On convexity of the probabilistic design problem for quadratic stabilizability

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 the paper is as follows: With nominally determined quadratic Lyapunov function V(x)=x/sup T/P, the set of controller gains /spl Kscr//sub /spl epsiv// guaranteeing quadratic Lyapunov instability risk level 0/spl les//spl epsiv//spl les/1 or less is convex. Hence, the 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 /spl epsiv/, the controller gains which are required can be much smaller than their counterpart obtained via classical robustness theory.