جفت های مجاز و بازخورد معادل- پارامترهای Youla در کنترل یادگیری تکراری

This paper revisits a well-known synthesis problem in iterative learning control, where the objective is to optimize a performance criterion over a class of causal iterations. The approach taken here adopts an infinite-time setting and looks at limit behavior. The first part of the paper considers iterations without current-cycle-feedback (CCF) term. A notion of admissibility is introduced to distinguish between pairs of operators that define a robustly converging iteration and pairs that do not. The set of admissible pairs is partitioned into disjoint equivalence classes. Different members of an equivalence class are shown to correspond to different realizations of a (stabilizing) feedback controller. Conversely, every stabilizing controller is shown to allow for a (non-unique) factorization in terms of admissible pairs. Class representatives are introduced to remove redundancy. The smaller set of representative pairs is shown to have a trivial parameterization that coincides with the Youla parameterization of all stabilizing controllers (stable plant case). The second part of the paper considers the general family of CCF-iterations. Results derived in the non-CCF case carry over, with the exception that the set of equivalent controllers now forms but a subset of all stabilizing controllers. Necessary and sufficient conditions for full generalization are given.

Some 20 years ago, Arimoto, Kawamura, and Miyazaki (1984) were among the first to develop a theory of learning specifically tailored to single-loop control problems. Upon observing the human tendency to learn from experience, they were led to ask whether it would be possible to implement a similar ability in the automatic operation of dynamical systems. In answer, they proposed a ‘betterment process’, now known as iterative learning control (ILC). The method proved effective and inspired a great number of researchers. Over the years, Arimoto's original algorithm has been modified and extended in a number of ways: assumptions have been relaxed, robustness has been improved, and convergence properties have been laid out in detail. See Moore, Dahleh, and Bhattacharyya (1992) and Amann, Owens, and Rogers (1994) for an overview of early results. Today, there is an extensive literature covering a wealth of different learning rules applicable to a wide range of systems, both linear and nonlinear. Recent surveys include Moore (1999), Chen and Wen (1999), and Xu and Tan (2003). In the early days, questions of analysis and synthesis were addressed almost exclusively within a time-domain framework which was built around the finite-trial-length postulate (Arimoto, 1998, chap. 1; Arimoto et al., 1984). This framework became the standard for many years and is still among the most commonly used today. Yet, over the course of two decades, a variety of other techniques have been introduced, some to considerable effect. This paper is about one such technique. The technique in question originates in the early nineties, when, following developments in the general field of control, people begin to view ILC as an H∞H∞ synthesis problem. The synthesis problem is to minimize a performance criterion, typically the mean-squared tracking error, over the space of bounded (real-rational) transfer functions, RH∞RH∞. See, for example, Padieu and Su (1990), Kavli (1992), Amann, Owens, Rogers, and Wahl (1996), and Moore et al. (1992). In this approach the finite-trial-length postulate is dropped and an extra assumption introduced, namely that learning operators be causal (recall that every element in RH∞RH∞ defines a causal bounded (finite-dimensional) LTI operator). At the time, few would have anticipated that as natural a role as causality plays in conventional feedback control, as restrictive and unnecessary it would prove in the context of ILC. The success of Arimoto's learning rule was known to be tied up with the availability of a future error (i.e. with a non-causal ingredient of some kind). And indeed, the fact that a learning operator need not be causal was well-established ( Moore et al., 1992 and Moore, 1999). Yet it would seem that the precise reason as to why causality should affect ILC performance the ways it does was not known. In contrast, the implications for compensator design were well-understood, owing largely to the classical work of Bode. By the time Goldsmith, 2001 and Goldsmith, 2002 introduced the notion of equivalent feedback, the fact that causality seriously impairs the achievable performance was widely acknowledged, and studies into the merits of non-causal ILC were well-underway. Indeed, the thesis that a non-causal approach would constitute ‘the only viable route for ILC’ ( Goldsmith, 2002, p. 708) had been voiced by others before. It would appear however that Goldsmith was the first to provide compelling evidence for it. The evidence has been contested (see Owens and Rogers, 2004 and Goldsmith, 2004) but as of yet the thesis has not been overthrown. The work presented in this paper builds on that of Goldsmith's. We provideseveral extensions, most notably a converse result, which states that the set of equivalent controllers is generally but a subset of all stabilizing controllers. Conditions under which both sets coincide are given. Also, we state precise conditions (as captured by our notion of admissibility) under which causal ILC and conventional feedback are equivalent and provide an example of a causal iteration with an equivalent controller that is destabilizing. Following Padieu and Su (1990), Kavli (1992), Moore et al. (1992), Moore (1993), de Roover (1996) and Amann et al. (1996), among others, this paper poses the problem of ILC as a two-parameter synthesis problem. The parameters are assumed causal bounded operators acting on the current input and current error, respectively. Our approach comprises the following steps. First, a notion of admissibility is introduced. This notion is used to single out ‘bad’ pairs of operators. Then the two-parameter problem is shown to be overparameterized; that is, different admissible pairs are shown to induce different sequences converging to the same fixed point. Redundancy is removed by grouping ‘equivalent pairs’ into equivalence classes, restricting attention to class representatives. Finally, the resulting one-parameter problem is shown to be a standard compensator design problem. The organization of the paper is as follows. Section 2 introduces the problem of ILC. Two problem cases are identified and discussed in subsequent sections: standard ILC in Section 3 and current-cycle-feedback-ILC (CCF-ILC) in Section 4. Section 5 closes with conclusions and recommendations.

In this paper we have characterized the set of admissible pairs for a class of linear learning rules. Specifically, for the stable plant case, we have shown that the set of admissible pairs is isomorphic to the set of stabilizing feedback controllers and we have explored the consequences of this result. For the case of an unstable plant under CCF, we showed that the set of admissible pairs is generally smaller than the set of stabilizing controllers. At a more general level, our analysis suggests that, under certain specific conditions detailed in the paper, causal ILC and conventional feedback are truly equivalent methods. If these conditions are not satisfied, equivalence cannot be established. From the results it is clear that causality constrains the application of ILC to a point that one can reasonably question its use. While the strength of ILC is in its potential to exploit information that is inaccessible to conventional controllers, this very potential does not materialize in causal ILC. In view of this we propose that future research should focus on non-causal ILC. The framework developed in this paper can readily be extended to incorporate non-causal operators (Verwoerd, 2005), and it would seem that extensions to nonlinear operators are also possible. Given in particular the body of literature on the Youla parameterization problem for nonlinear systems, it seems reasonable to expect that the correspondence between ILC and conventional feedback control extends beyond the realm of linear systems.