A randomized approximation algorithm for the minimal-norm static-output-feedback problem

A new randomized algorithm is suggested, for extracting static-output-stabilizing-feedbacks, with approximately minimal-norm, for LTI systems. The algorithm has two similar stages, where in the first one the feasibility problem is solved, and in the second one the optimization problem is solved. The formulation is unified for the feasibility and for the optimization problems, as well as for continuous-time or discrete-time systems. The method is demonstrated by applying it to the hard (conjectured to be NP-hard) problem of the minimal-gain static-output-stabilizing-feedback, and to the hard (conjectured to be NP-hard) problem of regional pole-placement via static-output-feedback in non-convex or unconnected regions. A proof of convergence (in probability) that captures the two rounds of the algorithm is given, and complexity analysis is provided, under some mild assumptions.

[1]  Ricardo C. L. F. Oliveira,et al.  Improved conditions for reduced-order 𠄋∞ filter design as a static output feedback problem , 2011, Proceedings of the 2011 American Control Conference.

[2]  Wim Michiels,et al.  Combining Convex–Concave Decompositions and Linearization Approaches for Solving BMIs, With Application to Static Output Feedback , 2011, IEEE Transactions on Automatic Control.

[3]  Yl L. Xu,et al.  Optimum design of active/passive control devices for tall buildings under earthquake excitation , 2002 .

[4]  Robert Schmid,et al.  Robust Pole Placement With Moore's Algorithm , 2013, IEEE Transactions on Automatic Control.

[5]  Pierre Apkarian,et al.  Nonsmooth H∞ synthesis , 2005, IEEE Trans. Autom. Control..

[6]  Claude J. P. Bélisle.  Convergence theorems for a class of simulated annealing algorithms on ℝd , 1992 .

[7]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

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

[9]  Robert L. Smith,et al.  Simulated annealing for constrained global optimization , 1994, J. Glob. Optim..

[10]  Minyue Fu,et al.  Pole placement via static output feedback is NP-hard , 2004, IEEE Transactions on Automatic Control.

[11]  YangKaiyang,et al.  Generalized pole placement via static output feedback , 2006 .

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

[13]  Pierre Apkarian,et al.  Robust pole placement in LMI regions , 1999, IEEE Trans. Autom. Control..

[14]  Mikhail V. Khlebnikov,et al.  An LMI approach to structured sparse feedback design in linear control systems , 2013, 2013 European Control Conference (ECC).

[15]  Jann N. Yang,et al.  H∞‐based control strategies for civil engineering structures , 2003 .

[16]  F. Leibfritz.  COMPleib: COnstrained Matrix–optimization Problem library – a collection of test examples for nonlinear semidefinite programs, control system design and related problems , 2006 .

[17]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.

[18]  Boris Polyak,et al.  Mixed LMI/randomized methods for static output feedback control design , 2010, Proceedings of the 2010 American Control Conference.

[19]  A. Bunse-Gerstner,et al.  Numerical Methods for Algebraic Riccati Equations , 1989 .

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

[21]  Pascal Gahinet,et al.  H/sub /spl infin// design with pole placement constraints: an LMI approach , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[22]  Billie F. Spencer,et al.  Controlling Buildings: A New Frontier in Feedback , 1998 .

[23]  Jinchao Xu,et al.  The method of alternating projections and the method of subspace corrections in Hilbert space , 2002 .

[24]  C. Badea,et al.  The rate of convergence in the method of alternating projections , 2010, 1006.2047.

[25]  Bruno Iannazzo,et al.  Numerical Solution of Algebraic Riccati Equations , 2012, Fundamentals of algorithms.

[26]  J. Helton,et al.  Linear matrix inequality representation of sets , 2003, math/0306180.

[27]  J. Tsitsiklis,et al.  NP-hardness of some linear control design problems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[28]  Heinz H. Bauschke,et al.  On the method of cyclic projections for convex sets in Hilbert space , 1994 .

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

[30]  A characterization of all the static stabilizing controllers for LTI systems , 2012 .

[31]  Tong Heng Lee,et al.  On the design of multivariable PID controllers via LMI approach , 2002, Autom..

[32]  Roger J.-B. Wets,et al.  Minimization by Random Search Techniques , 1981, Math. Oper. Res..

[33]  Vasile Sima,et al.  Performance survey of robust pole placement methods , 2014, 53rd IEEE Conference on Decision and Control.

[34]  O. Toker,et al.  On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[35]  Nick Lord,et al.  Abstract algebra with applications , 1993 .

[36]  P. Gahinet,et al.  H∞ design with pole placement constraints: an LMI approach , 1996, IEEE Trans. Autom. Control..

[37]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[38]  Andrei Gabrielov,et al.  Pole Placement by Static Output Feedback for Generic Linear Systems , 2002, SIAM J. Control. Optim..

[39]  P. Dorato,et al.  Static output feedback: a survey , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[40]  Robert Orsi,et al.  Generalized pole placement via static output feedback: A methodology based on projections , 2006, Autom..

[41]  Mehran Mesbahi,et al.  A semi-definite programming solution of the least order dynamic output feedback synthesis problem , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[42]  D. Henrion,et al.  Solving polynomial static output feedback problems with PENBMI , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[43]  P. Apkarian,et al.  Nonsmooth H ∞ synthesis , 2005 .

[44]  Patrick L. Odell,et al.  Matrix Theory: From Generalized Inverses to Jordan Form , 2007 .

[45]  Victor Y. Pan,et al.  Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding , 2002, J. Symb. Comput..

[46]  Michael L. Overton,et al.  Multiobjective robust control with HIFOO 2.0 , 2009, 0905.3229.