Abstract

Being one of the most fundamental concepts of systems and control theory, the controllability concept has been extensively studied, ever since conceived by Kalman, in various contexts including linear systems, nonlinear systems, infinite-dimensional systems, positive systems, switching systems, hybrid systems, and behavioral systems. Easily verifiable tests for global controllability have been hard to obtain with the exception of the classical results on finite-dimensional linear systems. In fact, even in the framework of smooth nonlinear systems, results on controllability are local in nature and there is no hope to come up with general algebraic characterizations of global controllability. Indeed, the problem of characterizing controllability for some classes of systems fall into the most undesirable category of problems from computational complexity point of view, namely undecidable problems. A remarkable example of such system classes is the so-called sign-systems which are the simplest instances of piecewise affine dynamical systems.

A piecewise affine dynamical system is a finite-dimensional nonlinear input/state/output dynamical system with the distinguishing feature that the functions representing the systems differential equations are piecewise affine functions. Any piecewise affine system can be considered as a collection of ordinary finite-dimensional linear input/state/output systems, together with a partition of the product of the state space and input space into polyhedral regions. Each of these regions is associated with one particular linear system from the collection. Depending on the region in which the state and input vector are contained at a certain time, the dynamics is governed by the linear system associated with that region. Thus, the dynamics switches if the state-input vector changes from one polyhedral region to another. Any piecewise affine systems is therefore also a hybrid system, i.e., a dynamical system whose time evolution is governed both by continuous as well as discrete dynamics.

In this talk, we investigate controllability and stabilizability conditions for continuous piecewise affine dynamical systems. Although every piecewise affine system is a nonlinear system, none of the existing results for smooth nonlinear systems can be applied to piecewise affine systems because of the lack of smoothness. The main results of this talk are algebraic necessary and sufficient conditions for controllability and stabilizability of continuous piecewise affine systems. These conditions are much akin to classical Popov-Belevitch-Hautus controllability/stabilizability conditions.