A Unified Approach to Fixed-Order Controller Design via Linear Matrix Inequalities

Abstract

We consider the design of fixed-order (or low-order) linear controllers which meet certain performance and/or robustness specifications. The following three problems are considered; covariance control as a nominal performance problem, 2-stabilization as a robust stabilization problem, and robust L∞ control problem as a robust performance problem. All three control problems are converted to a single linear algebra problem of solving a linear matrix inequality (LMI) of the type BGC + (BGC)T + Q < 0 for the unknown matrix G. Thus this paper addresses the fixed-order controller design problem in a unified way. Necessary and sufficient conditions for the existence of a fixed-order controller which satisfies the design specifications for each problem are derived, and an explicit controller formula is given. In any case, the resulting problem is shown to be a search for a (structured) positive definite matrix X such that X ∊C1, and X-1 ∊ C2 where C1 and C2 are convex sets defined by LMIs. Computational aspects of the nonconvex LMI problem are discussed. © 1995, OPA (Overseas Publishers Association).