Frequency-domain tests for robust control designs

In this paper, we explore a new approach to robust control designs, which allows to derive an easy to check frequency-domain test condition for the so-called generalized return difference of robust controllers for several classes of uncertain systems and does not require to solve matrix equations or inequalities typical for these problems. The basis of the approach is necessary and sufficient frequency-domain conditions for a given stabilizing state feedback to be a “locally minimax control”, satisfying a Bellman-Isaacs inequality associated to a certain linear-quadratic differential game. It is shown how robust control designs for nonlinear Lur´e systems with sector bounded uncertainty, linear systems with time-varying norm bounded uncertainty, and multivariable systems with uncertain interconnections reduce to the appropriate problems of the locally minimax control, and generalized return difference conditions for the required controllers are derived.