طراحی و ویژگی همگرایی تجزیه و تحلیل بهینه مقاوم از رویکردهای کنترل یادگیری تکراری

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.

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.

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 between the time-weighted norm and supreme norm. Finally according to the newly introduced Q-factor, convergence speed comparison were made among lower-order and higher-order ILC schemes, and the result was in favor of the lower order ILC. Using a highly nonlinear and non-affine dynamic system, a number of simulations were conducted, which confirmed the theoretical results.