Zygmunt Hasiewicz

On Nonparametric Identification of Wiener Systems

In this paper, a new method for the identification of the Wiener nonlinear system is proposed. The system, being a cascade connection of a linear dynamic subsystem and a nonlinear memoryless element, is identified by a two-step semiparametric approach. The impulse response function of the linear part is identified via the nonlinear least-squares approach with the system nonlinearity estimated by a pilot nonparametric kernel regression estimate. The obtained estimate of the linear part is then used to form a nonparametric kernel estimate of the nonlinear element of the Wiener system. The proposed method permits recovery of a wide class of nonlinearities which need not be invertible. As a result, the proposed algorithm is computationally very efficient since it does not require a numerical procedure to calculate the inverse of the estimate. Furthermore, our approach allows non-Gaussian input signals and the presence of additive measurement noise. However, only linear systems with a finite memory are admissible. The conditions for the convergence of the proposed estimates are given. Computer simulations are included to verify the basic theory

Nonlinear system identification by the Haar multiresolution analysis

The paper deals with the problem of reconstruction of nonlinearities in a certain class of nonlinear systems of composite structure from their input-output observations when prior information about the system is poor, thus excluding the standard parametric approach to the problem. The multiresolution idea, being the fundamental concept of modern wavelet theory, is adopted, and the Haar multiresolution analysis in particular is applied to construct nonparametric identification techniques of nonlinear characteristics. The pointwise convergence properties of the proposed identification algorithms are established. Conditions for the convergence are given; and for nonlinearities satisfying a local Lipschitz condition, the rate of convergence is evaluated, With applications in mind, the problem of data-driven selection of the optimum resolution degree in the identification procedure, essential for the multiresolution analysis, is considered as well. The theory is verified by computer simulations.

Combined parametric-nonparametric identification of Hammerstein systems

A novel, parametric-nonparametric, methodology for Hammerstein system identification is proposed. Assuming random input and correlated output noise, the parameters of a nonlinear static characteristic and finite impulse-response system dynamics are estimated separately, each in two stages. First, the inner signal is recovered by a nonparametric regression function estimation method (Stage 1) and then system parameters are solved independently by the least squares (Stage 2). Convergence properties of the scheme are established and rates of convergence are given.

Hammerstein system identification by non-parametric instrumental variables

A mixed, parametric–non-parametric routine for Hammerstein system identification is presented. Parameters of a non-linear characteristic and of ARMA linear dynamical part of Hammerstein system are estimated by least squares and instrumental variables assuming poor a priori knowledge about the random input and random noise. Both subsystems are identified separately, thanks to the fact that the unmeasurable interaction inputs and suitable instrumental variables are estimated in a preliminary step by the use of a non-parametric regression function estimation method. A wide class of non-linear characteristics including functions which are not linear in the parameters is admitted. It is shown that the resulting estimates of system parameters are consistent for both white and coloured noise. The problem of generating optimal instruments is discussed and proper non-parametric method of computing the best instrumental variables is proposed. The analytical findings are validated using numerical simulation results.

Nonparametric identification of nonlinearities in block-oriented systems by orthogonal wavelets with compact support

The paper addresses the problem of identification of nonlinear characteristics in a certain class of discrete-time block-oriented systems. The systems are driven by random stationary white processes (independent and identically distributed input sequences) and disturbed by stationary, white, or colored random noise. The prior information about nonlinear characteristics is nonparametric. In order to construct identification algorithms, the orthogonal wavelets of compact support are applied, and a class of wavelet-based models is introduced and examined. It is shown that under moderate assumptions, the proposed models converge almost everywhere (in probability) to the identified nonlinear characteristics, irrespective of the noise model. The rule for optimum model-size selection is given and the asymptotic rate of convergence of the model error is established. It is demonstrated that, in some circumstances, the wavelet models are, in particular, superior to classical trigonometric and Hermite orthogonal series models worked out earlier.

Identification of a linear system observed through zero-memory non-linearity

The paper deals with the identification problem for the Wiener-type system, where the memoryless output non-linearity is known but is not necessarily one-to-one or even monotonic. First, the deterministic identifiability test for the linear dynamic segment is derived assuming that the respective segment outputs are accessible for direct measurements. Next, using the results by Masry and Cambanis (1980), a least-squares type parameter estimation algorithm for linear dynamics is proposed for the case where only the noise-corrupted outputs of the overall non-linear tandem can be gained. A convergence with probability one is shown of the parameter estimate to the true value of the system parameter vector and some computational aspects of the proposed algorithm are discussed.

Non-parametric estimation of non-linearity in a cascade time-series system by multiscale approximation

Abstract The paper addresses the problem of using multiscale approximation for the identification of non-linearities in Hammerstein systems. The exciting signals are random, stationary and white, with a bounded (unknown) probability density function, and system outputs are corrupted by a zero-mean stationary random noise – white or coloured. The a priori information is poor. In particular, no parametric form of the non-linear characteristics is known in advance. To recover non-linearities, a class of non-parametric identification algorithms is proposed and investigated. The algorithms use only input–output measurements and are based on multiscale orthogonal approximations associated with scaling functions of compact support. We establish the pointwise weak consistency of such routines along with asymptotic rates of convergence. In particular, local ability of the algorithms to discover non-linear characteristics in dependence on local smoothness of the identified non-linearity, input density and the scaling function is examined. It is shown that under mild requirements the routines attain optimal rate of convergence. The form and convergence of the algorithms are insensitive to correlation of the noise.

Modular neural networks for non-linearity recovering by the Haar approximation

The paper deals with the design of a composite neural system for recovering non-linear characteristics from random input-output measurement data. It is assumed that non-linearity output measurements are corrupted by an additive zero-mean white random noise and that the input excitation is an i.i.d. random sequence with an arbitrary (and unknown) probability density function. A class of modular networks is developed. The class is based on the Haar approximation of functions with piecewise constant functions on a refinable grid and consists of the networks composed of perceptron-like modules connected in parallel. The networks provide a local mean value estimators of functions. The relationship between complexity and accuracy of modular networks is analysed. It is shown that under mild conditions on the non-linearities and input probability density functions the networks yield pointwise consistent estimates of non-linear characteristics, provided that complexity of the networks grows appropriately with the number of training data. Efficiency of the networks is examined and the asymptotic rate of convergence of the network estimates is established. Specifically, local ability of the networks to recover non-linear characteristics in dependence on local smoothness of the underlying non-linear function and the input probability density is discussed. Optimum complexity selection rules, guaranteeing the best performance of the networks, are given. Illustrative simulation examples are provided.

Computational algorithms for multiscale identification of nonlinearities in Hammerstein systems with random inputs

Simple and efficient computational algorithms for nonparametric wavelet-based identification of nonlinearities in Hammerstein systems driven by random signals are proposed. They exploit binary grid interpolations of compactly supported wavelet functions. The main contribution consists in showing how to use the wavelet values from the binary grid together with the fast wavelet algorithms to obtain the practical counterparts of the wavelet-based estimates for irregularly and randomly spaced data, without any loss of the asymptotic accuracy. The convergence and the rates of convergence are examined for the new algorithms and, in particular, conditions for the optimal convergence speed are presented. Efficiency of the algorithms for a finite number of data is also illustrated by means of the computer simulations.

Applicability of least-squares to the parameter estimation of large-scale no-memory linear composite systems

The parameter estimation of a large-scale interconnected static linear system is treated with the structure described by an interconnection matrix. The standard least-squares algorithm is investigated, and its ability to recover, with probability one. composite system parameters from the data accessible in the complex system is examined. The limit properties of the least-squares estimate in the particular context of a complex system are derived, and both local (referring to individual system elements) and structural (relating to the complex system in the large) necessary and sufficient applicability conditions are established.