Lecture Notes on Logically Switched Dynamical Systems
The subject of logically switched dynamical systems is a large one which overlaps with many areas including hybrid system theory, adaptive control, optimal control, cooperative control, etc. Ten years ago we presented a lecture, documented in [1], which a
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Introduction The subject of logically switched dynamical systems is a large one which overlaps with many areas including hybrid system theory, adaptive control, optimal control, cooperative control, etc. Ten years ago we presented a lecture, documented in [1], which addressed several of the areas of logically switched dynamical systems which were being studied at the time. Since then there have been many advances in many directions, far to many too adequately address in these notes. One of the most up to date and best written books on the subject is the monograph by Liberzon [2] to which we refer the reader for a broad but incisive perspective as well an extensive list of references. In these notes we will deal with two largely disconnected topics, namely switched adaptive control (sometimes called supervisory control) and “flocking” which is about the dynamics of reaching a consensus in a rapidly changing environment. In the area of adaptive control we focus mainly on one problem which we study in depth. Our aim is to give a thorough analysis under realistic assumptions of the adaptive version of what is perhaps the most important design objective in all of feedback control, namely set-point control of a singleinput, single output process admitting a linear model. While the non-adaptive version the set-point control problem is very well understood and has been so for more than a half century, the adaptive version still is not because there is no credible counterpart in an adaptive context of the performance theories which address the non-adaptive version of the problem. In fact, even just the stabilization question for the adaptive version of the problem did not really get ironed out until ten years ago, except under unrealistic assumptions which ignored the effects of noise and/or un-modeled dynamics. As a first step we briefly discuss the problem of adaptive disturbance rejection. Although the switching logic we consider contains no logic or discrete ∗
This research was supported by the US Army Research Office, the US National Science Foundation and by a gift from the Xerox Corporation.
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event sub-system, the problem nonetheless sets the stage for what follows. One of the things which turns out (in retrospect) to have impeded progress with adaptive control has been the seemingly benign assumption that the parameters of the (nominal) model of the process to be controlled are from a continuum of possible values. In Chap. 4 we briefly discuss the unpleasant consequences of this assumption and outline some preliminary ideas which might be used to deal with them. In Chap. 5 we turn to detailed discussion of a switched adaptive controller capable of causing the output of an imprecisely modeled process to approach and track a constant reference signal. The material in this chapter provides a clear example of what is meant by a logically switched dynamical system. Finally in Chap. 6 we consider several switched dynamical systems which model the behavior of a group of mobile autonomous agents moving in a rapidly
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