A survey of randomized algorithms for control synthesis and performance verification

In this paper, we present an overview of probabilistic techniques based on randomized algorithms for solving ''hard'' problems arising in performance verification and control of complex systems. This area is fairly recent, even though its roots lie in the robustness techniques for handling uncertain control systems developed in the 1980s. In contrast to these deterministic techniques, the main ingredient of the methods discussed in this survey is the use of probabilistic concepts. The introduction of probability and random sampling permits overcoming the fundamental tradeoff between numerical complexity and conservatism that lie at the roots of the worst-case deterministic methodology. The simplicity of implementation of randomized techniques may also help bridging the gap between theory and practical applications.

[1]  Giuseppe Carlo Calafiore,et al.  Randomized algorithms for probabilistic robustness with real and complex structured uncertainty , 2000, IEEE Trans. Autom. Control..

[2]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[3]  Roberto Tempo,et al.  On the Complete Instability of Interval Polynomials , 2007, Proceedings of the 45th IEEE Conference on Decision and Control.

[4]  Pramod P. Khargonekar,et al.  Randomized algorithms for robust control analysis and synthesis have polynomial complexity , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[5]  Giuseppe Carlo Calafiore,et al.  A probabilistic framework for problems with real structured uncertainty in systems and control , 2002, Autom..

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

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

[8]  L. Lorefice,et al.  Robust and Randomized Control Design of Mini-UAVs: The MH1000 Platform , 2007, 2007 American Control Conference.

[9]  G. Zames.  On the metric complexity of casual linear systems: ε -Entropy and ε -Dimension for continuous time , 1979 .

[10]  Yasuaki Oishi,et al.  Polynomial-time algorithms for probabilistic solutions of parameter-dependent linear matrix inequalities , 2007, Autom..

[11]  H. Woxniakowski.  Information-Based Complexity , 1988 .

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

[13]  T. W. Anderson,et al.  An Introduction to Multivariate Statistical Analysis , 1959 .

[14]  R. Tempo.  Robust estimation and filtering in the presence of bounded noise , 1987, 26th IEEE Conference on Decision and Control.

[15]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[16]  Li Qiu,et al.  Topics in Control and its Applications , 1999 .

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

[18]  Y. Oishi,et al.  A Mixed Probabilistic/Deterministic Approach to Fixed Order H∞ Controller Design , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[19]  Giuseppe Carlo Calafiore,et al.  Radial and Uniform Distributions in Vector and Matrix Spaces for Probabilistic Robustness , 1999 .

[20]  Tamer Basar,et al.  Randomized algorithms for quadratic stability of quantized sampled-data systems, , 2004, Autom..

[21]  Henryk Wozniakowski,et al.  A general theory of optimal algorithms , 1980, ACM monograph series.

[22]  J. Lamperti.  ON CONVERGENCE OF STOCHASTIC PROCESSES , 1962 .

[23]  R. Tempo,et al.  Probabilistic robust design of LPV control systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[24]  Robert F. Stengel,et al.  Some Effects of Parameter Variations on the Lateral-Directional Stability of Aircraft , 1980 .

[25]  Yasumasa Fujisaki,et al.  Guaranteed cost regulator design: A probabilistic solution and a randomized algorithm , 2007, Autom..

[26]  Fabrizio Dabbene,et al.  Control design with hard/soft performance specifications: a Q-parameter randomization approach , 2004 .

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

[28]  Henryk Wozniakowski,et al.  Information-based complexity , 1987, Nature.

[29]  T. Başar.  Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses , 2001 .

[30]  B. Ross Barmish,et al.  The uniform distribution: A rigorous justification for its use in robustness analysis , 1996, Math. Control. Signals Syst..

[31]  Mathukumalli Vidyasagar,et al.  Learning and Generalization: With Applications to Neural Networks , 2002 .

[32]  Hidenori Kimura,et al.  Computational complexity of randomized algorithms for solving parameter-dependent linear matrix inequalities , 2003, Autom..

[33]  Tansu Alpcan,et al.  Randomized algorithms for stability and robustness analysis of high-speed communication networks , 2005, IEEE Transactions on Neural Networks.

[34]  Fabrizio Dabbene,et al.  Computing Hard Bounds for Probabilistic Robustness , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[35]  Daniel Liberzon,et al.  Common Lyapunov functions and gradient algorithms , 2004, IEEE Transactions on Automatic Control.

[36]  Arkadi Nemirovski,et al.  Several NP-hard problems arising in robust stability analysis , 1993, Math. Control. Signals Syst..

[37]  M. Sznaier,et al.  An algorithm for sampling subsets of H/sub /spl infin// with applications to risk-adjusted performance analysis and model (in)validation , 2005, IEEE Transactions on Automatic Control.

[38]  R. Stengel.  Stochastic Optimal Control: Theory and Application , 1986 .

[39]  Roberto Tempo,et al.  Monte carlo and las vegas randomized algorithms for systems and control : An introduction , 2007 .

[40]  Mathukumalli Vidyasagar,et al.  Learning And Generalization , 2002 .

[41]  B. R. Barmish,et al.  Distributionally Robust Monte Carlo Simulation: A Tutorial Survey , 2002 .

[42]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[43]  Alberto L. Sangiovanni-Vincentelli,et al.  Maximizing the stability radius of a set of systems under real-time scheduling constraints , 2005, IEEE Transactions on Automatic Control.

[44]  Xavier Bombois,et al.  Least costly identification experiment for control , 2006, Autom..

[45]  H.J.A.F. Tulleken,et al.  Probabilistic robust controller design , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

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

[47]  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.

[48]  Carlos S. Kubrusly,et al.  Stochastic approximation algorithms and applications , 1973, CDC 1973.

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

[50]  Tamer Basar,et al.  Randomized algorithms for synthesis of switching rules for multimodal systems , 2005, IEEE Transactions on Automatic Control.

[51]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[52]  R. Tempo,et al.  Optimal algorithms theory for robust estimation and prediction , 1985 .

[53]  G Stix,et al.  Immuno-logistics. Moving vaccines from the lab to the bush and the street. , 1994, Scientific American.

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

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

[56]  Roberto Tempo,et al.  Monte Carlo and Las Vegas Randomized Algorithms for Systems and Control? , 2007, Eur. J. Control.

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

[58]  A. Nemirovskii.  Several NP-hard problems arising in robust stability analysis , 1993 .

[59]  R. Tempo,et al.  Probabilistic robustness analysis: explicit bounds for the minimum number of samples , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[60]  Roberto Tempo.  Robust estimation and filtering in the presence of bounded noise , 1987 .

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

[62]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

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

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

[65]  Valery A. Ugrinovskii,et al.  Randomized Algorithms for Robust Stability and Guaranteed Cost Control of Stochastic Jump Parameter Systems with Uncertain Switching Policies , 2005 .

[66]  Thomas D. Sandry,et al.  Probabilistic and Randomized Methods for Design Under Uncertainty , 2007, Technometrics.

[67]  Joseph F. Traub,et al.  Complexity and information , 1999, Lezioni Lincee.

[68]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[69]  Minyue Fu,et al.  Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis , 1998, Math. Control. Signals Syst..

[70]  Carl N. Nett,et al.  Control oriented system identification: a worst-case/deterministic approach in H/sub infinity / , 1991 .

[71]  Svatopluk Poljak,et al.  Checking robust nonsingularity is NP-hard , 1993, Math. Control. Signals Syst..

[72]  Mathukumalli Vidyasagar,et al.  Randomized algorithms for robust controller synthesis using statistical learning theory , 2001, Autom..