An efficient method to estimate the suboptimality of affine controllers

Abstract

Abstract. We consider robust feedback control of time-varying, linear discrete-time systems operating over a finite horizon. For such systems, we consider the problem of designing robust causal controllers that minimize the expected value of a convex quadratic cost function, subject to mixed linear state and input constraints. Determination of an optimal control policy for such problems is generally computationally intractable, but suboptimal policies can be computed by restricting the class of admissible policies to be affine on the observation. By using a suitable re-parameterization and robust optimization techniques, these approximations can be solved efficiently as convex optimization problems. We investigate the loss of optimality due to the use of such affine policies. Using duality arguments and by imposing an affine structure on the dual variables, we provide an efficient method to estimate a lower bound on the value of the optimal cost function for any causal policy, by solving a cone program whose size is a polynomial function of the problem data. This lower bound can then be used to quantify the loss of optimality incurred by the affine policy.

A Convex Information Relaxation for Constrained Decentralized Control Design Problems

“…This proof extends arguments originally developed in [32,Lem 4.4] to accommodate the more general setting considered in this note, where the affine controller Q is subject to a decentralized information constraint.…”

“…Problem (15) Equipped with this definition, we state the following result, which provides a finite-dimensional relaxation of problem (15) as a conic program. We note that the proposed conic relaxation is largely inspired by the duality-based relaxation methods originally developed in the context of centralized control design problems [30], [32]. We provide a proof of Theorem 2 in Appendix A, which extends these techniques to accommodate the added complexity of decentralized information constraints on the controller.…”

Robust Multi-Period OPF With Storage and Renewables

“…The solution of the AARC is expected to be suboptimal relative to the two-stage robust solution with complete recourse. However, research in [33] and [34] suggests that in some cases the affinely adjustable solution is indeed optimal, and experience with power system dispatch problems is encouraging [26]. Recent research on generalized decision rules [35] aims to reduce sub-optimality as compared to the robust solution via affine policies; however, its practical applicability to multistage power system optimization problems is yet to be determined.…”

Optimality of robust disturbance-feedback strategies

“…In the case that the objective function is convex in the disturbance, the disturbance set is polytopic, and the disturbance enter the system additively; the exact optimal robust feedback strategy can be determined by an enumerative, brute force method [3], which becomes computationally intractable when the time horizon of the problem is increased. Note that [4][5][6] have tackled similar problems in the past. However, those approaches require the strong assumption that the objective function is concave in the disturbances, while the assumptions in our paper are rather mild.…”