Multivariable System Identification - A Survey

Abstract

The paper discusses multivariable system structures,pararnetric and nonpararnetric identification techniques for multivariable systems. The first part considers the role of canonical forms in multivariable system identification and presents a broad survey of state-space and ARMA canonical forms. The concept of structural and parametriC invariants is discussed and explicit expressions for the canonically observable and controllable forms are given. A generalization of the canonical form concept is given with particular attention to the problem of invariants generation. The generation of canonical forms from the Hankel matrix is discussed. Explicit expressions for ARMA canonical forms using the same set of invariants as the previously discussed state-space canonical forms are given. This part of the paper ends with a critical appraisal of the trend towards using canonical forms for multivariable system identification. The second part starts with a discussion of problems associated with the identification of structural invariants. After that an alternative approach is presented, aiming at identifying some noncanonical representation. The problem of generating a well-conditioned basis in the predictor space using the Hankel matrix, concurrently with the model parameter estimation is discussed. That part of the paper ends with a discussion of problems connected with the choice of performance indices for the simultaneous identification of multivariable system structure and parameters. The third part of the paper describes nonparametric identification methods, based on infinite Markov parameter sequences. At first a general description of the linear time invariant multivariable system using the Hankel model is given. Classical results of the realization theory and the Ho-Kalman algorithm are presented. This follows with some important methods for the realization of infinite Harkov parameter sequences for noiseless as well as noise-corrupted sequences. A natural extension of the minimal realization problem is presented afterwadrs, leading to the concept of minimal partial realization of the finite series of Markov parameters. The problem of Markov parameter estimation is discussed in turn. Hethods for asymptotically unbiased and efficient estimation of Markov parameters are presented with special attention to the Gauss-Harkov approximated scheme and the problem of the noise covariance matrix reconstruction.