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On nonparametric identification with prediction of time-varying systems. (English) Zbl 0547.93071
This paper is concerned with the problem of nonparametric identification with prediction of time-varying systems of the form $$y_ n=R_ n(u_ n)+z_ n,\quad n=1,2,...,$$ where $$R_ n$$ are unknown Borel-measurable functions, $$u_ n$$ and $$y_ n$$ inputs and outputs, $$z_ n$$ measurement noises independent of $$u_ n$$ such that $$Ez_ n=0, Ez^ 2_ n=\rho^ 2_ n\leq\sigma^ 2, n=1,2,...$$. The input sequence $$u_ n$$ represents some random vectors with the same probability density function f. So the problem is to construct an on-line algorithm tracking for $$R_{n+k}(u)=E[y_{n+k}| u_{n+k}=u],\quad k\geq 0$$ based on the sequence of observations $$(u_ i,y_ i,\quad i=1,...,n).$$ In this paper two algorithms tracking for $$R_{n+k}(k\geq 0)$$ of the form $$\hat R_ n(u)=\hat h_ n(u)/f(u), \hat r_ n(u)=\hat h_ n(u)/\hat f(u)$$ are discussed respectively for the cases of unknown and known function f. The values $$\hat h{}_ n(u)$$ are estimates of $$h_ n(u)$$ with some definite properties, and $$\hat f{}_ n$$ is a nonparametric estimate of f. It is shown that under some conditions $$E[\hat R_ n(u)-R_{n+k}(u)]^ 2\to^{n}0$$ for fixed $$k\geq 0$$ and $$|\hat R_ n(u)- R_{n+k}(u)|\to^{n}0$$ with probability 1 for fixed $$k\geq 0$$. As a consequence it is shown that under the additional conditions $$\hat f_ n(u)\to^{n}f(u), | R_ n(u)\|\hat f_ n(u)- f(u)|\to^{n}0$$ (with probability 1), then $$|\hat r_ n(u)- R_{n+k}(u)|\to^{n}0,\quad k\geq 0$$ (with probability 1) at a fixed point $$u\in A$$.
Reviewer: D.Syzdykov

##### MSC:
 93E12 Identification in stochastic control theory 93C99 Model systems in control theory 93E25 Computational methods in stochastic control (MSC2010) 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 93E10 Estimation and detection in stochastic control theory 62M20 Inference from stochastic processes and prediction
##### Keywords:
nonparametric identification; on-line algorithm
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