A convex parameterization of risk-adjusted stabilizing controllers

The focal point of this paper is the synthesis of controllers under risk-specifications. In recent years there has been a growing interest in the development of techniques for controller design where, instead of requiring that the performance specifications are met for every possible value of admissible uncertainty, it is required that the risk of performance violation is below a small well defined risk level. In contrast to previous work, where the search for the controller gains is done using randomized algorithms, the results in this paper indicate that for a class of uncertain linear time invariant systems, the search for the “risk adjusted” controller can be done efficiently using deterministic algorithms. More precisely, for the case when the characteristic polynomial of the closed loop system depends affinely on the uncertainty, we provide a convex parameterization of “risk-adjusted” stabilizing controllers. These results are also extended to the case where the distribution of the uncertainty vector q is not available. The only assumption is that the distribution belongs to a class of distributions ℱ motivated by physical reasoning