Minimum realizations and system modeling. I. Fundamental theory and algorithms

Abstract

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...

Biomarkers of environmental benzene exposure.

“…5, 8/26/93, (or other dose surrogate, such as amount metabolized) over a population, rather than to obtain point estimates of dose. One approach to developing population PBPK models is to map a PBPK model * Data onto a compartmental model and thereby establish a relationship between the parameters in the two models (this can be accomplished by adopting system reduction methods (37) and lumping theory (38). The population pharmacokinetic model can then be developed based on the compartmental model using standard population pharmacokinetic software such as the nonlinear mixed effects model (NON-*, MEM) developed at the University of 200 250 300 California at San Francisco (39 Time, population that form heptatoxic benzene metabolite transparameters in a trans-muconaldehyde (40).…”

An eigensystem realization algorithm for modal parameter identification and model reduction

“…The triple [TAT' 1 , TB, CT' 1 ] will also be a realization. 6 However, the eigenvalues of the matrix A are preserved.…”

“…In view of Eqs. (4) through (6), the measurement function Y(k+\) can be obtained through either of two other algorithms [Eqs. (Al) 2] with realization (Al) will be transformed into the following equation:…”

“…By minimum realization is meant a model with the smallest state space dimension among systems realized that has the same input-output relations within a specified degree of accuracy. Questions regarding the minimum realization from various types of input-output data and the generation of a minimum partial realization are studied by Tether, 4 Silverman, 5 and Rossen and Lapidus 6 using Markov parameters. Rossen and Lapidus 7 successfully applied Ho-Kalman 3 and Tether 4 methods to chemical engineering systems.…”

Biomarkers of Environmental Benzene Exposure

“…5, 8/26/93, (or other dose surrogate, such as amount metabolized) over a population, rather than to obtain point estimates of dose. One approach to developing population PBPK models is to map a PBPK model * Data onto a compartmental model and thereby establish a relationship between the parameters in the two models (this can be accomplished by adopting system reduction methods (37) and lumping theory (38) …”