We propose a parallel algorithm for stabilizing large discrete-time linear control systems on a Beowulf cluster. Our algorithm rst separates the Schur stable part of the linear control system using an inverse-free iteration for the matrix disc function, and then computes a stabilizing 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 P{II processors and a Myrinet interconnection network show the parallelism and scalability of our approach.
|
739
|
Using MPI: Portable Parallel Programming with the Message Passing Interface, 2nd edition
– Gropp, Lusk, et al.
- 1999
|
|
534
|
PVM: Parallel Virtual Machine: A Users’ Guide and Tutorial for Networked Parallel Computing
– Geist, Beguelin, et al.
- 1994
|
|
122
|
Algebraic Riccati Equations
– Lancaster, Rodman
- 1995
|
|
109
|
The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution. Number 163
– Mehrmann
- 1991
|
|
58
|
et al., LAPACK Users’ Guide
– Anderson
- 1999
|
|
55
|
de Geijn. Using PLAPACK: Parallel Linear Algebra Package
– van
- 1997
|
|
51
|
Inverse free parallel spectral divide and conquer algorithms for nonsymmetric eigenproblems
– Bai, Demmel, et al.
- 1994
|
|
41
|
Quintana-Ort'i: Numerical Solution of Schur Stable Linear Matrix Equations on Multicomputers
– Benner, Quintana-Ort'i, et al.
- 1999
|
|
39
|
Quintana-Ort'i: Parallel Partial Stabilizing Algorithms for Large Linear Control Systems
– Benner, Castillo, et al.
- 1998
|
|
37
|
Solution of the Matrix Equation
– Bartels, Stewart
- 1972
|
|
33
|
The matrix sign function
– Kenney, Laub
- 1995
|
|
32
|
Algorithms for Linear-Quadratic Optimization, volume 200 of Pure and Applied Mathematics
– Sima
- 1996
|
|
31
|
model reduction and solution of the algebraic Riccati equation by use of the sign function
– Linear
- 1980
|
|
30
|
A parallel implementation of the nonsymmetric QR algorithm for distributed memory architectures
– Henry, Watkins, et al.
|
|
29
|
de Geijn. Parallelizing the QR Algorithm for the Unsymmetric Algebraic Eigenvalue Problem: Myths and Reality
– Henry, Van
- 1997
|
|
27
|
Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems
– Benner
- 1997
|
|
24
|
A Schur method for pole assignment
– Varga
- 1981
|
|
22
|
Parallel Algorithm for Solving Some Spectral Problems of Linear Algebra
– Malyshev
- 1993
|
|
18
|
Computing rank-revealing QR factorizations of dense matrices
– Bischof, Quintana-Orti
- 1996
|
|
17
|
et al. ScaLAPACK Users' Guide
– Blackford
- 1997
|
|
13
|
A BLAS 3 version of the QR factorization with column pivoting
– Quintana-Orti, Sun, et al.
- 1990
|
|
9
|
An analysis of the pole placement problem. II. The multi input case
– Mehrmann, Xu
- 1998
|
|
9
|
An algorithm for pole assignment of time invariant linear systems
– Miminis, Paige
- 1982
|
|
8
|
An analysis of the pole placement problem. I. The single-input case
– Mehrmann, Xu
- 1996
|
|
7
|
Singular Control Systems, volume 118
– Dai
- 1989
|
|
6
|
A stabilization algorithm for linear discrete constant systems
– Armstrong, Rublein
- 1976
|
|
6
|
Algorithms and Conditioning for Eigenvalue Assignment
– Arnold
- 1992
|
|
5
|
Stabilization of large linear systems
– He, Mehrmann
- 1994
|
|
2
|
A multishift Hessenberg method for pole assignment of single-input systems
– Varga
- 1996
|
|
1
|
Stabilization of Linear Systems
– Dragan, Halanay
- 1994
|
|
1
|
de Geijn. A note on parallel matrix inversion
– Quintana-Ort, Quintana-Ort, et al.
|
|
