17. 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 (or simply, a controller), \(K\), so that a family of real polynomials, affine in the parameters of the controllers, is Schur. An attractive feature of the paper is the systematic approximation of the set of all such stabilizing controllers, \(K\). This approximation is accomplished through the exploitation of the interlacing property of Schur polynomials and a systematic construction 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. We also show a related result, namely, that the set of rational proper stabilizing controllers for single-input single-output linear time invariant discrete-time plants will form a bounded set in the controller parameter space if and only if the order of the stabilizing cannot be reduced any further. Moreover, if the order of the controller is increased, the set of higher order controllers will necessarily be unbounded.