Stabilizing large‐scale generalized systems on parallel computers using multithreading and message‐passing

We discuss the parallelization of an efficient algorithm for the partial stabilization of large‐scale linear control systems in generalized state‐space form. The algorithm is composed of highly parallel iterative schemes that appear in the computation of certain matrix functions. Here we evaluate different approaches to exploit parallelism at two levels, based on threads and processes. Our experimental results on a cluster of symmetric multiprocessors and a CC‐NUMA platform show that the efficiency of the matrix operations underlying the iterative schemes carry over to the parallel implementation of the stabilization algorithm. Copyright © 2006 John Wiley & Sons, Ltd.

[1]  E. Armstrong,et al.  An extension of Bass' algorithm for stabilizing linear continuous constant systems , 1975 .

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

[3]  James Demmel,et al.  The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers , 1997, SIAM J. Sci. Comput..

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

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

[6]  Enrique S. Quintana-Ortí,et al.  Spectral division methods for block generalized Schur decompositions , 2004, Math. Comput..

[7]  Alan J. Laub,et al.  Solution of the Sylvester matrix equation AXBT + CXDT = E , 1992, TOMS.

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

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

[10]  Jack Dongarra,et al.  ScaLAPACK Users' Guide , 1987 .

[11]  Enrique S. Quintana-Ortí,et al.  Solving stable generalized Lyapunov equations with the matrix sign function , 1999, Numerical Algorithms.

[12]  Enrique Salvador Quintana Ortí.  Algoritmos paralelos para resolver ecuaciones matriciales de Riccati en problemas de control , 1997 .

[13]  Peter Benner.  Partial Stabilization of Generalized State‐Space Systems Using the Disk Function Method , 2003 .

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

[15]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[16]  M. Ikeda,et al.  Stabilization of Linear Systems , 1972 .

[17]  B. Wie,et al.  Approach to large space structure control system design using traditional tools , 1990 .

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