A unified approach to fixed order controller design via linear matrix inequalities

Considers 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, Q-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 obtains analytical solutions to 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 given, and all feasible controllers are parametrized explicitly. In any case, the resulting computational problem is shown to be a search for a (structured) positive definite matrix X such that X∈C₁ and X^-1∈C₁ where C₁ and C₂ are convex sets defined by LMIs.