Synthesis of Fixed Structure controllers for discrete time systems

In this paper, we develop a linear programming approach to the synthesis of stabilizing fixed structure controllers for a class of linear time invariant discrete-time systems. The stabilization of this class of systems requires the determination of a real controller parameter vector, $K$, so that a family of real polynomials, affine in the parameters of the controllers, is Schur. An attractive feature of the paper is the exploitation of the interlacing property of Schur polynomials (based on the characterization in terms of Tchebyshev polynomials) to systematically generate an arbitrarily large large number of 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 $P(z,K)$ Schur. Illustrative examples are provided to show the applicability of the proposed methodology.