A unified approach to fixed‐order controller design via linear matrix inequalities
Tetsuya Iwasaki, Robert E. Skelton · Mathematical Problems in Engineering · 1995
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,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 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.