The Scenario Approach to Robust Control Design

Abstract

Abstract-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.

Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

“…Combined with (12), this implies that for any k ∈ δ B (a ), (|δ B (a )| − 1)µ k = 0, and hence µ k = 0 for all k ∈ F . This same argument applies for all connected components, implying µ k = 0 for all k ∈ F C , and so by (12), λ a = α for all a ∈ C. Next, subtracting (8) for k * from each equation in (11) yields (λ a * − µ k * ) + k ∈δ B (a * ) µ k − µ k = 0, and hence µ k = 0 for all k ∈ N \ (F C ∪ T ) since λ a * − µ k * = 0 and µ k = 0 for all k ∈ δ B (a * ).…”

“…The SAA problem can also be used to derive statistical confidence intervals on the optimal value of the true problem [32]. Additional results on SAA for chance constraints can be found in [11,12,13]. To formulate (1) as an integer program, we introduce binary variables x a for a ∈ A, where x a = 1 if and only if arc a is selected.…”

Cutting-set methods for robust convex optimization with pessimizing oracles

“…These were followed by many results for specific problem classes or applications; see, e.g., the survey [22]; examples include robust linear programs [2,23,24], robust least-squares [25,26], robust quadratically constrained programs [27], robust semidefinite programs [28], robust conic programming [29], robust discrete optimization [30]. Work focused on specific applications includes robust control [31,32], robust portfolio optimization [33][34][35][36], robust beamforming [37][38][39], robust machine learning [40], and many others.…”

Fixed-order H∞ controller design for nonparametric models by convex optimization

“…Therefore, it is more reasonable to use a randomized approach to solve the SIP. According to the results of [23,24] with a reasonable number N of randomly chosen frequency samples, the optimal solution ρ * to the convex optimization problem will satisfy the constraints for all frequencies with a high probability level. In order to be more precise, let the violation probability V (ρ * ) be defined as the probability that for ω 0 ∈ R the convex constraints are not satisfied for ρ * .…”