The generalized eigenstructure problem in linear system theory

The algebraic theory of linear time-invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context. In this paper the numerical aspects of a certain class of such algorithms-dealing with what the author calls generalized eigenstructure problems-are discussed. Some new and/or modified algorithms are presented. Both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria.

[1]  B. McMillan.  Introduction to formal realizability theory — II , 1952 .

[2]  F. L. Bauer.  Optimally scaled matrices , 1963 .

[3]  J. Rice.  A Theory of Condition , 1966 .

[4]  L. Silverman.  Inversion of multivariable linear systems , 1969 .

[5]  Joos Vandewalle,et al.  On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion , 1970 .

[6]  G. Stewart,et al.  An Algorithm for the Generalized Matrix Eigenvalue Problem Ax = Lambda Bx , 1971 .

[7]  Gerald L. Thompson,et al.  THE ROOTS OF MATRIX PENCILS (Ay = Lambda By): EXISTENCE, CALCULATIONS, AND RELATIONS TO GAME THEORY. , 1972 .

[8]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[9]  G. Stewart,et al.  An Algorithm for Generalized Matrix Eigenvalue Problems. , 1973 .

[10]  A. Morse.  Structural Invariants of Linear Multivariable Systems , 1973 .

[11]  David Q. Mayne,et al.  An elementary derivation of Rosenbrock's minimal realization algorithm , 1973 .

[12]  P. Wedin.  Perturbation theory for pseudo-inverses , 1973 .

[13]  An algorithm for an invariant canonical form , 1974, CDC 1974.

[14]  Donna K. Dunaway.  Calculation of Zeros of a Real Polynomial Through Factorization Using Euclid’s Algorithm , 1974 .

[15]  E. Davison,et al.  Properties and calculation of transmission zeros of linear multivariable systems , 1974, Autom..

[16]  H. Rosenbrock.  Structural properties of linear dynamical systems , 1974 .

[17]  R. Ward.  The Combination Shift $QZ$ Algorithm , 1975 .

[18]  Jr. G. Forney,et al.  Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems , 1975 .

[19]  David Jordan,et al.  On computation of the canonical pencil of a linear system , 1976 .

[20]  B. Molinari.  A strong controllability and observability in linear multivariable control , 1976 .

[21]  G. W. Stewart,et al.  Algorithm 506: HQR3 and EXCHNG: Fortran Subroutines for Calculating and Ordering the Eigenvalues of a Real Upper Hessenberg Matrix [F2] , 1976, TOMS.

[22]  N. Karcanias,et al.  Poles and zeros of linear multivariable systems : a survey of the algebraic, geometric and complex-variable theory , 1976 .

[23]  David Jordan,et al.  On computation of the canonical pencil of a linear system , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[24]  P. Moylan.  Stable inversion of linear systems , 1977 .

[25]  Alan J. Laub,et al.  Calculation of transmission zeros using QZ techniques , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.

[26]  L. D. Jong,et al.  Towards a formal definition of numerical stability , 1977 .

[27]  T. Kailath,et al.  Generalized state-space systems , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[28]  T. Kailath,et al.  Properties of the system matrix of a generalized state-space system , 1980, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[29]  B. Moore,et al.  Singular value analysis of linear systems , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[30]  James Hardy Wilkinson,et al.  Linear Differential Equations and Kronecker's Canonical Form , 1978 .

[31]  David H. Owens,et al.  On structural invariants and the root-loci of linear multivariable systems , 1978 .

[32]  David G. Luenberger,et al.  Time-invariant descriptor systems , 1978, Autom..

[33]  T. Kailath,et al.  Properties of the system matrix of a generalized state-space system , 1978 .

[34]  B. Molinari.  Structural invariants of linear multivariable systems , 1978 .

[35]  L. Silverman,et al.  Stable extraction of Kronecker structure of pencils , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[36]  J. Dwight Aplevich,et al.  Tableau methods for analysis and design of linear systems , 1977, Autom..

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

[38]  James Hardy Wilkinson,et al.  Kronecker''s canonical form and the QZ algorithm , 1979 .

[39]  Israel Gohberg,et al.  Factorizations of Transfer Functions , 1980 .

[40]  R. V. Patel,et al.  Computation of matrix fraction descriptions of linear time-invariant systems , 1981 .

[41]  T. Kailath,et al.  A generalized state-space for singular systems , 1981 .

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