Wlodzimierz Greblicki

Identification of discrete Hammerstein systems using kernel regression estimates

In this note a discrete-time Hammerstein system is identified. The weighting function of the dynamical subsystem is recovered by the correlation method. The main results concern estimation of the nonlinear memoryless subsystem. No conditions concerning functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is presented.The algorithm is a nonparametric kernel estimate of regression functions calculated from dependent data. It is shown that the algorithm converges to the characteristic as the number of observations tend to infinity. For sufficiently smooth characteristics, the rate of convergence is O(n^{-2/5}) in probability.

Nonparametric identification of Wiener systems by orthogonal series

A Wiener system, i.e., a system comprising a linear dynamic and a nonlinear memoryless subsystems connected in a cascade, is identified. Both the input signal and disturbance are random, white, and Gaussian. The unknown nonlinear characteristic is strictly monotonous and differentiable and, therefore, the problem of its recovering from input-output observations of the whole system is nonparametric. It is shown that the inverse of the characteristic is a regression function and a class of orthogonal series nonparametric estimates recovering the regression is proposed and analyzed. The estimates apply the trigonometric, Legendre, and Hermite orthogonal functions. Pointwise consistency of all the algorithms is shown. Under some additional smoothness restrictions, the rates of their convergence are examined and compared. An algorithm to identify the impulse response of the linear subsystem is proposed. >

Stochastic approximation in nonparametric identification of Hammerstein systems

Derived from the idea of stochastic approximation, recursive algorithms to identify a Hammerstein system are presented. Two of them recover the characteristic of the nonlinear memoryless subsystem, while the third one estimates the impulse response of the linear dynamic part. The a priori information about both subsystems is nonparametric. Consistency in quadratic mean is shown, and the convergence rate is examined. Results of numerical simulation are also presented.

Non-parametric orthogonal series identification of Hammerstein systems

The non-linearity in a discrete system governed by the Hammerstein functional is identified. The system is driven by a random while input signal and the output is disturbed by a random white noise. No parametric a priori information concerning the non-linearity is available and non-parametric algorithms are proposed. The algorithms are derived from the trigonometric as well as Hermite orthogonal series. It is shown that the algorithms converge to the unknown characteristic in a pointwise manner and that the mean integrated square error converges to zero as the number of observations tends to infinity. The rate of convergence is examined. A numerical example is also given.

Continuous-time Hammerstein system identification

A continuous-time Hammerstein system, i.e., a system consisting of a nonlinear memoryless subsystem followed by a linear dynamic one, is identified. The system is driven and disturbed by white random signals. The a priori information about both subsystems is nonparametric, which means that functional forms of both the nonlinear characteristic and the impulse response of the dynamic subsystem are unknown. An algorithm to estimate the nonlinearity is presented and its pointwise convergence to the true characteristic is shown. The impulse response of the dynamic part is recovered with a correlation method. The algorithms are computationally independent. Results of a simulation example are given.

Nonparametric function recovering from noisy observations

Abstract We consider the nonparametric regression model Yi=g(xi)+ζi, where g is a bounded function, over the interval [0,1], to be estimated, xis are nonrandom and ζis are independent identically distributed random variables with Eζi=0. This paper studies the behavior of the general family of nonparametric estimates gn(x)=Σi=1nYiwni(x), where the weight functions {wni} are of the form wni(x)=wni(x;x1,…,xn), i=1,…,n. The family of estimates includes all known estimates proposed by Priestley and Chao (1972), Clark (1977), Gasser and Muller (1979), Cheng and Lin (1981) as well as Georgiev (1984b, 1985). Sufficient conditions for mean square and complete convergence are derived. New results for the Priestley-Chao and Gasser-Muller-Cheng-Lin estimates are obtained. Also proposed is a class of new nearest neighbor estimates of g. Finally, a simulation experiment demonstrates the remarkable success of the nearest neighbor technique with bandwidth depending on the local density of the design points.

Hammerstein system identification by non-parametric regression estimation

A discrete-time, multiple-input non-linear Hammerstein system is identified. The dynamical subsystem is recovered using the standard correlation method. The main results concern estimation of the non-linear memoryless subsystem. No conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given. The algorithm is simply a non-parametric kernel estimate of the regression function calculated from the dependent data. It is shown that the algorithm converges to the characteristic of the subsystem in the pointwise as well as the global sense. For sufficiently smooth characteristics, the rate of convergence is o(n-1/(2+d in probability, where d is the dimension of the input variable.

Nonparametric identification of a cascade nonlinear time series system

Abstract A nonparametric algorithm for identification of a nonlinear cascade time series system is presented. To recover the system nonlinearity the Legendre orthogonal polynomials based procedure is proposed. The procedure consistency and the rate of convergence are established. The identification of a linear part of the system is also studied. Both stationary and quasi-stationary linear subsystems are considered. The results are illustrated by numerical examples.

Dynamic system identification with order statistics

Systems consisting of linear dynamic and memory-less nonlinear subsystems are identified. The paper deals with systems in which the nonlinear element is followed by a linear element, as well as systems in which the subsystems are connected in parallel. The goal of the identification is to recover the nonlinearity from noisy input-output observations of the whole system; signals interconnecting the elements are not measured. Observed values of the input signal are rearranged in increasing order, and coefficients for the expansion of the nonlinearity in trigonometric series are estimated from the new sequence of observations obtained in this way. Two algorithms are presented, and their mean integrated square error is examined. Conditions for pointwise convergence are also established. For the nonlinearity satisfying the Lipschitz condition, the error converges to zero. The rate of convergence derived for differentiable nonlinear characteristics is insensitive to the roughness of the probability density of the input signal. Results of numerical simulation are also presented. >

Recursive nonparametric identification of Hammerstein systems

Abstract A discrete-time nonlinear Hammerstein system is identified. The dynamical subsystem is recovered using the standard correlation method. The main results concern the estimation of the nonlinear memoryless subsystem. The class of nonlinearities considered in the paper consists of a broad class of functions which cannot be parametrized. Two new algorithms of a recursive form for estimating the nonlinear characteristic are proposed. It is shown that they converge at all continuity points of the characteristic. The integrated absolute error also converges to zero. The algorithms are equivalent with respect to the asymptotical rate of convergence. The efficiency of the algorithms is discussed, and numerical examples are presented.