Optimization over state feedback policies for robust control with constraints

Abstract

This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.

A tractable approximation of chance constrained stochastic MPC based on affine disturbance feedback

“…The authors propose to have the control policy parameterized as an affine function of the disturbances, which leads to a convex set of feasible decision variables. This affine disturbance feedback parametrization is shown to be equivalent to the affine state feedback parametrization in [7] in the sense that is leads to the same control inputs. In [1], [9] and [7] bounded disturbances are assumed, whereas in [17] stochastic disturbances are considered.…”

“…This affine disturbance feedback parametrization is shown to be equivalent to the affine state feedback parametrization in [7] in the sense that is leads to the same control inputs. In [1], [9] and [7] bounded disturbances are assumed, whereas in [17] stochastic disturbances are considered. Unfortunately, with the method in [17] the problem that was originally an LP is turned into a second order cone problem.…”

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

“…The proposed methods could be robustified against additive disturbances using standard methods (Bemporad & Morari, 1999;Goulart, Kerrigan, & Maciejowski, 2006). Robustness is not explicitly included in the control law derivation as it is not a contribution of this paper, and for brevity.…”

Use of model predictive control and weather forecasts for energy efficient building climate control

“…The main benefit of using this affine disturbance feedback formulation is that the resulting optimization problem can be formulated as a convex problem, so that it can be solved using common commercial codes while providing a very good performance [17,18].…”