Randomized methods based on new Monte Carlo schemes for control and optimization

We address randomized methods for control and optimization based on generating points uniformly distributed in a set. For control systems this sets are either stability domain in the space of feedback controllers, or quadratic stability domain, or robust stability domain, or level set for a performance specification. By generating random points in the prescribed set one can optimize some additional performance index. To implement such approach we exploit two modern Monte Carlo schemes for generating points which are approximately uniformly distributed in a given convex set. Both methods use boundary oracle to find an intersection of a ray and the set. The first method is Hit-and-Run, the second is sometimes called Shake-and-Bake. We estimate the rate of convergence for such methods and demonstrate the link with the center of gravity method. Numerical simulation results look very promising.

[1]  Roberto Tempo,et al.  Randomized Algorithms in Systems and Control , 2013 .

[2]  Boris T. Polyak,et al.  The D-decomposition technique for linear matrix inequalities , 2006 .

[3]  Vladimir A. Yakubovich,et al.  Linear Matrix Inequalities in System and Control Theory (S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan) , 1995, SIAM Rev..

[4]  B. Barmish.  Necessary and sufficient conditions for quadratic stabilizability of an uncertain system , 1985 .

[5]  Reuven Y. Rubinstein,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.

[6]  Boris T. Polyak,et al.  Stability regions in the parameter space: D-decomposition revisited , 2006, Autom..

[7]  Eli Upfal,et al.  Probability and Computing: Randomized Algorithms and Probabilistic Analysis , 2005 .

[8]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[9]  Santosh S. Vempala,et al.  Solving convex programs by random walks , 2004, JACM.

[10]  Persi Diaconis,et al.  The Markov chain Monte Carlo revolution , 2008 .

[11]  Robert L. Smith,et al.  Direction Choice for Accelerated Convergence in Hit-and-Run Sampling , 1998, Oper. Res..

[12]  N GryazinaElena,et al.  Stability regions in the parameter space , 2006 .

[13]  Shankar P. Bhattacharyya,et al.  Robust Stabilization Against Structured Perturbations , 1987 .

[14]  Fabrizio Dabbene,et al.  A Randomized Cutting Plane Method with Probabilistic Geometric Convergence , 2010, SIAM J. Optim..

[15]  Roberto Tempo,et al.  Mixed Deterministic/Randomized Methods for Fixed Order Controller Design , 2008, IEEE Transactions on Automatic Control.

[16]  Marco C. Campi,et al.  Why Is Resorting to Fate Wise? A Critical Look at Randomized Algorithms in Systems and Control , 2010, Eur. J. Control.

[17]  F. Comets,et al.  Billiards in a General Domain with Random Reflections , 2006, math/0612799.

[18]  Lih-Yuan Deng,et al.  The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning , 2006, Technometrics.

[19]  Robert L. Smith,et al.  Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions , 1984, Oper. Res..

[20]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[21]  Semi-Markov Process.  ON SIMULATION OF RANDOM VECTORS WITH GIVEN DENSITIES IN REGIONS AND ON THEIR BOUNDARIES , 1994 .

[22]  László Lovász,et al.  Hit-and-run mixes fast , 1999, Math. Program..

[23]  D. J. Newman,et al.  Location of the Maximum on Unimodal Surfaces , 1965, JACM.

[24]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[25]  W. Gilks.  Markov Chain Monte Carlo , 2005 .

[26]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[27]  Konstantin Borovkov.  On a New Variant of the Monte Carlo Method , 1992 .

[28]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[29]  Sahjendra N. Singh,et al.  Nonlinear Control of Mismatched Uncertain Linear Systems and Application to Control of Aircraft , 1984 .

[30]  L. Ghaoui,et al.  A cone complementarity linearization algorithm for static output-feedback and related problems , 1997, IEEE Trans. Autom. Control..

[31]  V. Turchin.  On the Computation of Multidimensional Integrals by the Monte-Carlo Method , 1971 .

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