Realization theory and quadratic optimal controllers for systems defined over Banach and Frechét algebras

Abstract

If a system (A,B,C) depends in a bounded, measurable, continuous, smooth or even analytic manner on plant parameters, then a very important and natural problem to study is the existence of a feedback law F, which is a switching gain in the sense that F depends on (A,B,C) and in this way depends on the plant parameters, which stabilizes the system for all values of the parameter and which has the same kind of dependence on the paramters. Using the Gel'fand theory for Banach algebras, the author proved such a theorem in the bounded measurable case under reasonable hypothesis deriving this case and the continuous case from the analytic case by use of the operational calculus and earlier work on systems with parameters. Now, recent work by E. Kamen on the stability of half-plane digital filters shows that similar problems for other Banach algebras also arise in applications. This situation makes the construction of a realization theory for and of stabilizability criteria for systems defined over a Banach or Frechet algebra A desirable, and in this paper we develop such a theory with special emphasis on the construction of finitely-generated realizations, the existence of coprime factorizations for T(s) defined over A, and solvability of the quadratic optimal control problem and the associated algebraic Riccati equation over A.