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^TPx, 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 ε, the controller gains which are required can be much smaller than their counterparts obtained via classical robust theory.