Elsevier

Automatica

Volume 38, Issue 11, November 2002, Pages 1867-1880
Automatica

Robust optimal design and convergence properties analysis of iterative learning control approaches

https://doi.org/10.1016/S0005-1098(02)00143-7Get rights and content

Abstract

In this paper, we address four major issues in the field of iterative learning control (ILC) theory and design. The first issue is concerned with ILC design in the presence of system interval uncertainties. Targeting at time-optimal (fastest convergence) and robustness properties concurrently, we formulate the ILC design into a min–max optimization problem and provide a systematic solution for linear-type ILC consisting of the first-order and higher-order ILC schemes. Inherently relating to the first issue, the second issue is concerned with the performance evaluation of various ILC schemes. Convergence speed is one of the most important factors in ILC. A learning performance index—Q-factor—is introduced, which provides a rigorous and quantified evaluation criterion for comparing the convergence speed of various ILC schemes. We further explore a key issue: how does the system dynamics affect the learning performance. By associating the time weighted norm with the supreme norm, we disclose the dynamics impact in ILC, which can be assessed by global uniform bound and monotonicity in iteration domain. Finally we address a rather controversial issue in ILC: can the higher-order ILC outperform the lower-order ILC in terms of convergence speed and robustness? By applying the min–max design, which is robust and optimal, and conducting rigorous analysis, we reach the conclusion that the Q-factor of ILC sequences of lower-order ILC is lower than that of higher-order ILC in terms of the time-weighted norm.

Introduction

In the last two decades iterative learning control (ILC) has been extensively studied, achieves significant progress in both theory and application, and becomes one of the most active fields in intelligent control and system control (Arimoto, 1985; Kawamura, Miyazaki, & Arimoto, 1987a; Hara, Yamamoto, Omata, & Nakano, 1988; Bien & Huh, 1989; Sugie & Ono, 1991; Kuc, Lee, & Nam, 1992; Jang, Choi, & Ahn, 1995; Saab, 1995; Amann, Owens, & Rogers, 1996; Phan & Juang, 1996; Lucibello, Panzier, & Ulivi, 1997; Lee & Bien, 1997; Longman & Lo, 1997; Moore, 1998; Chien, 2000; Lee & Lee, 2000; de Roover, Bosgra, & Steinbuch, 2000; Wang, 2000; Norrlof & Gunnarsson, 2001; Ham, Qu, & Kaloust, 2001; Xu & Tan, 2002). On the other hand, there are still numerous open problems left to researchers for further exploitation. In this paper, we address four open and most important issues in the field of ILC theory and design:

  • 1.

    Can we design iterative learning controllers possessing robustness and optimality concurrently, in particular achieving fastest convergence in the presence of system uncertainties?

  • 2.

    Can we evaluate learning convergence speed for various ILC schemes in a rigorous and quantitative manner?

  • 3.

    How does the system dynamics affect the learning performance in iteration domain?

  • 4.

    Can a higher-order ILC scheme perform better than lower-order ILC schemes?

It has been made clear that the ILC convergence is solely related to the system direct feed-through term. When the feed-through term is associated with a interval uncertainty, i.e. with lower and upper bounds, the idea of traditional ILC design is to guarantee the convergence condition for the worst case (upper bound), hence leads to a slower convergence speed due to the conservative design. Targeting at the time-optimal property and meanwhile assuring the learning convergence, we formulate the ILC design into a min–max optimization problem and provide a systematic solution. The max operation maximizes the influence from the system uncertainty, and min operation minimizes a learning factor that determines the convergence speed.

When a highly nonlinear, uncertain and non-affine system is under iterative learning control, it is very hard to guarantee that the performance of one ILC scheme is better than that of another one “uniformly” for all iterations. What is possible and more practical is to look for such indices that can capture the essential nature of an ILC scheme for “most” iterations. More rigorously speaking, one ILC scheme is thought of performing better than another in certain aspect, if the corresponding index of the former is better than that of the latter for infinitely many iterations except for a finite number of iterations. For this purpose, a learning performance index—Q-factor—is introduced, which provides a rigorous and quantified evaluation criterion for comparing the convergence speed of sequences generated by different ILC schemes. A lower Q-factor means a faster convergence speed for most iterations. Using Q-factor, it is easy to derive a “characteristic equation” that specifies the convergence speed for an iterative learning process.

In most ILC design and analysis, the system dynamic effect is neglected while a time weighted norm is used. In this paper, we investigate the relationship between the system dynamic influence and the time-weighted norm. In order to quantify the dynamic impact to the learning process, we introduce two indices with supreme norm—the global uniform bound of tracking error in iteration domain, and monotonicity period. The former describes the worst case error bound, and the latter specifies the maximum tracking interval in which the tracking error decreases monotonically in terms of the supreme norm. By means of these two indices, the system dynamic impact, which is hidden and suppressed by the time-weighted norm, is clearly exhibited.

The last issue is rather controversial in ILC. Intuitively, a higher-order ILC, that employs preceding control information of more than one iteration, should be able to improve learning performance as more of preceding control information is used. However, a simple linear combination of preceding control information may not provide new information. Note that most higher-order ILC schemes proposed hitherto are of linear type. What is more, for a convergent ILC sequence, in most iterations the latest should be the most accurate and the rest are less. A linear combination of less accurate ones may further degrade the performance. To answer this question, rigorous analysis and fair comparisons are indispensable. In the last part of this paper we analyze and compare the learning convergence speed associated with linear first-order and higher-order ILC schemes. Based on the min–max design and Q-factor, we are able to conduct a quantitative comparison and reach the following conclusion. Under the same interval uncertainty and applying the same min–max design which is robust and optimal, the Q-factor of ILC sequences of lower-order ILC is lower than that of higher-order ILC in terms of the time-weighted norm. In the sequel, the first-order ILC achieves the fastest convergence speed in the iteration domain in the sense of Q-factor.

This paper is organized as follows. The learning control problem is formulated in Section 2. Section 3 presents the convergence analysis and robust optimal design for the first-order ILC scheme under interval uncertainty. Section 4 explores the dynamic impact in iteration domain. Section 5 compares the learning convergence speed for higher-order ILC schemes.

Section snippets

Problem formulation

Consider the nonlinear dynamic system (1),ẋ(t)=f(x(t),u(t),t)x(0)=x0,y(t)=g(x(t),u(t),t),where t∈[0,T],x(t)∈Rn,y(t)∈R and uR, f(·) and g(·) are partially unknown functions. The system is satisfying the following assumptions.

Assumption 1

Denoting Ω≜Rn×R×[0,T],0<α1⩽∂g/∂u⩽α2 and ||∂g/∂x||⩽βx,∀(x,u,t)∈Ω. Here α1, α2 are known constants and βx is an unknown constant.

Remark 1

∂g/∂u is equivalent to system direct feed-through term and represents the system gain. 0<α1∂g/∂u warrants no singularity in the system control.

Convergence analysis and robust optimal design for first-order ILC

In this section, we investigate the convergence properties and provide a systematic way to design the first-order ILC scheme in a robust and optimal manner.

Analysis of dynamic impact in iteration domain

In above discussion, δ1 represents the system dynamic impact from x to the ILC process. In this section, we analyze the dynamic impact in iteration domain. By investigating the role of λ, in particular with λ=0 which is essentially the supreme norm, we show the global uniform bound of the tracking error in iteration domain, as well as the monotonic convergent interval.

Convergence speed analysis of higher-order ILC

The idea of the higher-order ILC is straightforward: using control information of more than one iteration to improve learning control performance, such as robustness and convergence speed. However, it is still not clear whether the higher-order ILC scheme can work better than the lower-order. In this section, different ILC schemes are compared in terms of convergence speed. First, a second-order ILC scheme is compared with the first-order ILC scheme, then an (m+1)th order ILC scheme, ∀m⩾2, is

Conclusion

In this paper a systematic design, min–max design, was first proposed for various linear-type ILC schemes, from the first order to mth order, which takes into account both optimality and system interval type uncertainties. Next we investigated three important ILC performance factors: the convergence speed, the global uniform tracking error bound, and the maximum monotonicity period. The system dynamic impact to the iteration learning process was made clear through analyzing the relationship

Dr. Jian-Xin Xu received his Bachelor degree from Zhejiang University, China in 1982. He attended the University of Tokyo, Japan, where he received his Master's and Ph.D. degrees in 1986 and 1989, respectively. All his degrees are in Electrical Engineering. He worked for 1 year in the Hitachi research Laboratory, Japan and for more than 1 year in Ohio State University, USA as a Visiting Scholar. In 1991 he joined the National University of Singapore, and is currently an associate professor in

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    Dr. Jian-Xin Xu received his Bachelor degree from Zhejiang University, China in 1982. He attended the University of Tokyo, Japan, where he received his Master's and Ph.D. degrees in 1986 and 1989, respectively. All his degrees are in Electrical Engineering. He worked for 1 year in the Hitachi research Laboratory, Japan and for more than 1 year in Ohio State University, USA as a Visiting Scholar. In 1991 he joined the National University of Singapore, and is currently an associate professor in the Department of Electrical & Computer Engineering. His research interests lie in the fields of learning control, variable structure control, fuzzy logic control, discontinuous signal processing, and applications to motion control and process control problems. Up to now he has 70 peer-reviewed papers published or to appear, 10 chapters in edited books, and 130 papers in prestigious conference proceedings. He co-edited three books: “Iterative Learning Control” published by Kluwer Academic Press in 1998, “Advances in Variable Structure Systems” by World Scientific in 2000 and “Variable Structure Systems—Towards 21st Century” by Springer-Verlag in 2001. He is a senior member of IEEE.

    Ms. Ying Tan received her Bachelor degree from Tianjin University, China in 1995. She attended the Institute of Systems Engineering, Tianjin University, China, where she received his Master's in 1998. In 1998, she joined the National University of Singapore and finished her Ph.D. study in 2001. She is currently a research engineer in the Department of Electrical & Computer Engineering. Her research interests are in the fields of learning control, neural network, fuzzy logic control and wavelet network.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by the Associate Editor Tong Heng Lee under the direction of Editor Robert R. Bitmead

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