Minimum realizations and system modeling: II. Theoretical and numerical extensions

Abstract

The ideas and techniques discussed in Part I of this Review' are extended to the modeling of nonlinear lumped and distributed systems. Troublesome numerical and computational difficulties are examined, and recommendations for the alleviation of these difficulties are made. Particular emphasis is placed on the stability of system models constructed with the Tether method. Highly successful modeling of realistic systems are presented numerically. R. H. ROSSEN and L. LAPIDUS Department of Chemicd EngineeringPrinceton University Princeton, New Jersey 08540Numerous algorithms for the construction of statespace representations of linear systems have appeared in the recent literature. However, with only a few exceptions (Kallina, 1970;Rossen, 1972), the analysis of these procedures has been limited to the area of linear lumped parameter systems, and the reported application of the algorithms has been restricted to abstract systems. Therefore, the difficulties which arise in the realization of real physical processes have received little investigation.Part I of this Review was devoted to a presentation of the basic concepts underlying the theory of minimum realizations and a summary of the publications relating to that theory and to the algorithms derived from it. In Part 11, we will investigate the application of the partial realization procedure developed by Tether to the modeling of processes which have been studied in the literature. Observations will be made concerning the applicability of the procedure to different classes of chemical engineering systems, and techniques will be offered for the alleviation of some of the difficulties relating to that application.The Tether algorithm is used because it was developed for the construction of the system model directly from the impulse response curve of the system and is therefore easily adapted to the modeling of nonlinear systems or of distributed systems. It has also been determined that the Tether procedure can be superior to the procedures developed by other researchers when the construction of an exact model for the system is impossible and when a partial realization is required. SIGNIFICANCE AND CONCLUSIONSThe minimum realization algorithm originally proposed by Ho and Kalman (1966) and modified by Tether (1970) is shown to be an excellent tool for generating linear state-space representations of linear and nonlinear systems based solely on their responses to impulse forcing functions. In many respects, it is superior to the frequently encountered modeling techniques based on moment analysis and frequency response analysis. The algorithm requires less input/output data than the other methods and by treating all of the input/output pairs simultaneously, the algorithm is able to extract more information from the available data. Also by not specifying the dimension of the realization a priori, the algorithm retains greater flexibility than the moment methods.It is suggested here that maximum effectiveness can be gained from the algorithm if the two input sequences o...

Archives of Control Sciences

“…The calculation method used consists in approximation of the instantaneous power waveforms of individual units by the waveforms calculated on the basis of relationship (22). The latter waveforms are the superposition of modal compo-nents determined by the sought eigenvalues and their participation factors.…”

“…has the order equal to m. Therefore, based on the measurement data, a system of minimum order m that is fully controllable and observable should be created [3,21,22]. For this purpose, one can use the algorithm of Ho and Kalman [21] and determine the matrices:…”

“…The responses to some small disturbances that appear automatically in PS (without introducing a disturbance), after some time since the occurrence of the disturbance, are similar in shape to the response to the Dirac pulse and can be calculated with satisfactory accuracy based on formula (22). In the case of such disturbances, in formula (22), instead of the participation factors there are the amplitudes of individual modal components which are approximately proportional to the correlation coefficients. This approximate proportionality between the respective participation factors and correlation coefficients also applies in the case of introducing a purposeful disturbance into PS.…”

Iterative method of identifying Multiple-Input-Multiple-Output systems in state-space

“…The Subspace Identification Method (SIM) stems from the classical state-space methods, which had the first influence relied on the realization theory that has arisen in Ho and Kalmans work in 1966 (Ho;Kalman, 1966), and indeed from the works already mentioned of Tether and Silverman. A review article stated the minimum realizations and system modeling (Rossen;Lapidus, 1972), consolidating the subject also crucial for SIM. The Kung's paper (Kung, 1978) relevantly contributed to the state-space realization theories in the joint concept of Markov parameters, Hankel matrix, and SVD.…”