Solving Large-Scale Robust Stability Problems by Exploiting the Parallel Structure of Polya's Theorem

In this paper, we propose a distributed computing approach to solving large-scale robust stability problems on the simplex. Our approach is to formulate the robust stability problem as an optimization problem with polynomial variables and polynomial inequality constraints. We use Polya's theorem to convert the polynomial optimization problem to a set of highly structured linear matrix inequalities (LMIs). We then use a slight modification of a common interior-point primal-dual algorithm to solve the structured LMI constraints. This yields a set of extremely large yet structured computations. We then map the structure of the computations to a decentralized computing environment consisting of independent processing nodes with a structured adjacency matrix. The result is an algorithm which can solve the robust stability problem with the same per-core complexity as the deterministic stability problem with a conservatism which is only a function of the number of processors available. Numerical tests on cluster computers and supercomputers demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors and analyze systems with 100+ dimensional state-space. The proposed algorithms can be extended to perform stability analysis of nonlinear systems and robust controller synthesis.

[1]  Michael L. Overton,et al.  Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results , 1998, SIAM J. Optim..

[2]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[3]  E. Joffrin,et al.  A control-oriented model of the current profile in tokamak plasma , 2007 .

[4]  Masakazu Kojima,et al.  Generalized Lagrangian Duals and Sums of Squares Relaxations of Sparse Polynomial Optimization Problems , 2005, SIAM J. Optim..

[5]  Robert J. Vanderbei,et al.  An Interior-Point Method for Semidefinite Programming , 1996, SIAM J. Optim..

[6]  H. James Hoover,et al.  Limits to Parallel Computation: P-Completeness Theory , 1995 .

[7]  Pedro Luis Dias Peres,et al.  A less conservative LMI condition for the robust stability of discrete-time uncertain systems , 2001, Syst. Control. Lett..

[8]  Masakazu Kojima,et al.  SDPARA: SemiDefinite Programming Algorithm paRAllel version , 2003, Parallel Comput..

[9]  A. Antoulas,et al.  A Survey of Model Reduction by Balanced Truncation and Some New Results , 2004 .

[10]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

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

[12]  Laxmikant V. Kale,et al.  The Charm Parallel Programming Language and System: Part I - Description of Language Features , 1994 .

[13]  P. Tsiotras,et al.  Multi-Parameter Dependent Lyapunov Functions for the Stability Analysis of Parameter-Dependent LTI Systems , 2005, Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, 2005..

[14]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[15]  P. Parrilo,et al.  Symmetry groups, semidefinite programs, and sums of squares , 2002, math/0211450.

[16]  Ricardo C. L. F. Oliveira,et al.  Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations , 2007, IEEE Transactions on Automatic Control.

[17]  J. Doyle,et al.  Essentials of Robust Control , 1997 .

[18]  J. Doyle,et al.  Quadratic stability with real and complex perturbations , 1990 .

[19]  Madhu V. Nayakkankuppam,et al.  Solving large-scale semidefinite programs in parallel , 2007, Math. Program..

[20]  Didier Henrion,et al.  GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi , 2003, TOMS.

[21]  Graziano Chesi,et al.  Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach , 2005, IEEE Transactions on Automatic Control.

[22]  G. Stengle.  A nullstellensatz and a positivstellensatz in semialgebraic geometry , 1974 .

[23]  W. Walker,et al.  Mpi: a Standard Message Passing Interface 1 Mpi: a Standard Message Passing Interface , 1996 .

[24]  Pierre-Alexandre Bliman,et al.  A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters , 2003, SIAM J. Control. Optim..

[25]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[26]  S. J. Benson,et al.  DSDP3: dual scaling algorithm for general positive semidefinite programming. , 2001 .

[27]  P. Parrilo.  Exploiting Algebraic Structure in Sum of Squares Programs , 2005 .

[28]  B. Buchberger,et al.  Gröbner bases and applications , 1998 .

[29]  Pierre-Alexandre Bliman,et al.  An existence result for polynomial solutions of parameter-dependent LMIs , 2004, Syst. Control. Lett..

[30]  G. Amdhal,et al.  Validity of the single processor approach to achieving large scale computing capabilities , 1967, AFIPS '67 (Spring).

[31]  D. Kamenetsky.  Symmetry Groups , 2003 .

[32]  Pedro L. D. Peres,et al.  An LMI approach to compute robust stability domains for uncertain linear systems , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[33]  P. Gahinet,et al.  A linear matrix inequality approach to H∞ control , 1994 .

[34]  Xiong Zhang,et al.  Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization , 1999, SIAM J. Optim..

[35]  Fernando Paganini,et al.  A Course in Robust Control Theory , 2000 .

[36]  Brian Borchers,et al.  Implementation of a primal–dual method for SDP on a shared memory parallel architecture , 2007, Comput. Optim. Appl..

[37]  Matthew M Peet,et al.  A parallel-computing solution for optimization of polynomials , 2010, Proceedings of the 2010 American Control Conference.

[38]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

[39]  Pablo A. Parrilo,et al.  Introducing SOSTOOLS: a general purpose sum of squares programming solver , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[40]  Masakazu Muramatsu,et al.  SparsePOP: a Sparse Semidefinite Programming Relaxation of Polynomial Optimization Problems , 2005 .

[41]  Andrew Bartlett,et al.  Robust Control: Systems with Uncertain Physical Parameters , 1993 .

[42]  Ricardo C. L. F. Oliveira,et al.  Robust LMIs with parameters in multi-simplex: Existence of solutions and applications , 2008, 2008 47th IEEE Conference on Decision and Control.

[43]  Shankar P. Bhattacharyya,et al.  Robust Control: The Parametric Approach , 1995 .

[44]  Steven J. Deitz,et al.  High-level programming language abstractions for advanced and dynamic parallel computations , 2005 .

[45]  Masakazu Kojima,et al.  Exploiting sparsity in primal-dual interior-point methods for semidefinite programming , 1997, Math. Program..

[46]  P. Peres,et al.  Stability of polytopes of matrices via affine parameter-dependent lyapunov functions : Asymptotically exact LMI conditions , 2005 .

[47]  Matthew M. Peet,et al.  Exponentially Stable Nonlinear Systems Have Polynomial Lyapunov Functions on Bounded Regions , 2007, IEEE Transactions on Automatic Control.

[48]  BorchersBrian,et al.  Implementation of a primaldual method for SDP on a shared memory parallel architecture , 2007 .

[49]  Franz Rendl,et al.  A Spectral Bundle Method for Semidefinite Programming , 1999, SIAM J. Optim..

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

[51]  Victoria Powers,et al.  A quantitative Pólya's Theorem with zeros , 2009, J. Symb. Comput..

[52]  Kartik Krishnan Sivaramakrishnan,et al.  A parallel interior point decomposition algorithm for block angular semidefinite programs , 2010, Comput. Optim. Appl..

[53]  David J. N. Limebeer,et al.  Linear Robust Control , 1994 .

[54]  P. Gahinet,et al.  Affine parameter-dependent Lyapunov functions and real parametric uncertainty , 1996, IEEE Trans. Autom. Control..

[55]  Renato D. C. Monteiro,et al.  Primal-Dual Path-Following Algorithms for Semidefinite Programming , 1997, SIAM J. Optim..

[56]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[57]  Edward R. Scheinerman.  Mathematics: A Discrete Introduction , 2000 .

[58]  SivaramakrishnanKartik Krishnan.  A parallel interior point decomposition algorithm for block angular semidefinite programs , 2010 .