Improved Sample Size Bounds for Probabilistic Robust Control Design: A Pack-Based Strategy T. Alamo a , R. Tempo b and E.F. Camacho a Abstract— This paper deals with probabilistic methods and randomized algorithms for robust control design. The main contribution is to introduce a new technique, denoted as “pack- based strategy”. When combined with recent results available in the literature, this technique leads to significant improvements in terms of sample size reduction. One of the main results is to show that for fixed confidence δ, the required sample size increases as 1/ǫ, where ǫ denotes the guaranteed accuracy. Using this technique for non-convex optimization problems involving Boolean expressions consisting of polynomials, we prove that the number of required samples grows with the accuracy parameter ǫ as 1 ǫ ln 1 ǫ . Index Terms— Probabilistic robustness, Randomized algo- rithms, Robust control, Robust convex optimization I. I NTRODUCTION Uncertainty randomization is now widely accepted as an effective tool in dealing with control problems which are computationally difficult, see e.g. [14]. In particular, regarding synthesis of a controller to achieve a given perfor- mance, two complementary approaches, sequential and non- sequential, have been proposed in recent years. For sequential methods, the resulting algorithms are based on stochastic gradient [8], [9], [12] or ellipsoid iterations [10]; see also [3], [7] for other classes of sequential algo- rithms. Convergence properties in finite-time are in fact one of the focal points of these papers. Various control problems have been solved using these sequential randomized algo- rithms, including robust LQ regulators and uncertain Linear Matrix Inequalities. A classic approach for non-sequential methods is based upon statistical learning theory, see [15] and [16] for further details. In particular, the use of this theory for feedback design of uncertain systems has been initiated in [17]; subsequent work along this direction include [18] and [11]. However, the sample size bounds derived in these papers, which guarantee that the obtained solution meets a given probabilistic specification, may be too conservative for being practically useful in a systems and control context. For non-sequential methods, a novel and successful paradigm, denoted as the scenario approach, has been introduced in [5], [6]. In this approach, the original robust control problem is reformulated in terms of convex optimization with sampled constraints which are randomly generated. a Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 Sevilla. Spain. e-mail: {alamo,eduardo}@cartuja.us.es b IEIIT-CNR, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy. e-mail: roberto.tempo@polito.it The main result of the aforementioned papers is to derive an explicit bound on the number of randomly generated con- straints which guarantees a required probabilistic accuracy of the resulting solution. This paper is focused on non-sequential methods. A new and simple technique, denoted as “pack-based strategy” is introduced. This technique is used to derive improved bounds for the scenario approach. We also present a result, which is based on this “pack-based strategy” for non-convex feasibility and optimization problems. II. NOTATION AND PROBLEM STATEMENT In this paper, L, M , N and n θ represent positive integers. Given a sample space W, a collection of N i.i.d. samples w= {w (1) ,...,w (N) } is said to belong to the set W N = W×···×W (N times). For x ∈ IR, x> 0, ⌈x⌉ is the minimum integer greater than or equal to x, ln (·) is the natural logarithm and e is the Euler number. We consider an optimization problem affected by uncer- tain parameters w bounded in the set W. In addition, the design parameters are parameterized by means of a vector of decision variables θ, denominated “design parameter” and restricted to the set Θ ⊆ IR n θ . In many situations, the robust synthesis problem can be recast as the following robustness problem: min θ∈Θ c ⊤ θ subject to f (θ, w) ≤ 0, for all w ∈W, (1) where f (θ, w) : Θ ×W → [−∞, ∞] is a measurable function. Unfortunately, many classical robustness problems can be solved efficiently (i.e., in polynomial-time) only under rather strong assumptions, for example, if f (θ,w) is convex in θ for all elements w ∈W and W has finite cardinality. In this paper, we consider problems in which the function f is convex in θ and W may have infinite cardinality. The convexity assumption is now stated precisely. Assumption 1: [convexity] Let Θ ⊂ IR n θ be a convex and closed set, and let W⊆ IR nw . We assume that f (θ,w): Θ ×W→ [−∞, ∞] is convex in θ for any fixed value of w ∈W. Given θ ∈ Θ, checking if f (θ, w) ≤ 0, for all w ∈W is often an NP-hard problem [4]. Thus, it is of interest to consider the notion of probability of violation. To this end, we assume that the vector w is random with a given probability measure Pr W over the support set W. Definition 1: [probability of violation] Consider a proba- bility measure Pr W over W and let θ ∈ Θ be given. The Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007 FrC11.4 1-4244-1498-9/07/$25.00 ©2007 IEEE. 6178