Iterative Learning Control Design for Switched Systems
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PICAL ISSUE
Iterative Learning Control Design for Switched Systems P. V. Pakshin∗,a and J. P. Emelianova∗,b ∗
Arzamas Polytechnic Institute of R.E. Alekseev Nizhny Novgorod State Technical University, Arzamas, Russia e-mail: a [email protected], b [email protected] Received July 23, 2019 Revised October 21, 2019 Accepted January 30, 2020
Abstract—In this paper, discrete-time linear systems with parameter switching in the repetitive mode are considered. A new iterative learning control design method is proposed. This method is based on the construction of an auxiliary 2D model in the form of a discrete repetitive process; the stability of the auxiliary model guarantees the convergence of the learning process. Stability conditions are derived using the divergent method of Lyapunov vector functions. The concept of the average dwell time in pass direction is introduced. An example that demonstrates the capabilities and features of the new method is presented. Keywords: iterative learning control, discrete-time systems, switched systems, repetitive processes, 2D systems, stability, dissipativity, vector Lyapunov function DOI: 10.1134/S0005117920080081
1. INTRODUCTION In modern control theory, switched systems are often understood as a class of models of dynamic systems consisting of a finite number of subsystems, of which only one is currently functioning, called the active subsystem, and the choice of the active subsystem is determined by some logical rule. The simplest example is a multi-mode system, in which subsystems are interpreted as separate modes of this system. Subsystems are usually described by an indexed set of differential or difference equations. The class of switched systems has been intensively studied in recent decades and continues to be actively developed nowadays, which is motivated by numerous applications in engineering, physics, biology, economics, and other fields, as well as by theoretical problems still open in this field of research. Like for other classes of control systems, the theory of stability and stabilization is of top priority, where a number of interesting and important results have been established. For an initial acquaintance with these results, in the first place the reader is recommended the monograph [1], the surveys [2, 3], and the recent books [4, 5]. Since the 1960s the theory of the so-called 2D systems began to develop actively. Its appearance was motivated by the problems of image processing and multidimensional electrical circuits, where the Roesser and Fornasini–Marchesini models [6] were constructed and subsequently became the classical 2D models; also, see a rich bibliography in the book [6]. A significant upsurge in the development of the theory of 2D systems was connected with the work of Arimoto [7], who first presented a theoretical justification of iterative learning control (ILC) algorithms for robots performing repetitive operations and revealed the natural 2D nature of the control process. (It includes a dynamic process on each separate repetition, also cal
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