Stochastic approximation algorithms for risk-adjusted quadratic stability

This paper concentrates on a risk-adjusted version of the well known quadratic stability problem for uncertain linear systems. For a wide class of probability density functions, we provide algorithms for solving the following problem: With nominally determined quadratic Lyapunov function V(x)=x/sup T/Px, find a state feedback gain which maximizes the probability of quadratic stability. This so-called Probabilistic Design Problem has been previously shown to be a convex program. In this paper we provide stochastic approximation algorithms which converge to its solution. It is demonstrated that for small values of the risk probability /spl epsi/, the controller gains which are required can be much smaller than their counterparts obtained via classical robust theory.