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**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 |