The purpose of this paper is to develop a basic method of approximating identification in a function space having a Schauder decomposition. Consider the time dependent nonlinear systems x.(t)=A(t)x(t)+f(t,x(t);B(t)), x(0)=x0 and y(t)=hx(t)+e(t). Our estimation problem is to find time dependent parameters A(t) and B(t) in the function space c[0, 1] such that J(A,B)=∫01y(t)−hx(t;A,B)2dt12 is a mininum. It can be discretized to the problem: find A^n and B^n in the range of Pn such that JA^n,B^n is a minimum, where Pn is a Schauder decomposition for the function Space Z[0, 1] (Z[0, 1] contains C[0, 1]).The convergence of approximation is descussed. Let Z[0, 1] be a Banach space with a Schauder decomposition Pn, let (A^n,B^n) be a sequence of best approximation estimate. If (A^,B^) is the limit of (A^n,B^n) in Z[0, 1] Then xA^n,B^n−xA^,B^→0, J(A^,B^)=limn→∞J(A^n,B^n) and inf J(A,B)=JA^,B^. Finally numerical example of the problem is included.
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