Canonical matrix fraction and state-space descriptions for deterministic and stochastic linear systems

Several results exposing the interrelations between state-space and frequency-domain descriptions of multivariable linear systems are presented. Three canonical forms for constant parameter autoregressive-moving average (ARMA) models for input-output relations are described and shown to corrrespond to three particular canonical forms for the state variable realization of the model. Invariant parameters for the partial realization problem are characterized. For stochastic processes, it is shown how to construct an ARMA model, driven by white noise, whose output has a specified covariance. A two-step procedure is given, based on minimal realization and Cholesky-factorization algorithms. Though the goal is an ARMA model, it proves useful to introduce an artificial state model and to employ the recently developed Chandrasekhar-type equations for state estimation. The important case of autoregressive processes is studied and it is shown how the Chandrasekhar-type equations can be used to obtain and generalize the well known Levinson-Wiggins-Robinson (LWR) recursion for estimation of stationary autoregressive processes.

[1]  N. Levinson.  The Wiener (Root Mean Square) Error Criterion in Filter Design and Prediction , 1946 .

[2]  Bernard Friedland,et al.  Least Squares Filtering and Prediction of Nonstationary Sampled Data , 1958, Inf. Control..

[3]  James Durbin,et al.  The fitting of time series models , 1960 .

[4]  V. Belevitch.  ON NETWORK ANALYSIS BY POLYNOMIAL MATRICES , 1963 .

[5]  P. Whittle.  On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix , 1963 .

[6]  E. Robinson,et al.  Recursive solution to the multichannel filtering problem , 1965 .

[7]  Β. L. HO,et al.  Editorial: Effective construction of linear state-variable models from input/output functions , 1966 .

[8]  Elijah Polak,et al.  An algorithm for reducing a linear, time-invariant differential system to state form , 1966 .

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

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

[11]  T. Kailath.  Application of a Resolvent Identity to a Linear Smoothing Problem , 1969 .

[12]  Thomas Kailath,et al.  Fredholm resolvents, Wiener-Hopf equations, and Riccati differential equations , 1969, IEEE Trans. Inf. Theory.

[13]  V. Popov.  Some properties of the control systems with irreducible matrix — Transfer functions , 1970 .

[14]  H. Rosenbrock,et al.  State-space and multivariable theory, , 1970 .

[15]  A new canonical form of state-variable equations and its application in the determination of a mathematical model of an unknown system† , 1971 .

[16]  A. Bryson,et al.  Discrete square root filtering: A survey of current techniques , 1971 .

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

[18]  G. Wilson.  The Factorization of Matricial Spectral Densities , 1972 .

[19]  M. Makhlouf.  An algebra for the analysis of linear stationary and non-stationary multivariable systems using the time-domain infinite matrices† , 1972 .

[20]  J. Rissanen,et al.  1972 IFAC congress paper: Partial realization of random systems , 1972 .

[21]  J. Rissanen.  A fast algorithm for optimum linear predictors , 1973 .

[22]  Jan C. Willems,et al.  On the identifiability of linear dynamical systems , 1973 .

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

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

[25]  Thomas Kailath,et al.  Some new algorithms for recursive estimation in constant linear systems , 1973, IEEE Trans. Inf. Theory.

[26]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part VII: Some applications of vector autoregressive-moving average models , 1973 .

[27]  Keith Glover.  Structural Aspects of System Identification , 1973 .

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

[29]  David Q. Mayne.  The design of linear multivariable systems , 1973 .

[30]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part VI: Discrete-time innovations representations and recursive estimation , 1973 .

[31]  J. Rissanen.  Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials , 1973 .

[32]  Thomas Kailath,et al.  A view of three decades of linear filtering theory , 1974, IEEE Trans. Inf. Theory.

[33]  M. Morf,et al.  Some new algorithms for recursive estimation in constant, linear, discrete-time systems , 1974 .

[34]  J. Rissanen.  Basis of invariants and canonical forms for linear dynamic systems , 1974, Autom..

[35]  T. Kailath,et al.  The shift-invariance approach to continuous-time fast estimation algorithms , 1974, CDC 1974.

[36]  A. Lindquist.  A New Algorithm for Optimal Filtering of Discrete-Time Stationary Processes , 1974 .

[37]  Howard L. Weinert,et al.  Structure determination and parameter identification for multivariable stochastic linear systems , 1975 .

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