Z t  >   Uk1 ðsÞUk1 ðsÞ ds:

ð21Þ

0

According to the update law (6) and in reference to [27], the following relationship also holds:

 1  > 1 Uk Uk  U>k1 Uk1 ¼ ð^hk  ^hk1 Þ> ð^hk þ ^hk1  2hÞ 2q 2q 1 ¼ ð^hk  hÞ> ð^hk  ^hk1 Þ q > 1 ^ hk  ^hk1 ð^hk  ^hk1 Þ  2q

ð22Þ

Consider updating law (6), the above equation can be expanded as

Combining (8) into (14) yields

DV 2k ðtÞ ¼ 

4g 3

Substituting (8) into the above equation and considering (16), the above equation can be rearranged as:

ð14Þ

0

4

jrk ðsÞj3 ds þ

0

0

ð11Þ

The difference of the second energy function between kth and (k  1) th iterations can be expressed as:

DV 2k ðtÞ

¼c

DV 4k ðtÞ ¼ :

t

1

0

:

Z

It is worth noting that the upper bound, bd, of the unknown disturbance is inserted into the above inequality. The difference of the third energy function between kth and (k  1)th iterations has the following form:

t

The differences of the energy functions between two adjacent iterations will be first derived; then, the convergence of the output tracking error will be evaluated. In the following derivations, the reset condition shown in Assumption 3 will be used. Specifically, when t = 0, the energy function defined in (9) becomes Vk(0) = 0 if vk(0) is selected as 0. Then, DVk(0) = Vk(0)  Vk1(0) = 0. In what follows, the convergence of DVk(t), for 0 < t 6 T, is investigated. The difference of the first energy function between kth and (k  1)th iterations is represented by DV 1k ðtÞ ¼ V 1k ðtÞ  V 1k1 ðtÞ, and has the following form: DV 1k ðtÞ ¼



4a3 g 3

jrk ðsÞj3 sgn ðrk ÞU>k ðsÞnðxk ; sÞds þ

0

0

V k ðtÞ ¼ V 1k ðtÞ þ V 2k ðtÞ þ V 3k ðtÞ þ V 4k ðtÞ;

ð17Þ

ð16Þ

   1 1 > 1 4qg Uk Uk  U>k1 Uk1 ¼  U>k ðtÞnðxk ; tÞ jrk j3 sgnðrk Þ þ qcrk 2q q 3 1 ^ ^ > ^ ^ hk  hk1 ðhk  hk1 Þ  2q 1 4g ¼  jrk j3 sgnðrk ÞU>k nðxk ; tÞ 3 1  crk U>k nðxk ; tÞ  ð^hk  ^hk1 Þ> ð^hk  ^hk1 Þ: 2q ð23Þ Therefore, (21) is related to sliding surface dynamics in the following manner:

W. Chen et al. / Mechatronics 22 (2012) 588–593

Z  1 4g t  DV 4k ðtÞ ¼  jrk j3 sgn ðrk ÞU>k ðsÞnðxk ; sÞ ds 3 0 Z t Z t   1 c rk ðsÞU>k ðsÞnðxk ; sÞ ds  ð^hk 2q 0 0  h^k1 Þ> ðh^k  h^k1 Þds:

ð24Þ

On the basis of the differences between two iterations, we go further to prove the convergence of both the sliding surface dynamics and the output tracking error. Let c ¼ b1 ; b2 ¼ 43g, and bd c ¼ 4a31 g, the sum of the differences of the total energy function can be obtained by adding all of them:

DV k ðtÞ ¼ DV 1k ðtÞ þ DV 2k ðtÞ þ DV 3k ðtÞ þ DV 4k ðtÞ 6

v k1 ðtÞv k1 ðtÞ 2

rk1 ðtÞrk1 ðtÞ

c 

Z

2 t

0

1

 a3 c

jrk ðsÞj3 ds  ca1 Z

Z

t

4

jrk ðsÞj3 ds

rk ðsÞrk ðsÞds þ

0

Z

t

4bd g 3

Z

t

V m ¼ Rm Im þ Lm 1 3

jrk ðsÞj ds

Z 0

t

rk ðsÞrk ðsÞds; 8 jrk ðtÞj >

bd

a3

dIm þ K m xm ; dt

ð27Þ

ð25Þ

where xm is the angular rate of the motor shaft, Rm is the armature resistance, Lm is the armature inductance, and Km is the back-emf constant. The SRV02 motor has a very low inductance and since it is much lower than the motor resistance, i.e. Lm  Rm, then, the circuit equation can be simplified by setting Lm = 0. Eq. (27) becomes

ð26Þ

V m ¼ Rm Im þ K m xm :

0

The above inequality can be further simplified as

DV k ðtÞ 6 a3 c

4.2. Modeling SRV02

5

jrk ðsÞj3 ds

4b g 6 a3 c rk ðsÞrk ðsÞds þ d 3 0 Z t 4 4a3 g  jrk ðsÞj3 ds: 3 0

gears. The SRV02-based testbed is connected to a Quanser Q4 terminal board with Matlab/Simulink Real-Time Workshop (RTW) based software and data acquisition card. Matlab/Simulink blocks can communicate with the data acquisition card through the WinCon blockset. The SRV02 rotary plant can be used stand-alone for several experiments including modeling, position control, speed control, and balance control, etc. Detailed introduction of the SRV02 plant can be found in [28].

The SRV02 rotary plant consists of a DC motor and gearbox. As such, we should construct DC motor armature circuit and gear train models. Apply Kirchoff’s Voltage Law, we can establish the electrical equation describing the amount of current that runs through the motor leads, Im, when a motor armature voltage, Vm, is applied:

0

t

0 t

4a3 g 3

4

 gjrk1 ðtÞj3 

Z

591

which is negative definite when rk(t) – 0, t 2 [0, T]. This concludes that the energy function Vk(t) is convergent. In addition, positive definiteness of Vk(t) can ensure the convergence of the sliding surface dynamics rk(t) to the region of jrk ðtÞj ¼ abd3 . Since the sliding surface dynamics (2) is selected to be Hurwitz; then, the outputtracking error is convergent asymptotically. This completes the proof of the Theorem 1. h

ð28Þ

Armature current can be solved by the above equation:

Im ¼

Vm Km  xm : Rm Rm

ð29Þ

For motor shaft, Newton’s second law of motion can be written as:

  dxm J eq þ Beq xm ¼ sm  sml : dt

ð30Þ

Remark 3. By selecting a large a3, the region of convergence can be made small.

Symbols Jeq and Beq are equivalent moment of inertia and viscous friction, respectively. They have the following expressions:

Remark 4. The region of convergence of the robust ILC based on second-order SMC is greater than that for traditional SMC. The advantage is that the robust ILC is continuous for practical implementation.

J eq ¼ J m þ J l

Remark 5. Future research could be pursued to use estimated system states for the design of the robust ILC.

 2 N1 N3 ; N2 N4  2 N1 N3 ; Beq ¼ Bm þ Bl N2 N4

ð32Þ

where Jm and Jl are moments of inertia of motor shaft and load, and Ni,i = 1, . . . , 4, is the gear number of teeth.

4. Experiments on a testbed – Quanser SRV02 rotary plant 4.1. Introduction to the experimental platform The Quanser’s SRV02 is a general-purpose HIL (hardware-inthe-loop) experiment platform for rapid prototyping of real-time control of mechatronics systems. This lab system can be used as a research platform to test various linear and nonlinear control schemes. The experimental setup includes a computer, a Universal Power Module (power amplifier), and an SRV02 high-gear configuration as shown in Fig. 1. The SRV02 plant consists of a DC motor, a tacho for speed measurement of the load gear, and an optical encoder for digital position measurement. The DC motor is encased in a solid aluminum frame and equipped with a planetary gearbox; that is, the motor has its own internal gearbox that drives external

ð31Þ

Fig. 1. The testbed setup used in ILC experiments.

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W. Chen et al. / Mechatronics 22 (2012) 588–593

Table 1 SRV02 plant specifications. Description

Value

Vm Rm Lm Kt km Jm N2/N1 N4/N3 Bcq Jcq

Motor nominal input voltage Motor armature resistance Motor armature inductance Motor torque constant Back-emf constant Motor shaft moment of inertia Internal gearbox ratio External high-gear configuration ratio Viscous damping coefficient Equivalent high-gear moment of inertia without external load Maximum input current Mass of disc load Radius of disc load Motor efficiency

6V 2.6 X 0.18 mH 7.68E03 N m 7.68E03 V/(rad/s) 3.90E7 kg m2 14 5 0.015 N m/(rad/s) 2.08E03 kg m2 1.0 A 0.04 kg 0.05 m 0.69

15

Control Input Signal

Symbol

Imax Md Rd Vm

Control Input after 40 Iterations

20

10

5

0

−5

0

2

4

6

8

10

time Fig. 3. Control input at the 40th iteration.

Fig. 2. Block diagram of the proposed ILC.

On the right hand side of Eq. (30), sml is the load torque reflected   N1 N3 sl , where sl is the load torque. N2 N4

back to motor shaft with sml ¼

Fig. 4. RMS of position output error for 40 iterations.

Finally, relating motor torque to armature current leads to:

sm ¼ gm K t Im ;

ð33Þ

where Kt is the current-torque constant, and gm is the motor efficiency. Substitute (29) into (30), and consider (31)–(33), it is obtained:

This brings rapid prototyping and experimental capabilities to control algorithms verification.

! dxm gm K t K m Beq g K s þ xm þ m t V m  ml : ¼ dt J eq Rm J eq J eq Rm J eq

4.3. Experiment results

ð34Þ

Following the system’s formulation in (1), the SRV02 model can be described in state-space form

dhm ¼ xm ; dt ! dxm gm K t K m Beq g K s þ xm þ m t V m  ml ; ¼ dt J eq Rm J eq J eq Rm J eq

ð35Þ

where hm is the motor shaft angular position. The armature input voltage can be used as system control input, and motor angular rate is regarded as system output. All parameters are provided in Table 1 based on SRV02 specifications [28]. In the experiment, load disc position and angular rate will be observed.  To this end, we need to find load disc angular rate by xl ¼ NN12 NN34 xm . We are now ready to test our designed ILC schemes on this realtime experimental platform, with almost no programming effort.

This paper provides a new design of ILC for the purpose of output tracking using second-order sliding mode technique. The proposed robust ILC can be designed according to (5)–(7). To make the design clearer and easier, a block diagram is also employed, as shown in Fig. 2, to depict the configuration of the proposed ILC. To help describe the continuity and reduced chattering, the control input signal generated under Matlab environment is shown in Fig. 3; it is clear that the control signal is continuous and chattering free. In this paper, a class of simple zero-phase filters is considered to suppress high frequency noise. We operated the SRV02 for 40 iterations. The experimental results are demonstrated in the following figures. Fig. 4 shows that the root mean squares (RMSs) of the position error, after 40 iterations, gradually tends to zero. Fig. 5 displays that the position output converges to the desired trajectory at the 40th iteration. At last, Fig. 6 demonstrates accurate tracking of the speed output to the desired trajectory at the 40th iteration. It is concluded that the effectiveness of the proposed synthesized ILC has been verified and validated on SRV02 plant.

W. Chen et al. / Mechatronics 22 (2012) 588–593

593

We are also grateful to all of anonymous reviewers who have provided us with valuable and constructive comments and suggestions. References

Fig. 5. Position output and the desired trajectory at the 40th iteration.

Fig. 6. Speed output and the desired trajectory at the 40th iteration.

5. Conclusions This paper has proposed a robust Iterative Learning Control (ILC) strategy in a class of nonlinear systems for the purpose of output tracking. The continuous second-order sliding-mode control is synthesized into the design of the ILC, leading to the ILC more robust to noises and disturbances than the saturation approximation. In addition, the continuousness of the robust ILC is enhanced by the integral used to attenuate the effect of the disturbances. Therefore, the control signal is completely continuous. The experimental results have clearly exhibited the excellent output-tracking performance by the proposed robust ILC. Acknowledgments Authors appreciate Dr. Jun Pan’s help in conducting an experimental test. He is a professor at Automation Laboratory in Northwestern Institute of Mechanical and Electrical Engineering, Shaanxi, China.

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