Parallel Partial Stabilizing Algorithms for Large Linear Control Systems

In this paper we present parallel algorithms for stabilizing large linear control systems on multicomputers. Our algorithms first separate the stable part of the linear control system and then compute a stabilizing feedback for the unstable part. Both stages are solved by means of the matrix sign function which presents a high degree of parallelism and scalability.The experimental results on an IBM SP2 platform show the performance of our approach.

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

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

[3]  C. Langenhop.  On the stabilization of linear systems , 1964 .

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

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

[6]  A. Varga.  Technical Notes and Correspondence A Multishift Hessenberg Method for Pole Assignment of Single-Input Systems , 1996 .

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

[8]  Jack Dongarra,et al.  A User''s Guide to PVM Parallel Virtual Machine , 1991 .

[9]  Vasile Sima,et al.  An efficient Schur method to solve the stabilizing problem , 1981 .

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

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

[12]  P. Lancaster,et al.  The Algebraic Riccati Equation , 1995 .

[13]  T. Chan.  Rank revealing QR factorizations , 1987 .

[14]  Anthony Skjellum,et al.  Using MPI - portable parallel programming with the message-parsing interface , 1994 .

[15]  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..

[16]  V. Mehrmann,et al.  AN ANALYSIS OF THE POLE PLACEMENT PROBLEM II. THE MULTI-INPUT CASE∗ , 1997 .

[17]  R. Byers,et al.  The Matrix Sign Function Method and the Computation of Invariant Subspaces , 1997, SIAM J. Matrix Anal. Appl..

[18]  C. Paige,et al.  An algorithm for pole assignment of time invariant linear systems , 1982 .

[19]  R. Byers.  Solving the algebraic Riccati equation with the matrix sign function , 1987 .

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

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

[22]  D. Kleinman,et al.  An easy way to stabilize a linear constant system , 1970 .

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

[24]  Enrique S. Quintana-Ortí,et al.  Stabilizing Large Control Linear Systems on Multicomputers , 1996, VECPAR.

[25]  A. Varga.  A Schur method for pole assignment , 1981 .

[26]  Jack Dongarra,et al.  PVM: Parallel virtual machine: a users' guide and tutorial for networked parallel computing , 1995 .

[27]  Gene H. Golub,et al.  Matrix computations , 1983 .

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

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

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

[31]  James Demmel,et al.  Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I , 1993, PPSC.

[32]  William Gropp,et al.  Skjellum using mpi: portable parallel programming with the message-passing interface , 1994 .

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

[34]  Mark Edwin Arnold,et al.  Algorithms and conditioning for eigenvalue assignment , 1993 .

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

[36]  A. Laub,et al.  Rational iterative methods for the matrix sign function , 1991 .