The Linear Quadratic Regulator with Chance Constraints

Abstract

This paper is concerned with the design of linear state feedback control laws for linear systems with additive Gaussian disturbances. The objective is to find the feedback gain that minimizes a quadratic cost function in closed-loop operation, while observing chance constraints on the input and/or the state. It is shown that this problem can be cast as a semi-definite program (SDP), in which the chance constraints appear as linear or bilinear matrix inequalities. Both individual chance constraints (ICCs) and joint chance constraints (JCCs) can be considered. In the case of ICCs only, the resulting SDP is linear and can be solved efficiently as a convex optimization program. In the presence of JCCs the SDP becomes bilinear, however it can still be solved efficiently by an iterative algorithm, at least to a local optimum. The application of the method is demonstrated for several numerical examples, underscoring its flexibility and ease of implementation.