Geometric methods for the classification of linear feedback systems

Abstract

The precise determination of the nature and form of system invariants under output feedback is of fundamental interest in classical and modern control theory. An algebraic solution of this problem would entail, for example, a description of the generators and relations of the ring of output feedback invariants. Geometrically, we seek to describe the basic properties of the quotient space whose points consist of feedback equivalence classes of the system, recovering the algebraic theory in terms of the function theory on this space. Naturally, any description should be made in system theoretic terms. In this paper we construct and study, in the scalar case, the quotient manifold for linear systems modulo the full output feedback group using classical algebraic geometry. As a corollary, we obtain explicit generators for the associated ring of invariants which we then interpret in terms of classical control theory, e.g. in terms of root-locus plots. Turning to the question of the existence of continuous canonical forms, we show that these exist globally, only when the system order is odd. In the even case, the obstruction to the existence of globally defined continuous forms is described in terms of the values of the system Cauchy index. This is illustrated for low order examples and certain questions concerning relations among the generators is treated using methods of the Schubert calculus. The complete set of invariants for the full output feedback group is obtained and our results concerning the existence and nonexistence of canonical forms is also compared to related results contained in the literature.