Many worst-case robust control problems cannot be solved due to computational intractability.
In this talk, a new probabilistic solution framework is proposed for robust control analysis and synthesis problems that can be expressed in the form of robust convex optimization. This includes for instance the wide class of NP-hard control problems representable by means of parameter-dependent linear matrix inequalities (LMIs).
By appropriate sampling of the constraints, one obtains a convex optimization problem (the scenario problem) that can be easily solved through standard optimization techniques. The solution of the scenario problem 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. Explicit and efficient bounds on the number of samples required to attain a-priori specified levels of probabilistic guarantee of robustness are given.
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.
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