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H∞-control with regional pole constraints

In this paper, the problem of static state feedback H∞ control with various regionl stability constraints for the closed-loop system is considered. A general framework for solving this problem is established using the Lagrange multiplier method in conjunction with the generalized Lyapunov theory and the Riccati equation based H∞ optimization characterization. Necessary conditions are given for the existence of the desired optimal feedback gains. Furthermore, as a simplest regional stability case, the stability degree problem is studied with two different approaches, one an indirect overbounding method which extends the standard H∞ problem formulation with an α-shift (degree of stability) and the other using the direct optimization of the H∞ performance measure with stability degree as a constraint under the general framework. The conservatism of the former approach and the generality of the latter approach are clearly demonstrated. It is also shown that similar problem formulation can be used to address the LQR problem with α-stability without alteration of the original performance measure. Finally the theory is illustrated with examples and the computational issues are discussed.