A subspace approach to linear dynamical systems

Abstract

In this paper we present an approach to linear dynamical systems which combines the positive features of two well known formulations, namely, standard state space theory (see for e.g., Wonham, 1978 [9]) and behavioural systems theory (see Polderman and Willems, 1997 [4]). Our development is also ‘geometric’ in the tradition of Wonham and others. But, instead of using explicit linear maps, we work with linear relations implicitly which amounts to working with subspaces. One of our primary motivations is computational efficiency—all our computations can be performed on the system as it is without elimination of variables and further (unlike the ‘behaviourists’ who manipulate matrices with polynomial entries) we work only with real matrices.Using our formulation we derive the standard vector space results on controlled and conditioned invariant subspaces of linear dynamical systems. Duality, which is a distinctive feature of state space theory but not of the behavioural view point, comes out naturally in our approach too through the use of the adjoint. We illustrate our ideas for an important class of dynamical systems viz., electrical networks.The theory proposed in this paper gives a unified description of both the standard linear dynamical systems and the linear singular systems (or the linear descriptor systems) (see for e.g., F. Gantmacher 1959 [1] and Lewis, 1986 [2]). Therefore, the algorithms described for the invariant spaces in this paper are also applicable to linear singular systems.