Probabilistic controller analysis and synthesis for quadratic performance: the method of HPD inscription

Probabilistic control of LTI plants for quadratic performance is considered. Here the plant is subject to parametric uncertainty such that the uncertain plant parameters are jointly Gaussian. The method proposed here $the Highest Posterior Density (HPD) inscription method - is based on the fact that stability and quadratic performance impose linear constraints on the eigenvalues of the solution of a Lyapunov equation. The HPD inscription method uses two approximations to evaluate the probability of stability and performance. First, the dependence of the Lyapunov eigenvalues with respect to the uncertain plant parameters is linearized. Since the latter are assumed Gaussian, the linearized Lyapunov eigenvalues are also Gaussian, and have ellipsoidal HPD regions. The second approximation is to use for probability of stability and performance the contents of the largest HPD ellipsoid of linearized Lyapunov eigenvalues that can be inscribed in the convex stability/performance region. Results are presented for control analysis and design in the case of state feedback. These results are illustrated on a simple example.