Linear systems with chance constraints: Constraint-admissible set and applications in predictive control

Maximal constraint-admissible sets have been widely used in the study of linear systems with hard constraints. This paper proposes a generalization of the maximal constraint-admissible set to the case where chance or probabilistic constraints are present in a linear system. Properties of the probabilistic constraint-admissible set are discussed and it is shown that the maximal chance constraint-admissible set is not time invariant. An inner approximation to the maximal set is then proposed to ensure its invariance property. This approximate set is then applied in the design of a model predictive controller for a linear system with additive disturbances and chance constraints. Feasibility and stability of the resultant closed-loop system are discussed.

I. Introduction

Constraint-admissible invariant (CAI) set plays an important role in the study of constrained systems. These sets have been used in many approaches for the control of constrained systems. For example, CAI sets are used as terminal sets in Model Predictive Control (MPC) [1], [2], [3], [4], [5], [6], they are also used to characterize the domain of attractions of nonlinear control law [7], [8] for system with varying set point. Other uses of CAI set also exist [9], [10], [11].