Rational approximation of distributed parameter systems
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RATIONAL APPROXIMATION OF DISTRIBUTED PARAMETER SYSTEMS
UDC 62–52:519.6:519.711
V. F. Gubarev
A new approach to the construction of approximate models is proposed and justified for a wide class of distributed and lumped parameter systems. The approach is based on the iterative identification method that is especially efficient under uncertainty and errors in available data. Keywords: modeling, iterative identification, approximation, uncertainty, Green’s function. INTRODUCTION In modeling, predicting, and controlling processes in complex systems, an appropriate mathematical model is necessary. It should be adequate to the purposes and problems to be solved. At the same time, the model should not be too complex and difficult to use. For example, the greater the dimension of variables in the equations describing controlled processes, the more complicated the control law synthesized on their basis. At the same time, a small-dimensional approximate model yields in many cases a simple control, quite comprehensible in practice. Moreover, in describing complex real objects, the uncertainties not taken into account by the model always remain, and errors are possible in quantitative estimates or measurements. Finite-dimensional approximations are used even for numerical solution of, for example, problems of mathematical physics. Therefore, simplified models adequate to the problems to be solved with their help are more preferable in applications since they substantially expand the use of mathematical methods in various fields, especially for processes in distributed-parameter systems [1, 2]. The present paper deals with approximating models for infinite-dimensional systems, convenient and comprehensible to solve modeling, control, and prediction problems. A unified approach to the rational approximation of space-time systems is developed based on identification methods. 1. PROBLEM FORMULATION What is required first of all for mathematical modeling of a complex real system is to select a model structure for it. This is impossible to do without a priori information on and general idea of the object. If it is distributed in space and its parameters or characteristics vary in time, we deal with a space-time system, which can be characterized by a scalar function in the elementary case and by a vector function w in the general case. This function depends on a space variable z taking values in an open domain F, including the boundary ¶F, and on a time variable t taking values on the numerical semiaxis t ³ t 0 . As a rule, processes in such a system are also defined by boundary and initial conditions. In what follows, we will consider only linear systems, which is also the initial a priori information. Then a unified standard form, fitting well the existing methods of simulation and identification, can be used to describe a very wide class of systems in various fields of application. In [3], Butkovskii proposed the idea of such standardization, and in [4], he collected and structured a wide class of distributed and lumped param
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