Efficient robust optimization for robust control with constraints

• Published in: Mathematical programming Vol. 114; no. 1; pp. 115 - 147
• Main Authors: Goulart, Paul J., Kerrigan, Eric C., Ralph, Daniel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.07.2008; Springer; Springer Nature B.V
• Subjects: Applied sciences; Calculus of Variations and Optimal Control; Optimization; Codes; Combinatorics; Computer science; control theory; systems; Control system synthesis; Control theory. Systems; Decomposition; Exact sciences and technology; Flows in networks. Combinatorial problems; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimal control; Optimization techniques; Robust control; System theory; Theoretical; Robust control; Receding horizon control; Robust optimization; Constrained control; Optimal control; Predictive control; State feedback; Cost minimization; Optimal policy; Feedback regulation; Matrix factorization; Approximation algorithm; Combinatorial optimization; Linear control; Convex programming; Model predictive control; Control constraint; Finite horizon; Cost function; Quadratic cost; State constraint; Mathematical programming; Discrete time systems; Interior point method; Quadratic programming; Constrained optimization; Closed feedback; Linear system; Function minimization; Hypercube; Primal dual method; Discrete time; Time optimal control; Riccati equation; Non convex analysis
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-007-0096-6

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.