Linear Controller Design for Chance Constrained Systems

TL;DR

This paper presents a convex SDP-based method for designing stationary linear controllers that optimize long-term quadratic costs while satisfying chance constraints on the system's steady-state behavior, accommodating Gaussian disturbances.

Contribution

It introduces a novel approach to stationary chance constrained control design using convex SDP with linear matrix inequalities, extending to output feedback.

Findings

Controller synthesis as convex SDP with LMIs.

Applicable to both single and joint chance constraints.

Method effective for Gaussian disturbances and stationary systems.

Abstract

This paper is concerned with the design of a linear control law for linear systems with stationary additive disturbances. The objective is to find a state feedback gain that minimizes a quadratic stage cost function, while observing chance constraints on the input and/or the state. Unlike most of the previous literature, the chance constraints (and the stage cost) are not considered on each input/state of the transient response. Instead, they refer to the input/state of the closed-loop system in its stationary mode of operation. Hence the control is optimized for a long-run, rather than a finite-horizon operation. The controller synthesis can be cast as a convex semi-definite program (SDP). The chance constraints appear as linear matrix inequalities. Both single chance constraints (SCCs) and joint chance constraints (JCCs) on the input and/or the state can be included. If the disturbance is Gaussian, additionally to WSS, this information can be used to improve the controller design. The presented approach can also be extended to the case of output feedback. The entire design procedure is flexible and easy to implement, as demonstrated on a short illustrative example.