Quadratic stabilization of uncertain linear systems via fixed-order guaranteed cost controllers

An approach is developed to synthesize a fixed-order dynamic stable controller which quadratically stabilizes a linear plant with norm-bounded time-varying uncertainty and minimizes an upper bound of the quadratic cost functional. First, it is shown that a sufficient condition for the quadratic stability and guaranteed cost bound of the closed-loop system is described in terms of a Riccati inequality. Secondly, the set of all fixed-order controllers is parametrized in terms of the solutions to the decoupled Riccati inequalities. Furthermore, the stability of such controllers can be ensured. This problem of finding the controller gains can be reduced to one of optimization problems subject to inequality constraints, which consists of the convex optimization problem and a special class of nonconvex optimization problems. Therefore, the fixed-order controller gains are evaluated by using two computational approaches. Finally, an illustrative example is demonstrated.