Partial stabilisation of large-scale discrete-time linear control systems

We propose a parallel algorithm for stabilising large, discrete-time, linear control systems on a Beowulf cluster. Our algorithm first separates the Schur stable part of the linear control system using an inverse free iteration for the matrix disc function and then computes a stabilising feedback matrix for the unstable part. This stage requires the numerical solution of a Stein equation. This linear matrix equation is solved using the sign function method after applying a Cayley transformation to the original equation. The experimental results on a cluster composed of Intel PII processors and a Myrinet interconnection network show the parallelism and scalability of our approach.

[1]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[2]  E. Armstrong,et al.  A stabilization algorithm for linear discrete constant systems , 1976 .

[3]  J. D. Roberts,et al.  Linear model reduction and solution of the algebraic Riccati equation by use of the sign function , 1980 .

[4]  L. Dai,et al.  Singular Control Systems , 1989, Lecture Notes in Control and Information Sciences.

[5]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[6]  A. Malyshev.  Parallel Algorithm for Solving Some Spectral Problems of Linear Algebra , 1993 .

[7]  A. Laub,et al.  The matrix sign function , 1995, IEEE Trans. Autom. Control..

[8]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[9]  Robert A. van de Geijn,et al.  Parallelizing the QR Algorithm for the Unsymmetric Algebraic Eigenvalue Problem: Myths and Reality , 1996, SIAM J. Sci. Comput..

[10]  Volker Mehrmann,et al.  AN ANALYSIS OF THE POLE PLACEMENT PROBLEM. I. THE SINGLE-INPUT CASE , 1996 .

[11]  Robert A. van de Geijn,et al.  Using PLAPACK - parallel linear algebra package , 1997 .

[12]  J. Demmel,et al.  An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems , 1997 .

[13]  V. Mehrmann,et al.  Stabilization of large linear systems , 1998 .

[14]  Christian H. Bischof,et al.  Computing rank-revealing QR factorizations of dense matrices , 1998, TOMS.

[15]  Christian H. Bischof,et al.  A BLAS-3 Version of the QR Factorization with Column Pivoting , 1998, SIAM J. Sci. Comput..

[16]  V. Dragan,et al.  Stabilization of Linear Systems , 1999 .

[17]  Robert A. van de Geijn,et al.  Specialized Parallel Algorithms for Solving Lyapunov and Stein Equations , 2001, J. Parallel Distributed Comput..

[18]  Robert A. van de Geijn,et al.  A Note On Parallel Matrix Inversion , 2000, SIAM J. Sci. Comput..

[19]  Jack J. Dongarra,et al.  A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures , 2002, SIAM J. Sci. Comput..

[20]  Enrique S. Quintana-Ortí,et al.  NUMERICAL SOLUTION OF DISCRETE STABLE LINEAR MATRIX EQUATIONS ON MULTICOMPUTERS , 2002, Parallel Algorithms Appl..

[21]  Enrique S. Quintana-Ortí,et al.  Parallel Partial Stabilizing Algorithms for Large Linear Control Systems , 2000, The Journal of Supercomputing.