This review article surveys representative literature on minimum re&-zations and system modeling. This theory makes possible the construction of minimum size state-space models directly from experimental input/output data. Theoretical developments, numerical algorithms, and the connection with other methods of identification of linear systems are covered. Partial minimum realizations which are extensions of the concepts developed are also included. A numerical example illustrates many of the techniques. SCOPEOne of the most important and interesting problems of concern to the chemical engineer is the modeling or identification of real dynamical systems. Such modeling must be done in the face of significant nonlinearities and high dimensionality, both items being major components in chemical engineering systems. The results of such analysis can, of course, be used in a descriptive and/or predictive sense for optimizing and controlling the actual behavior of a system.There currently exists a relevant theory for linear, constant, dynamical systems which makes possible the construction of state-space models directly from input/output experimental data. This theory, termed realization theory, has generated a more fundamental understanding of the relationships between state-space, transfer function, and input/output descriptions of linear systems.Underlying many, if not all, of the important principles of realization theory are the dual concepts of controllability and observability. The importance of these concepts, first recognized by Kalman (1963) and Gilbert (1963), in the construction of state-space representations of minimum dimension and in the stability analysis of physical processes has been documented (Kalman, 1965a, b;Roberts, 1969). Techniques developed within the last few years have allowed the extension to the construction of linear, constant-coefficient, dynamical models of nonlinear systems. Although subject to many of the same limitations that restrict other linearization procedures, these techniques have proven to be quite useful in the modeling of many varieties of chemical engineering processes (Kallina, 1970;Rossen, 1972).In this paper, realization techniques which operate in the state or output space and yield linear models from input/output data will be developed. The methods will not only furnish a linear representation of a system but will also develop a minimal model in the sense of specifying the minimum number of system parameters, a minimal realization. The black box approach is used in these methods because only the innut/output data are used in the construction of the models.It is important to realize that such model construction methods can form the basis for a rational choice of model size for real linear systems. In the second paper in this series, new developments which can be auplied to nonlinear lumped and distributed parameter systems will be outlined. As such, the theory forms a viable algorithm for real chemical engineering systems. CONCLUSIONS AND SIGNIFICANCETo help the reader...

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...