Application of probabilistically constrained linear programs to risk-adjusted controller design

The focal point of this paper is the Probabilistically Constrained Linear Program (PCLP) and how it can be applied to control system design under risk constraints. The PCLP is the counterpart of the classical linear program, where it is assumed that there is random uncertainty in the constraints and, therefore, the deterministic constraints are replaced by probabilistic ones. It is shown that for a wide class of distributions, called log-concave symmetric distributions, the PCLP is a convex program. A deterministic equivalent of the PCLP is presented which provides insight on numerical implementation. Finally, this concept is applied to control system design. It is shown how the PCLP can be applied to the design of controllers for discrete-time systems to obtain a closed loop system with a well-defined risk of violating the so-called property of super stability. Furthermore, we address the problem of risk-adjusted pole placement.