A Behavioral Approach to Control of Distributed Systems

Abstract

This paper develops a theory of control for distributed systems (i.e., those defined by systems of constant coefficient partial differential operators) via the behavioral approach of Willems. The study here is algebraic in the sense that it relates behaviors of distributed systems to submodules of free modules over the polynomial ring in several indeterminates. As in the lumped case, behaviors of distributed ARMA systems can be reduced to AR behaviors. This paper first studies the notion of AR controllable distributed systems following the corresponding definition for lumped systems due to Willems. It shows that, as in the lumped case, the class of controllable AR systems is precisely the class of MA systems. It then shows that controllable 2-D distributed systems are necessarily given by free submodules, whereas this is not the case for n-D distributed systems, n ≥ 3. This therefore points out an important difference between these two cases. This paper then defines two notions of autonomous distributed systems which mimic different properties of lumped autonomous systems. Control is the process of restricting a behavior to a specific desirable autonomous subbehavior. A notion of stability generalizing bounded input-bounded output stability of lumped systems is proposed and the pole placement problem is defined for distributed systems. This paper then solves this problem for a class of distributed behaviors.

OreModules: A Symbolic Package for the Study of Multidimensional Linear Systems

“…Let us summarize some of them in Table 1. We refer the reader to [15,28,26,29,31,32,33,34,37,38,47,49] for the precise definitions of the properties listed in the second and third column of Table 1.…”

“…• Check whether or not a multidimensional linear system is controllable in the sense of [14,15,26,30,29,31,32,47,49] or compute the autonomous elements of the system [30,31,32,47,49]. …”

“…Based on U. Oberst's ideas, the behavioural approach to multidimensional linear systems has been successfully developed in the recent years. See [30,29,36,47,49] and the references therein.…”

Behavioral Systems Theory: A Survey

“…In (Pillai and Shankar, 1999), we find a neat analytic interpretation of controllability in the multidimensional continuous case: B is controllable (i.e., it has an image representation) if and only if for all w 1 , w 2 ∈ B and all open sets U 1 , U 2 ⊂ R n whose closures are disjoint there exists w ∈ B such that…”

Time-relevant 2D behaviors

“…Systems described by linear constant coefficient PDE's or partial difference equations have received a great deal of attention in the last decades, and many interesting results have been obtained using the behavioral approach, see for instance [29], [20], [31]. Most of the contributions concern the case where all the independent variables are treated on an equal footing.…”