Improved sample size bounds for probabilistic robust control design: A pack-based strategy

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 delta, the required sample size increases as 1/isin, where isin 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 isin as 1/isin In 1/isin.

[1]  Giuseppe Carlo Calafiore,et al.  The scenario approach to robust control design , 2006, IEEE Transactions on Automatic Control.

[2]  F. Dabbene,et al.  An Iterative Localization Method for Probabilistic Feasibility of Uncertain LMIs , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[3]  Dmitry Panchenko,et al.  Improved sample complexity estimates for statistical learning control of uncertain systems , 2000, IEEE Trans. Autom. Control..

[4]  R. Tempo,et al.  Randomized Algorithms for Analysis and Control of Uncertain Systems , 2004 .

[5]  T. Alamo,et al.  A sequentially optimal randomized algorithm for robust LMI feasibility problems , 2007, 2007 European Control Conference (ECC).

[6]  Eduardo F. Camacho,et al.  Statistical Learning Theory: A Pack-based Strategy for Uncertain Feasibility and Optimization Problems , 2008, Recent Advances in Learning and Control.

[7]  Michel Verhaegen,et al.  An ellipsoid algorithm for probabilistic robust controller design , 2003, Syst. Control. Lett..

[8]  Giuseppe Carlo Calafiore,et al.  Stochastic algorithms for exact and approximate feasibility of robust LMIs , 2001, IEEE Trans. Autom. Control..

[9]  Roberto Tempo,et al.  Probabilistic robust design with linear quadratic regulators , 2001, Syst. Control. Lett..

[10]  R. Tempo,et al.  Probabilistic robustness analysis: explicit bounds for the minimum number of samples , 1997 .

[11]  Vincent D. Blondel,et al.  Probabilistic solutions to some NP-hard matrix problems , 2001, Autom..

[12]  Mathukumalli Vidyasagar,et al.  Statistical learning theory and randomized algorithms for control , 1998 .

[13]  Mathukumalli Vidyasagar,et al.  A Theory of Learning and Generalization: With Applications to Neural Networks and Control Systems , 1997 .

[14]  Giuseppe Carlo Calafiore,et al.  Uncertain convex programs: randomized solutions and confidence levels , 2005, Math. Program..

[15]  T. Alamo,et al.  Revisiting statistical learning theory for uncertain feasibility and optimization problems , 2007, 2007 46th IEEE Conference on Decision and Control.

[16]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[17]  Roberto Tempo,et al.  Probabilistic design of LPV control systems , 2003, Autom..

[18]  John N. Tsitsiklis,et al.  A survey of computational complexity results in systems and control , 2000, Autom..