On convexity of the Probabilistic Design Problem for quadratic stabilizability

This paper concentrates on a risk-adjusted version of the well known quadratic stabilization problem for uncertain linear systems. For a wide class of probability density functions and state equation structures for the uncertain parameters, the main result of this paper is as follows: With nominally determined quadratic Lyapunov function V(x) = x^TPx, the set of controller gains K_ε guaranteeing quadratic Lyapunov instability risk level 0≤ε≤1 or less is convex. Hence, this so-called Probabilistic Design Problem reduces to a convex program. One of the ramifications of this result involves the issue of high-gain control. It is demonstrated that for small values of the risk probability ε, the controller gains which are required can be much smaller than their counterpart obtained via classical robustness theory.