The scenario approach to robust control design

• Published in: IEEE transactions on automatic control Vol. 51; no. 5; pp. 742 - 753
• Main Authors: Calafiore, G.C., Campi, M.C.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.05.2006; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive control; Applied sciences; Computer science; control theory; systems; Constraint optimization; Control system synthesis; Control theory. Systems; Design engineering; Exact sciences and technology; Linear matrix inequalities; Lyapunov method; Measurement standards; Optimization; Polynomials; Probabilistic methods; Probabilistic robustness; Probability theory; randomized algorithms; Robust control; robust convex optimization; Robustness; Sampling; Sampling methods; Studies; Uncertainty; Control system analysis; Probabilistic approach; Control synthesis; Minimization; Randomized algorithm; uncertainty; Convex programming; Probabilistic robustness; Polynomial time; robust convex optimization; Robust control; Uncertain system; randomized algorithms; NP hard problem; Linear matrix inequality; Robustness; Sampling
• DOI: 10.1109/TAC.2006.875041
• ISSN: 0018-9286

This paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms. This includes the wide class of NP-hard control problems representable by means of parameter-dependent linear matrix inequalities (LMIs). It is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem (the scenario problem) whose solution is approximately feasible for the original (usually infinite) set of constraints, i.e., the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased. We provide an explicit and efficient bound on the number of samples required to attain a-priori specified levels of probabilistic guarantee of robustness. A rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial-time solution, if robustness is intended in the proposed risk-adjusted sense.