On the numerical computation of a structural decomposition in systems and control

In this paper, we develop a new numerical method for a special coordinate basis of a linear time invariant system. Such a special coordinate basis is essentially a structural decomposition which explicitly displays the finite and infinite zero structures, as well as the invertibility structures of the given system. The technique is playing important roles in numerous topics in system and control theory, such as robust control, H/sub /spl infin// and H/sub 2/ optimal control almost disturbance decoupling, and zero placement of linear systems, just to name a few. Our method consists of three steps: reduction by orthogonal transformations, reduction by generalized Sylvester equations, and extraction of infinite zero structure. The performance of our method is illustrated by some numerical examples.

[1]  P. Dooren.  A Generalized Eigenvalue Approach for Solving Riccati Equations , 1980 .

[2]  Ali Saberi,et al.  Special coordinate basis for multivariable linear systems—finite and infinite zero structure, squaring down and decoupling , 1987 .

[3]  Paul Van Dooren,et al.  Computation of structural invariants of generalized state-space systems , 1994, Autom..

[4]  Ali Saberi,et al.  Loop Transfer Recovery: Analysis and Design , 1993 .

[5]  Saberi,et al.  Squaring down of non-strictly proper systems , 1990 .

[6]  P. Dooren.  The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[7]  P. Dooren,et al.  An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects , 1997 .

[8]  Ben M. Chen,et al.  On blocking zeros and strong stabilizability of linear multivariable systems , 1992, Autom..

[9]  P. Dooren.  Reducing subspaces: Definitions, properties and algorithms , 1983 .

[10]  James Demmel,et al.  The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part II: software and applications , 1993, TOMS.

[11]  Ali Saberi,et al.  Theory of LTR for non-minimum phase systems, recoverable target loops, and recovery in a subspace Part 1. Analysis , 1991 .

[12]  Bo Kågström,et al.  An Algorithm for Numerical Computation of the Jordan Normal Form of a Complex Matrix , 1980, TOMS.

[13]  P. Sannuti,et al.  Theory of LTR for nonminimum phase systems, recoverable target loops, recovery in a subspace. I. Analysis , 1990, 29th IEEE Conference on Decision and Control.

[14]  Paul Van Dooren,et al.  Computation of zeros of linear multivariable systems , 1980, Autom..

[15]  Ali Saberi,et al.  Special coordinate basis for order reduction of linear multivariable systems , 1990 .

[16]  Bo Kågström,et al.  Algorithm 560: JNF, An Algorithm for Numerical Computation of the Jordan Normal Form of a Complex Matrix [F2] , 1980, TOMS.

[17]  Ben M. Chen,et al.  Loop Transfer Recovery , 1993 .

[18]  Ben M. Chen,et al.  Construction and parameterization of all static and dynamic H2 -optimal state feedback solutions, optimal fixed modes and fixed decoupling zeros , 1993, IEEE Trans. Autom. Control..

[19]  Ali Saberi,et al.  Time-scale structure assignment in linear multivariable systems using high-gain feedback , 1989 .

[20]  U.-L. Ly,et al.  Exact computation of the infimum in H/sub infinity /-optimization via output feedback , 1992 .

[21]  James Demmel,et al.  The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part I: theory and algorithms , 1993, TOMS.

[22]  Delin Chu,et al.  Numerically Reliable Computing for the Row by Row Decoupling Problem with Stability , 2001, SIAM J. Matrix Anal. Appl..

[23]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[24]  P. Dooren.  The generalized eigenstructure problem in linear system theory , 1981 .

[25]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[26]  Bo Kågström,et al.  LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs , 1994, TOMS.

[27]  P. Dooren.  Reducing subspaces : computational aspects and applications in linear system theory , 1982 .

[28]  Bo Kågström,et al.  RGSD an algorithm for computing the Kronecker structure and reducing subspaces of singular A-lB pencils , 1986 .

[29]  Ali Saberi,et al.  Design for H∞ almost disturbance decoupling problem with internal stability via state or measurement feedback―singular perturbation approach , 1992 .

[30]  Ali Saberi,et al.  A non-iterative method for computing the infimum in H∞ -optimization , 1992 .

[31]  Paul Van Dooren,et al.  Computational aspects of the Jordan canonical form , 1990 .

[32]  James Demmel,et al.  Accurate solutions of ill-posed problems in control theory , 1988 .

[33]  Ali Saberi,et al.  Explicit expressions for cascade factorization of general nonminimum phase systems , 1992 .

[34]  A. Varga.  A Descriptor Systems Toolbox for MATLAB , 2000, CACSD. Conference Proceedings. IEEE International Symposium on Computer-Aided Control System Design (Cat. No.00TH8537).