Computation of matrix fraction descriptions of linear time-invariant systems

In this paper, algorithms are presented for obtaining relatively prime matrix fraction descriptions (MFD's) of linear time-invariant systems. Orthogonal coordinate transformations are used to reduce a given state-space model to one whose state matrix is in "block Hessenberg" form. The observability (controllability) indexes of the system are also obtained in the process. A recursive algorithm is then described for obtaining a relatively prime MFD from the block Hessenberg representation. It is then shown that some simplification can be made in the recursive algorithm by first reducing the block Hessenberg form, by means of a nonsingular (triangular) coordinate transformation, to a more compact form, such as the "block Frobenius" form. A permutation of the state variables of the block Frobenius form yields a canonical representation similar to the Luenberger canonical form. Some numerical and other properties of the algorithms are discussed, and the use of the algorithms is illustrated by numerical examples. The numerical performance of the algorithms is also compared with that of the structure algorithm of Wolovich and Falb.

[1]  A. Laub,et al.  The singular value decomposition: Its computation and some applications , 1980 .

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

[3]  Michael K. Sain,et al.  A Free-Modular Algorithm for Minimal Design of Linear Multivariable Systems , 1975 .

[4]  William A. Wolovich,et al.  On the Structure of Multivariable Systems , 1969 .

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

[6]  W. Wolovich.  The determination of state-space representations for linear multivariable systems , 1973 .

[7]  R. V. Patel,et al.  Computation of minimal-order state-space realizations and observability indices using orthogonal transformations , 1981 .

[8]  Charles A. Desoer,et al.  Zeros and poles of matrix transfer functions and their dynamical interpretation , 1974 .

[9]  E. Davison,et al.  A minimization algorithm for the design of linear multivariable systems , 1973 .

[10]  L. Silverman,et al.  Input-output structure of linear systems with application to the decoupling problem , 1971 .

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

[12]  L. Foster.  A practical solution to the minimal design problem , 1979 .

[13]  Thomas Kailath,et al.  Fast and stable algorithms for minimal design problems , 1977 .

[14]  N. Munro,et al.  Minimal realisation of transfer-function matrices using the system matrix , 1971 .

[15]  B. Dickinson,et al.  A minimal realization algorithm for matrix sequences , 1973, CDC 1973.

[16]  Matrix fraction descriptions of linear time-invariant systems , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[17]  R. Kálmán.  Mathematical description of linear dynamical systems , 1963 .

[18]  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.

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

[20]  William R. Perkins,et al.  Generalized Hessenberg transformations for reduced-order modelling of large-scale systems† , 1978 .

[21]  J. H. Wilkinson,et al.  Similarity reduction of a general matrix to Hessenberg form , 1968 .

[22]  D. Luenberger.  Canonical forms for linear multivariable systems , 1967, IEEE Transactions on Automatic Control.