Segregated Linear Decision Rules for Distributionally Robust Control With Linear Dynamics and Quadratic Cost

Abstract

In this article, we investigate the multistage stochastic constrained control problem with linear dynamics and quadratic costs when only partial information of the disturbance distribution (i.e., the first two moments) is known. We adopt a distributionally robust chance constraint (DRCC)-based approach, where the DRCC holds (with high probability) as long as the true distribution of uncertainty belongs to a distribution family or the ambiguity set (constructed based on the known information). We approximate the DRCC with the worst-case conditional value-at-risk (CVaR) constraint, which bounds the expected constraint violation with respect to all distributions in the ambiguity set. The worst-case CVaR problem is known to be computationally tractable under linear decision rules (LDRs). To improve the performance of LDRs, we propose to apply the segregated linear decision rules (SLDRs) on dynamical control systems with the worst-case CVaR approximation. To deal with the tractability issue of the worst-case CVaR constraint, we construct a special group of segregation (of the random disturbance) that is shown to be without loss of optimality. The proposed segregation method enables us to establish the equivalence between the stochastic control problem (with worst-case CVaR constraints and under SLDRs) and a tractable semidefinite program.