## A unified approach to fixed-order controller design via linear matrix inequalities.(English)Zbl 0918.93014

Summary: We consider the design of fixed-order (lor 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, $${\mathcal Q}$$-stabilization as a robust stabilization problem, and robust $$L_\infty$$ 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 $$X\in{\mathcal C}_1$$ and $$X^{-1}\in{\mathcal C}_2$$ where $${\mathcal C}_1$$ and $${\mathcal C}_2$$ are convex sets defined by LMIs. Computational aspects of the nonconvex LMI problem are discussed.

### MSC:

 93B51 Design techniques (robust design, computer-aided design, etc.) 93B35 Sensitivity (robustness) 93D21 Adaptive or robust stabilization 15A39 Linear inequalities of matrices
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