This chapter focuses on selected problems of complex systems identification. The first part of the chapter is devoted to identification problems in general. The tasks of determination of the plant parameters and choice of the best model are given. Then, authors describe problems of complex systems, i.e.: identification with use of limited measurements, global identification and two-stage identification. The last one is presented in details. In order to illustrate proposed methods, an adaptive system with two-stage identification and its application to biomedical problem is presented.
TopBackground
The identification problem is to determine the model of the investigated process on the basis of measurement data collected on during the experiment. More precisely, for the identification plant (see Figure 1.) the problem is to find the relation between input values u(1), u(2), …, u(S) and output values y(1), y(2), …, y(L). For further consideration let us assume that input and output are S- and L-dimensional vectors, respectively. We denote them in the form:
(1)Figure 1. Identification plant with input u and output y
where:
u – plant input vector,
, U is a
S-dimensional input space, R is the set of real numbers,
y – plant output vector,
, Y is a
S-dimensional output space.
It is assumed that there exists plant characteristic:
, (2) but it is not known. The problem is to determine relation between input u and output y on the basis of experimental data. Denote by yn – result of n-th output measurement for the given input un, n=1, 2,…, N where N is the number of the measurements. Results of measurements are collected in the following matrices:
(3) where:
UN,
YN are matrices of input and output measurements, respectively. The identification problem is to find (see Figure 2.) the identification algorithm which allows getting the plant characteristic (plant model). The identification algorithm depends on the knowledge about investigated plant.