Paradigm of nonstochastic approach to system identification

Abstract

The concept of a complex system in this work is understood as a large set of dynamic interconnected systems, the exact mathematical model of which is not known or has a very large dimension. In such situation the use of standard methods for synthesizing feedback becomes difficult or even im­possible due to the degeneracy of the corresponding mathematical problems. One way out of this situation is to build an approximation model of reduced dimension. This can be done using a system of initial equations, if they are available, or using identification methods based on measurements of output and input variables acting on the system. In this case, the process of con­structing a mathematical model is reduced to a sequential enumeration of possible models of increasing complexity. As a criterion for the adequacy of the model, the norm of deviation of the output of the adjusted model from the measured value of the output of the system under study is considered. The article deals with the construction of linear models, the complexity of which is determined by their dimension. In the framework of nonstochastic approach it is developed the methodological and mathematical basis for model reconstruction which describes processes in complex systems. Asymptotic modelling allows for such system to form model classes appropriate to solve identification problem. Precise description corresponds to infinite expansion so the model quality is improved when its dimension is increased. However errors in available data do not allow increase their dimension limitlessly due to ill-conditionality of the identification problem beginning from some dimension. Regularization procedure permits to determine the effective approximate solution of identification problem which for nonstochastic case is in agreement with errors in data. Properties and peculiarities of the proposed approach are illustrated by simulation results.