On limitations in suboptimal decentralized control derived from the generalized structured singular value

We study how the maximum/minimum gain of a decentralized or sparse controller over a frequency interval influences the maximum/minimum gain of the corresponding closed-loop transfer function over that interval. In doing so, we derive lower bounds on the best closed-loop performance achievable by decentralized and sparse controllers satisfying additional frequency domain restrictions, and sufficient conditions for a structured controller to guarantee a desired level of closed-loop performance. This results in graphical tests for the controller’s singular value plots. These bounds are obtained by using tools inspired by generalizations of the structured singular value, and can be computed by solving Linear Matrix Inequalities (LMI). We close by applying our results to an active sound control problem.