Iterative Learning Control With Switching Gain ... - Ryerson University

P. R. Ouyang Assistant Professor Mem. ASME Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto, ON, M5B 2K3, Canada e-mail: [email protected]

B. A. Petz Graduate Student Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto, ON, M5B 2K3, Canada e-mail: [email protected]

F. F. Xi Professor Mem. ASME Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto, ON, M5B 2K3, Canada e-mail: [email protected]

1

Iterative Learning Control With Switching Gain Feedback for Nonlinear Systems Iterative learning control (ILC) is a simple and effective technique of tracking control aiming at improving system tracking performance from trial to trial in a repetitive mode. In this paper, we propose a new ILC called switching gain PD-PD (SPD-PD)-type ILC for trajectory tracking control of time-varying nonlinear systems with uncertainty and disturbance. In the developed control scheme, a PD feedback control with switching gains in the iteration domain and a PD-type ILC based on the previous iteration combine together into one updating law. The proposed SPD-PD ILC takes the advantages of feedback control and classical ILC and can also be viewed as online-offline ILC. It is theoretically proven that the boundednesses of the state error and the final tracking error are guaranteed in the presence of uncertainty, disturbance, and initialization error of the nonlinear systems. The convergence rate is adjustable by the adoption of the switching gains in the iteration domain. Simulation experiments are conducted for trajectory tracking control of a nonlinear system and a robotic system. The results show that fast convergence and small tracking error bounds can be observed by using the SPD-PD-type ILC. 关DOI: 10.1115/1.4002384兴 Keywords: iterative learning control, PD feedback control, switching gain, nonlinear system, trajectory tracking, convergence

Introduction

Iterative learning control 共ILC兲 关1兴 based on the system’s repetitive operations has drawn increasing attention because of its simple control structure and good tracking performances. The basic principle behind ILC is to use information collected from previous executions of the same task repetitively to form the control action for the current operation in order to improve tracking performances from iteration to iteration. Examples of systems that operate in a repetitive manner include robot arm manipulators, assembly line tasks, chemical batch processes, reliability testing rigs, and so on. In each of these tasks, the system is required to perform the same action over and over again with high precision. In general, ILC improves the tracking performance by a selftuning process without using system model. The key purpose of using ILC is to achieve high performance after a few iterations. One advantage of ILC is that there is no requirement for the dynamic model of the controlled system. The classic type of ILC algorithms is a feed-forward control system that is theoretically capable of reducing the tracking error to zero as the number of iterations increases toward infinity 关1–13兴. According to the learning action type, ILC can be classified as P-type 关2,3兴, D-type 关1,4,5兴, PD-type 关6,7兴, and PID-type 关8,9兴. A PD-type averaged iterative learning control is proposed for linear systems in Ref. 关10兴. Some detailed surveys for ILC can be found in Refs. 关11–13兴. Another direction in the development of ILC is using the current iteration tracking errors to form the control input that can be viewed as online ILC 关14–21兴. P-type online ILC systems based on the use of current tracking errors were proposed 关14–16兴. A similar control algorithm 关17兴 was used for discrete time systems. D-type online ILC 关4,14兴, PD-type online ILC 关10,15,18,19兴, and PID-type online ILC 关20,21兴 were developed and applied into

different systems control. As demonstrated in these papers, the iterative learning control with a current tracking error signal can achieve fast convergence rates by selecting high feedback control gains. It is well known that PD/PID feedback control is widely used in industrial applications such as robot systems and process control systems to achieve good performance. But PD/PID feedback control cannot achieve exactly the desired tracking performance because a nonzero error signal is required to activate the feedback control. Therefore, PD/PID control alone is not adequate for achieving a perfect trajectory tracking performance, especially when the system has nonlinearities and uncertainties. The combination of feedback control and iterative learning control is a promising technique to achieve good tracking performance and to speed up the convergence process. P-type feedback plus P-type ILC control algorithms 共P-P type兲 for continuous and discrete time-varying systems were proposed 关22–24兴. A P-P ILC algorithm was developed for kinematic path tracking of mobile robots in Ref. 关25兴. High-order D-D ILCs 关26,27兴 and PD-PD ILCs 关28,29兴 were developed. A P-D ILC was proposed in Ref. 关30兴 for discrete linear time-invariant systems. In this paper, a switching gain PD-type feedback control plus a PD-type ILC 共SPD-PD兲 is proposed for trajectory tracking control of time-varying nonlinear systems to expedite the convergence rate, to deal with the uncertainty and disturbance, and to improve the tracking performance from operation to operation. It can be seen that the SPD-PD ILC is an online-offline learning control. This paper is organized as follows. First, a general problem that will be controlled by the developed SPD-PD ILC is introduced; then, the SPD-PD ILC is proposed, and its convergence analysis is conducted; after that, two nonlinear systems are used to examine the effectiveness of the proposed SPD-PD ILC. Finally, some conclusions are drawn based on the simulation results.

2 Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 12, 2009; final manuscript received July 1, 2010; published online October 13, 2010. Assoc. Editor: Henryk Flashner.

Problem Statement

In this paper, the considered problem is a nonlinear timevarying system with nonrepetitive uncertainty and disturbance as follows:

Journal of Computational and Nonlinear Dynamics Copyright © 2011 by ASME

JANUARY 2011, Vol. 6 / 011020-1

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再

x˙k共t兲 = f共xk共t兲,t兲 + B共t兲uk共t兲 + ␩k共t兲 y k共t兲 = C共t兲xk共t兲 + ␰k共t兲

冎

共1兲

where k denotes the iteration index. x 苸 Rn, u 苸 Rr, and y 苸 Rm are the state, control input, and control output of the system, respectively. f共x共t兲 , t兲 苸 Rn is the system function, B共t兲 苸 Rn⫻r is the input matrix, C共t兲 苸 Rm⫻n is the output matrix, ␩k共t兲 苸 Rn is the uncertainty of the system, and ␰k共t兲 苸 Rm is the disturbance. In this paper, to evaluate tracking convergence, the following notational conventions for norms are adopted:

We assume that in each iteration, the repeatability of the initial state setting is satisfied within the following admissible deviation level: 储xk共0兲 − x0共0兲储 ⱕ ␧x

共4兲

for k = 1,2, . . .

where ␧x is a small positive constant that represents the acceptable accuracy of the designed state vector and x0共0兲 represents the desired initial state value. For briefness of the convergence analysis, some notations are introduced first and used in the following sections: Bd1 = max 储Kd1共0兲C共t兲储,

储f储 = max 兩f i兩

Bd2 = max 储Kd2C共t兲储

t苸关0,T兴

t苸关0,T兴

1ⱕiⱕn

冉兺 冊

B pd1 = max 储K p1共0兲C + Kd1共0兲C˙储,

n

储M共t兲储 = max

1ⱕiⱕm

储h共t兲储␭ = sup e t苸关0,T兴

−␭t

兩mi,j兩

j=1

储h共t兲储,

␭⬎0

␰

i

i

␰

To control the nonlinear system stated in Eq. 共1兲, we propose the following SPD-PD ILC law: uk+1共t兲 = uk共t兲 + K p1共k + 1兲ek+1共t兲 + Kd1共k + 1兲e˙k+1共t兲 + K p2共t兲ek共t兲 + Kd2共t兲e˙k共t兲

共2兲

with the feedback control gains given by

再

K p1共k兲 = s共k兲K p1共0兲 Kd1共k兲 = s共k兲Kd1共0兲

冎

with s共k + 1兲 ⬎ s共k兲

共3兲

where ek+1共t兲 = y d共t兲 − y k+1共t兲, ek共t兲 = y d共t兲 − y k共t兲, e˙k+1共t兲 = y˙ d共t兲 − y˙ k+1共t兲, and e˙k共t兲 = y˙ d共t兲 − y˙ k共t兲 are position errors and velocity errors of the output vector for the 共k + 1兲th iteration and the kth iteration, respectively. K pi 苸 Rm⫻r and Kdi 苸 Rm⫻r are the proportional and derivative gain matrices, respectively. Equation 共3兲 represents the switching gains of the feedback control, and s共k兲 ⬎ 1 is a monotonically increasing function of the iteration index. From Eq. 共2兲, we can see that the proposed SPD-PD ILC law consists of two control loops. The first loop includes a PD feedback controller with switching gains in the iteration domain, and the second loop is a standard PD-type ILC. Therefore, SPD-PD ILC is effectively a hybrid control that aims to take the advantages offered by both feedback control and ILC. The key purpose of introducing switching gains in the feedback loop is to expedite the convergence of the iterative operations and avoid vibration of systems. Also, one can see that the proposed control algorithm is an extension of the learning control algorithm developed in Ref. 关19兴. 011020-2 / Vol. 6, JANUARY 2011

BKp2 = max 储K p2储

BKd1 = max 储Kd1共0兲储,

BKd2 = max 储Kd2储

t苸关0,T兴

共1兲 The desired trajectory y d共t兲 is first-order continuous for t 苸 关0 , T兴. 共2兲 The output matrix C共t兲 is first-order continuous for t 苸 关0 , T兴. 共3兲 The function f共x共t兲 , t兲 is globally, uniformly Lipschitz in x for t 苸 关0 , T兴. That means 储f共xk+1共t兲 , t兲 − f共xk共t兲 , t兲储 ⱕ c f 储xk+1共t兲 − xk共t兲储, where k is the iteration index and c f 共 ⬎0兲 is the Lipschitz constant. 共4兲 Uncertainty and disturbance terms ␩k共t兲 and ␰k共t兲 are bounded as follows: ∀t 苸 关0 , T兴 and ∀k, we have 储␩k共t兲储 ⱕ b , 储␰ 共t兲储 ⱕ b , and 储␰˙ 共t兲储 ⱕ b ˙ .

BB = max 储B共t兲储,

KB1 =

t苸关0,T兴

t苸关0,T兴

Bd1 + c f B pd1 , ␭ − cf

␳1 = max 储共I + Kd1共0兲CB兲−1储, t苸关0,T兴

␳=

t苸关0,T兴

BC = max 储C共t兲储,

t苸关0,T兴

3

t苸关0,T兴

BKp1 = max 储K p1共0兲储, t苸关0,T兴

where f = 关f 1 , . . . , f n兴T is a vector, M = 关mi,j兴 苸 Rm⫻n is a matrix, and h共t兲共t 苸 关0 , T兴兲 is a real function, where T is the time period of a repetitive task. To restrict the discussion, the following assumptions are made for the system.

␩

˙储 B pd2 = max 储K p2C + Kd2C

t苸关0,T兴

␳1 , 1 − ␳1BsBBKB1

KB2 =

Bs = max储s共k兲储

Bd2 + c f B pd2 ␭ − cf

␳2 = max 储Im − Kd2CB储 t苸关0,T兴

␤ = ␳2 + BBKB2

Main Results and Convergence Analysis

THEOREM. For the nonlinear time-varying system (1), if the SPD-PD-type iterative learning control law (2) is applied and the switching gain algorithm (3) is adopted, then the final state error and the output tracking error are bounded, and the boundednesses are given by

冦

lim 储␦xk共t兲储␭ ⱕ

k→⬁

lim 储ek共t兲储␭ ⱕ

k→⬁

冉

1 ␳⌽ + Tb␩ + ␧x ␭ − c f 1 − ␳␤

冉

冊

冊

BC ␳⌽ + Tb␩ + ␧x + b␰ ␭ − c f 1 − ␳␤

冧

共5兲

where ⌽ = 共BsKB1 + KB2 + BsBd1 + Bd2兲b␩ + 共BsBKp1 + BKp2兲b␰˙ + 共BsBKd1 + BKd2兲b␰ + 共BsKB1 + KB2兲␧x. Provided the control gain Kd1共0兲 and the learning gain Kd2共t兲 are selected such that Im + Kd1共0兲B共t兲C共t兲 is nonsingular, and

冦

max 储共I + Kd1共0兲B共t兲C共t兲兲−1储 = ␳1 ⬍ 1

t苸关0,T兴

max 储I − Kd2共t兲B共t兲C共t兲储 = ␳2 ⬍ 1

t苸关0,T兴

冧

共6兲

Also, we propose the following initial state learning algorithm: xk+1共0兲 = 共1 + B共0兲Kd1共0兲C共0兲兲−1兵xk共0兲 + B共0兲Kd1共0兲y d共0兲 + B共0兲Kd2共0兲共y d共0兲 − y k共0兲兲其

共7兲

In this paper, the ␭-norm is used to examine the convergence of the tracking error for the proposed SPD-PD ILC algorithm. First of all, a relation between norm and ␭-norm is represented by Lemma 1. LEMMA 1. Suppose that x共t兲 = 关x1共t兲 , x2共0兲 , . . . , xn共t兲兴T is defined in t 苸 关0 , T兴. Then,

冉冕

t

0

冊

1 储x共␶兲储d␶ e−␭t ⱕ 储x共t兲储␭ ␭

for ␭ ⬎ 0

共8兲

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Proof.

冉冕

冊 冕

t

储x共␶兲储d␶ e

0

−␭t

xk共t兲 = xk共0兲 +

t

=

储x共␶兲储e

−␭␶ −␭共t−␶兲

e

0

ⱕ sup 兵储x共t兲储e−␭t其 t苸关0,T兴

ⱕ sup 兵储x共t兲储e

冕

t苸关0,T兴

d␶ e−␭共t−␶兲d␶

再

ek+1 = C共t兲␦xk+1 − ␰k+1

1 − e−␭t 其 ␭

冕

˙ 共t兲␦x − ␰˙ e˙k+1 = C共t兲␦x˙k+1 + C k+1 k+1

From Eq. 共2兲, we have

共f共xd共␶兲, ␶兲 + B共␶兲ud共␶兲兲d␶

共18兲

共f共xd共␶兲, ␶兲 − f共x共␶兲, ␶兲兲d␶ +

0

冕

t

共B共t兲␦uk − ␩k兲d␶ + ␦xk共0兲

0

共19兲 Applying 共3兲 to Eq. 共19兲, we get

冕

␦xk ⱕ

t

c f ␦ x kd ␶ +

冕

t

共B共t兲␦uk − ␩k兲d␶ + ␦xk共0兲

共20兲

0

Equation 共20兲 can be written in the norm form as 共10兲

储 ␦ x k储 ⱕ

冕

t

c f 储␦xk储d␶ +

0

冕

t

共储B共t兲␦uk储 + 储␩k储兲d␶ + 储␦xk共0兲储

0

共21兲

␦uk+1 = ␦uk − s共k + 1兲K p1共0兲ek+1共t兲 − s共k + 1兲Kd1共0兲e˙k+1共t兲 − K p2共t兲ek共t兲 − Kd2共t兲e˙k共t兲

共11兲

␦uk+1 = ␦uk − K p2兵C␦xk − ␰k其 − Kd2兵C␦x˙k + C˙␦xk − ␰˙ k其 − s共k 共12兲

冎

共13兲

储␦xk+1储␭ ⱕ

共I + s共k + 1兲Kd1共0兲CB兲␦uk+1 = 共I − Kd2CB兲␦uk − s共k + 1兲共K p1共0兲C + Kd1共0兲C˙兲␦xk+1 − 共K p2C

⫻共C␦ f k+1 − C␩k+1 − ␰˙ k+1兲

共1 − ␳1BsBBKB1兲储␦uk+1储␭ ⱕ ␳1共␳2 + BBKB2兲储␦uk储␭ + ␳1⌽ Equation 共24兲 can be simplified as 储␦uk+1储␭ ⱕ ␳␤储␦uk储␭ + ␳⌽

− Kd2共C␦ f k − C␩k − ␰˙ k兲 + s共k

lim 储␦uk共t兲储␭ =

共14兲

From 共3兲 we have 储␦ f k储 ⱕ c f 储␦xk储 and 储␦ f k+1储 ⱕ c f 储␦xk+1储. As s共k + 1兲 ⬎ 1, to choose a proper control gain K p1共0兲, and from Eq. 共6兲 we can ensure

共25兲

Applying Eq. 共15兲, Eq. 共14兲 can be rewritten in the ␭-norm and simplified as 储␦uk+1储␭ ⱕ ␳1兵␳2储␦uk储␭ + Bs共B pd1 + Bd1c f 兲储␦xk+1储␭ + 共B pd2 + Bd2c f 兲 ⫻储␦xk储␭ + 共BsBd1 + Bd2兲b␩ + 共BsBKp1 + BKp2兲b␰ 共16兲

␳⌽ 1 − ␳␤

共26兲

From Eq. 共26兲, we can see that the control input is bounded and is close to the desired control input. Submitting Eq. 共26兲 into Eq. 共22兲, we can get

k→⬁

共15兲

Journal of Computational and Nonlinear Dynamics

k→⬁

lim 储␦xk共t兲储␭ =

储共I + s共k + 1兲Kd1共0兲CB兲−1储 ⬍ 储共I + Kd1共0兲CB兲−1储 = ␳1 ⬍ 1

For the kth iteration, the state vector can be written as

共23兲

From Eq. 共6兲, we have ␳1␳2 ⬍ 1. If we choose ␭ ⬎ ␳1BB共c f 共BsB pd1 + B pd2兲 + BsBd1 + Bd2兲 / 1 − ␳1␳2 + c f , then we can guarantee ␳␤ ⬍ 1. From Eq. 共25兲, we can get

+ Kd2C˙兲␦xk − s共k + 1兲Kd1共0兲

BKd2兲b␰˙ 其

BB T ␧x 储␦uk+1储␭ + b␩ + ␭ − cf ␭ − cf ␭ − cf

共24兲

Submitting Eq. 共12兲 into Eq. 共13兲 and reorganizing gets

+ 1兲K p1共0兲␰k+1 + K p2␰k

共22兲

Submitting Eqs. 共22兲 and 共23兲 into Eq. 共16兲 and simplifying it yields

Also from Eq. 共1兲, we can get the following equations:

␦x˙k = ␦ f k + B␦uk − ␩k ␦x˙k+1 = ␦ f k+1 + B␦uk+1 − ␩k+1

BB T ␧x 储 ␦ u k储 ␭ + b␩ + ␭ − cf ␭ − cf ␭ − cf

For the 共k + 1兲th iteration, we can get a very similar result:

+ 1兲K p1共0兲兵C␦xk+1 − ␰k+1其 − s共k + 1兲Kd1共0兲兵C␦x˙k+1 + C˙␦xk+1 − ␰˙ k+1其

According to the definition of ␭-norm, for ␭ ⬎ c f , applying Lemma 1 to Eq. 共21兲 obtains 储 ␦ x k储 ␭ ⱕ

Submitting Eqs. 共9兲 and 共10兲 into Eq. 共11兲 gets

+ 共BsBKd1 +

t

t

0

The derivative of the tracking errors can be represented as

再

冕

From Eqs. 共17兲 and 共18兲, we get

共9兲

冎

␩k共t兲d␶

0

␦xk =

˙ 共t兲␦x − ␰˙ e˙k = C共t兲␦x˙k + C k k

t

0

xd共t兲 = xd共0兲 +

冎

冕

共17兲

We define the following four variables. ␦xk
xd − xk, ek
y d − y k, ␦uk
ud − uk, and ␦ f k = f d − f共xk兲. From Eq. 共1兲, we can calculate the tracking errors as ek = C共t兲␦xk − ␰k

共f共xk共␶兲, ␶兲 + B共␶兲uk共␶兲兲d␶ +

From Eq. 共1兲, we also have

t

1 ⱕ 储x共t兲储␭ ␭

再

t

0

0

−␭t

冕

冉

1 ␳⌽ + Tb␩ + ␧x ␭ − c f 1 − ␳␤

冊

共27兲

Equation 共27兲 proves that the state error is bounded. Finally, from Eq. 共9兲, we can get lim 储ek共t兲储␭ =

k→⬁

冉

冊

␳⌽ BC + Tb␩ + ␧x + b␰ ␭ − c f 1 − ␳␤

共28兲

Equation 共28兲 demonstrates that the output error is bounded. Remark 1. If the initial state updating law 共7兲 is used, we will ensure limk→⬁ xk共0兲 = x0共0兲. In this manner, we can get limk→⬁ ␧x = 0. Therefore, JANUARY 2011, Vol. 6 / 011020-3

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lim 储ek共t兲储␭ =

k→⬁

冉

冊

␳⌽ BC + Tb␩ + b␰ ␭ − c f 1 − ␳␤

共29兲

Remark 2. If there is no uncertainty and disturbance in Eq. 共1兲, then the final tracking error bound becomes lim 储ek共t兲储␭ =

k→⬁

B C␧ x ␭ − cf

共30兲

Remark 3. If the initial state updating law 共7兲 is applied and there is no uncertainty and disturbance, then the final tracking error is limk→⬁储ek共t兲储␭ = 0. Such a conclusion can be derived directly from Remarks 1 and 2. Remark 4. The convergence condition 共6兲 does not include the proportional gains. Therefore, it provides some extra freedom for the choices of K p1 and K p2 in the proposed control law 共2兲.

4

Simulation Study

4.1 Simulation of Nonlinear System With Uncertainty and Disturbance. In this section, we apply the proposed SPD-PD ILC algorithm for the trajectory tracking control of a nonlinear system to improve its tracking performance through repetitive operations. The simulation example used in Ref. 关26兴 is adopted for the purpose of comparison. The nonlinear system is described by

冋 册冋 x˙1k共t兲

=

x˙2k共t兲

sin共x2k共t兲兲 1 + sin共x1k共t兲兲 2 − 5t

冋 册冋 册 冋 册 冋 册冋 册 +

y 1k共t兲

y 2k共t兲

=

1 0

u1k共t兲

0 2

u2k共t兲

4 0

x1k共t兲

0 1

x2k共t兲

册冋 册 冋 册 冋 册 x1k共t兲

x2k共t兲

− 3 − 2t

cos共2␲ f 0t兲

+ 共0.5 + k␣0兲

+ 共0.5 + k␣0兲

2 cos共4␲ f 0t兲 sin共2␲ f 0t兲

2 sin共4␲ f 0t兲

共31兲 With f = 5 Hz. The desired tracking trajectories are set as y 1d共t兲 = y 2d共t兲 = 12t2共1 − t兲

for t 苸 关0,1兴

共32兲

To test the robustness of the proposed SPD-PD ILC algorithm, several simulation experiments are conducted using different control gains and learning gains. For all the cases, the initial states are set as x1共0兲 = 0.3 and x2共0兲 = −0.3. This means that there are some initial state errors. Also, we assume that matrix B in the initial state learning schedule 共7兲 is not accurate 共the estimated value is 0.4B兲. In the following sections, the classic PD ILC is obtained from control law 共2兲 by setting K p1共0兲 = Kd1共0兲 = 0. 4.1.1 Example 1: Repetitive Uncertainty and Disturbance. In the first simulation experiment, we set ␣0 = 0, which means that the uncertainty and disturbance are repetitive from iteration to iteration. The following feedback control gains and learning control gains are chosen: For classic PD ILC:

K p2 = diag兵1,1其,

For SPD-PD ILC:

proposed SPD-PD ILC and the classic PD ILC from iteration to iteration. From this figure, we can see that the SPD-PD ILC algorithm can obtain a very fast convergence rate 共seven iterations兲 and very small and monotonic decreased tracking errors. But for the classic PD ILC case, although the best learning gain Kd2 is used 共as ␳2 = 0兲, the tracking error bounds were still in a good-badgood mode before reaching a stable boundedness, and more iterations 共18 iterations兲 were needed in order to obtain a relatively acceptable tracking performance. We can see that the tracking errors using PD ILC are still relatively large compared with those using the SPD-PD ILC. Similar results were shown in Ref. 关26兴 where more than 20 iterations are needed to achieve a stable tracking performance. This example results demonstrate that the SPD-PD ILC is more powerful in terms of reducing the tracking error and facilitating the convergence. 4.1.2 Example 2: Varying Uncertainty and Disturbance. In this example, we consider a more general situation where the uncertainty and disturbance are increased from iteration to iteration by set ␣0 = 0.05. All the feedback control gains and iterative learning gains are set the same as in example 1. Figure 2 shows the simulation results. In Fig. 2, we can see that a good tracking performance can be achieved using the SPD-PD ILC algorithm, even in the situations where the uncertainty and disturbance are varied from iteration to iteration. Such a feature can be attributed to the switching gain feedback control, which can compensate for the disturbance by increasing the control gains from iteration to iteration. But for the classic PD ILC algorithm, there are large stable tracking error bounds 共around 0.1 for e1 and 0.2 for e2兲. That is because the classic PD ILC cannot compensate for the current iteration disturbance due to the limitation of the offline learning strategy. 4.1.3 Comparison on Different Feedback Control Gains. To adjust the final tracking error bounds, different feedback control

Kd2 = diag兵0.25,0.5其

K p1共0兲 = diag兵0.5,2.5其,

Kd1共0兲 = diag兵1,5其, K p2 = 0.6ⴱ diag兵1,1其,

Fig. 1 Maximum error bounds for example 1

s共k兲 = k

Kd2 = 0.6ⴱ diag兵0.25,0.5其

Referring to Eq. 共1兲, we have BC = diag兵4 , 2其 for this nonlinear system. Therefore, the perfect learning gain is Kd2 = diag兵0.25, 0.5其 according to Eq. 共6兲, and that is used in the classic PD ILC. According to the chosen gains, from Eq. 共6兲 we have ␳2 = 0 for the classic ILC, and ␳1 = 0.2 and ␳2 = 0.4 for the SPD-PD ILC. This means that inaccurate knowledge of matrices B and C are considered in the learning control gain design for the SPD-PD ILC. Figure 1 shows the maximum tracking error bounds for the 011020-4 / Vol. 6, JANUARY 2011

Fig. 2 Maximum error bounds for example 2

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Table 1 Comparison of maximum error bounds under different control gains Example 1 Control

Example 2

Iteration

max兩e1兩

max兩e2兩

Iteration

max兩e1兩

max兩e2兩

9 7 6

0.0005 0.0003 0.0003

0.0007 0.0007 0.0006

10 7 6

0.0058 0.0039 0.0032

0.0034 0.0022 0.0017

Low gains Mid gains High gains

Table 2 Comparison maximum error bounds for SPD-PD and PD-PD learning control Example 1

Example 2

Control

Iteration

max兩e1兩

max兩e2兩

Iteration

max兩e1兩

max兩e2兩

PD-PD SPD-PD

7 7

0.0023 0.0003

0.0038 0.0007

7 7

0.0241 0.0039

0.0125 0.0022

gains are used for the SPD-PD ILC algorithm. In this example, the iterative learning control gains are set the same as the previous two examples. The following feedback control gains are used in the simulation experiments. For all three different gain cases, the switching function is set as s共k兲 = k. The middle gains are chosen the same as in example 1, High gains:

K p1共0兲 = 1.5 diag兵0.5,2.5其,

Kd1共0兲 = 1.5 diag兵1,5其 Middle gains: Low gains:

K p1共0兲 = diag兵0.5,2.5其,

K p1共0兲 = 0.5 diag兵0.5,2.5其,

Kd1共0兲 = diag兵1,5其

proportional feedback gain K p1 and iterative learning gain K p2 are not factors for guaranteeing convergence as they are not included in the convergence condition 共6兲. Therefore, it is a good point to examine the effect of proportional gains on the tracking performance improvement. Figure 4 shows one simulation result of the tracking error bounds from iteration to iteration using SPD-PD ILC, SD-PD ILC 共K p1 = 0兲, SPD-D ILC 共K p2 = 0兲, and SD-D ILC 共K p1 = K p2 = 0兲. It is clearly shown that K p gains have effects only on the first few 共three in this simulation兲 iterations. After that, K p gains make little contribution to the convergence rate of the system. The simulation results show that the final tracking error bounds using these four learning control laws are almost the same.

Kd1共0兲 = 0.5 diag兵1,5其

Table 1 lists the maximum final tracking error bounds under different feedback control gains in the SPD-PD ILC. From this table, we can see that, with the increase in the feedback control gains, the final tracking errors become smaller, and the convergence rate becomes faster. It demonstrates that the control parameter ␳1 has significant effect to the convergence rate. Table 2 shows the final tracking error bounds 共stable boundary兲 for SPD-PD ILC compared with a fixed gain PD-PD ILC 共s共k兲 = 1兲 using the same learning gains. From this table, it is clearly shown that the SPD-PD ILC can obtain much better tracking performance 共at least five times兲 even with the existence of the nonrepetitive uncertainty and disturbance 共example 2兲. Figure 3 shows the tracking error bound results from iteration to iteration based on SPD-PD ILC and the fixed gain PD-PD ILC. In Table 2 and Fig. 3, we can see that the tracking performances can be improved if high feedback control gains are used in the developed SPD-PD ILC algorithm. It also clearly shows that the tracking error bounds monotonically decrease from iteration to iteration.

4.2 Simulation of Trajectory Tracking for a Planar Robotic System. It is well known that the dynamic model of a robotic system can be expressed as a second-order differential equation. If we introduce some state variables, the second-order differential equation can be easily transformed to a first-order nonlinear system expressed in Eq. 共1兲. Therefore, the proposed SPD-PD ILC can be applied to tracking control of the robotic system. In this paper, a 2DOF revolute planar robot discussed in Ref. 关31兴 is simulated for the trajectory tracking control. Table 3 lists the structural parameters of the robotic system. The mass center of each link is set at the center of the link. The desired trajectories and the disturbances for both links are chosen as

4.1.4 The Effect of Proportional Gains. From the convergence analysis conducted in the previous section, we can see that the

Fig. 4 Effect of proportional control gains to convergence rate Table 3 Planar robot parameters Link

Mass 共kg兲

Length 共m兲

Inertia 共kg m2兲

1 2

10.0 5.0

1.0 0.5

0.83 0.30

Fig. 3 Comparison of SPD-PD and PD-PD learning control

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Fig. 5 Tracking performance and controlled torque for joint 1 „the first, third, and fifth iterations…

冋 册冋 册 冋册 冋 q1d q2d

=

sin 3t

cos 3t

,

0.3 sin t ␩1 =␣ 0.1共1 − e−t兲 ␩2

册

Fig. 6 Tracking performance and controlled torque for joint 2 „the first, third, and fifth iterations…

for t 苸 关0,5兴 共33兲

4.2.1 Tracking Performance Improvement From Iteration to Iteration. In this simulation experiment, the feedback control gains and iterative learning gains are set as K p1共0兲 = Kd1共0兲 = diag兵20,10其 and K p2 = Kd2 = diag兵10,5其 The switching function is set as s共k兲 = 2k, and the disturbance factor is ␣ = 10. Figures 5 and 6 show the trajectory tracking performance improvements and the required control torques from iteration to iteration for joints 1 and 2, respectively. At the first iteration, there are very large tracking errors for both joints, and the required control torques are irregular. After three iterations, the tracking performances improved dramatically. After five iterations, we can see that the controlled joint motions are very close to the desired ones, while the required torques show the desired periodic features. Figure 7 shows the maximum tracking error bounds from iteration to iteration. In Fig. 7, we can see that the tracking errors decrease monotonically from iteration to iteration. Simulation re-

Fig. 7 Tracking error bounds from iteration to iteration

sults show that only 14 iterations are needed to achieve a very good tracking performance 共maximum errors are 1.75⫻ 10−4 and 4.75⫻ 10−5 for joints 1 and 2 from Table 4, respectively兲, while the maximum error bounds are 0.0041 and 0.0046 for joints 1 and 2 after 30 iterations using the control law developed in Ref. 关31兴. Therefore, the SPD-PD ILC has a fast convergence rate, small

Table 4 Maximum tracking errors for two joints Maximum tracking error for joint 1

Maximum tracking error for joint 2

Iteration

␣=0

␣ = 10

␣ = 50

␣=0

␣ = 10

␣ = 50

1 3 6 9 12 14

1.6807 0.3579 0.0137 0.001 3.13⫻ 10−4 1.76⫻ 10−4

1.7112 0.3578 0.0136 0.001 3.14⫻ 10−4 1.74⫻ 10−4

1.8404 0.3565 0.0135 0.001 3.04⫻ 10−4 1.82⫻ 10−4

0.5812 0.1096 0.0034 2.93⫻ 10−4 8.81⫻ 10−5 4.75⫻ 10−5

0.6026 0.1122 0.0033 2.92⫻ 10−4 8.90⫻ 10−5 4.75⫻ 10−5

0.7363 0.1213 0.0034 2.93⫻ 10−4 8.68⫻ 10−5 5.17⫻ 10−5

011020-6 / Vol. 6, JANUARY 2011

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tracking error bounds, and no vibration in the actuators. These figures demonstrate the effectiveness of the SPD-PD ILC for the trajectory tracking control of robotic systems. 4.2.2 Robustness for Rejecting Disturbance. To verify the effectiveness and robustness of the SPD-PD algorithm for the rejection of disturbances, different levels of disturbances are introduced in the dynamic model by adjusting the parameter ␣ in Eq. 共33兲. Table 4 lists the maximum tracking error bounds for three different levels of disturbances. It shows that all the finally tracking error bounds after a few iterations 共nine iterations in this example兲 are almost the same. From this table, we can conclude that the proposed SPD-PD ILC algorithm is robust to the disturbances.

5

Conclusions

In this paper, a new iterative learning control called SPD-PD ILC is proposed, which is a combination of a PD feedback control with switching control gains in the iteration domain and a PD-type ILC with previous iteration information in the updating control law. The proposed SPD-PD ILC takes the advantages offered by feedback control and iterative learning control. The proposed control law uses the current and previous system information to enhance the stability characteristics and quickly drive the tracking trajectories to the desired ones within bounds. The new SPD-PD ILC achieves tracking accuracy with very fast convergence rate and is robust against unpredictable disturbances and uncertainties by adjusting the feedback control gains. In addition, the SPD-PD ILC can provide extra degrees of freedom for the choices of the learning gains. The final tracking error bounds and the convergence rate can be adjusted by the switching gain of the PD feedback control that makes this control scheme more promising from a practical viewpoint. Nonlinear systems and a planar robotic system are used as examples to examine the effectiveness of the proposed SPD-PD ILC. Real applications of the SPD-PD ILC should be a future work.

Acknowledgment This research is supported by the Natural Sciences and Engineering Research Council of Canada 共NSERC兲 through a Discovery Grant.

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