Tractable Stochastic Model Predictive Control using Conditional Value at Risk Optimization

Abstract

A numerically tractable Stochastic Model Predictive Control (SMPC) strategy using Conditional Value at Risk (CVaR) optimization for discrete-time linear time-invariant systems, with state and input constraints, subject to additive uncertainty, is presented. SMPC strategies make use of the probabilistic description of uncertainty to define chance constraints which allow a certain admissible level of constraint violation. SMPC strategies require the initial state of a system to be within a particular set, referred to as feasibility set, probabilistically, such that the derived control input, when applied to the system, gives rise to states that are also within the feasibility set satisfying all chance constraints on the system. This leads to recursive feasibility of the SMPC strategy. Such strategies are restrictive in nature when the uncertainty in the system is unbounded, as in the case of White Gaussian noise. In such a case, the feasibility set is very small and leads to a strategy that is very conservative. To reduce this conservatism, some constraint violations are permitted. However, such violations affect the closed-loop behaviour of the system leading to performance degradation. This performance degradation can be quantified as a penalty on the system for violating constraints, and intuitively, it can be thought of as a risk taken by the system in that undesirable state. An approach following the exact penalty method is proposed using the CVaR function to determine the penalty cost. The same optimal solution as the original constrained problem is obtained from a single unconstrained minimization. Since accurate computation of the expected value of risk using the CVaR function is not possible, a scenario-based approximation of the CVaR is used to obtain an overall tractable and computationally efficient SMPC strategy. An extensive simulation study of the double integrator system is provided to present the functionality of the proposed method.