Mathematical Problems in Engineering
Volume 1 (1995), Issue 1, Pages 59-75

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

T. Iwasaki and R. E. Skelton

Space Systems Control Laboratory, Purdue University, West Lafayette 47907-1293, IN, USA

Received 25 April 1994

Copyright © 1995 T. Iwasaki and R. E. Skelton. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 X𝒞1 and X1𝒞2 where 𝒞1 and 𝒞2 are convex sets defined by LMIs. Computational aspects of the nonconvex LMI problem are discussed.