Consistent Approximations of the Zeno Behaviour in Affine-Type Switched Dynamic Systems

Vadim Azhmyakov

doi:10.1155/2016/2091526

Abstract

This paper proposes a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno behaviour. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real world processes do not present the corresponding behaviour, mathematical models of many engineering systems may be Zeno due to the used formal abstraction. We propose two useful approximative approaches to the Zeno dynamics, namely, an analytic technique and a variational description of this phenomenon. A generic trajectory associated with the Zeno dynamics can finally be characterized as a result of a specific projection or/and an optimization procedure applied to the original dynamic model. The obtained analytic and variational techniques provide an effective methodology for constructive approximations of the general Zeno-type behaviour. We also discuss shortly some possible applications of the proposed approximation schemes.

Highlights

The study of switched and general hybrid dynamic systems has gained lots of interest in recent years
This contribution initiates theoretic investigations devoted to the constructive approximations of the Zeno dynamics in switched systems
The abstract idea and the proposed approximative approach are based on some novel analytic techniques and involve a specific application of the optimal control methodology

Summary

Introduction

The study of switched and general hybrid dynamic systems has gained lots of interest in recent years (see, e.g., [1,2,3,4,5,6,7,8,9,10,11]). In the face of a recent progress in switched systems theory, there are numbers of fundamental properties of these systems that have not been investigated in sufficient detail Among others, these include a formal description of the so-called Zeno executions (see [2, 4, 10, 18,19,20,21,22]). An analytic approach based on the modern variational analysis and consistent approximation techniques have not been sufficiently advanced to the Zeno switched systems setting and to the corresponding control design procedures. In this connection we mention shortly the sliding mode control processes.

Zeno Executions in Affine Switched Systems

Consistent Approximations of the Zeno Dynamics

The Zeno-Type Dynamics Determined by a Switching Manifold

Approximations of the Zeno Dynamics Involving the Optimal Control Methodology

Concluding Remarks