A linear programming approach to the synthesis of fixed structure controllers

Many practical applications require the design of fixed order and structure feedback controllers. A broad class of fixed structure controller synthesis problems can be reduced to the determination of a real controller parameter vector (or simply, a controller), K = (k/sub 1/, k/sub 2/, ...k/sub l/), so that a given set of real or complex polynomials of the form P(s, K) := P/sub o/(s)+ k/sub 1/P/sub 1/(S) +... + k/sub l/P/sub l/(s) is Hurwitz. The stability of the closed loop system is addressed by requiring a real polynomial to be Hurwitz, while several performance criteria can be addressed by requiring a complex polynomial to be Hurwitz. We consider only real polynomials, P(S, K); the extension to complex polynomials is direct. A novel feature of this paper is the exploitation of the interlacing property (IP) of Hurwitz polynomials to synthesis, by systematically generating sets of linear inequalities in K. The union of the feasible sets of linear inequalities provides an approximation of the set of all controllers, K, which render these P(s, K) Hurwitz. We show that this approximation can be made as accurate as desired. The main tools that are used in the construction of the sets of linear inequalities are the Hermite-Biehler theorem, Descartes' rule of signs and its generalization due to Poincare. We provide examples of the applicability of the proposed methodology to the synthesis and design of fixed order stabilizing controllers.