Grzegorz Mzyk

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.

A Censored Sample Mean Approach to Nonparametric Identification of Nonlinearities in Wiener Systems

A new, censored sample mean nonparametric identification algorithm for estimation of a nonlinear characteristic in Wiener system using properly preselected input-output data is proposed. Conditions imposed on the unknown characteristic are weak. In particular, its invertibility and global continuity are not required. The algorithm is based on computation of local sample-mean of proper output measurements. The mean square consistency of the estimate is proved for each continuity point of the unknown characteristic and the issue of the asymptotic convergence rate is discussed. Computer simulations are included to illustrate efficiency of the method also for small and moderate number of data.

Nonlinearity Recovering in Hammerstein System from Short Measurement Sequence

The problem of data pre-filtering for nonparametric identification of Hammerstein system from short (finite) data set is considered. The two-stage method is proposed. First, the linear dynamic block is identified using instrumental variables technique, and the inverse of the obtained model is used for output filtering. Next, the standard procedure of nonparametric regression function estimation (kernel-based, or using orthogonal series expansion) is applied, involving the filtered output sequence instead of the original one. It is shown, that for small and moderate number of data, the estimation error can be significantly reduced in comparison with standard nonparametric methods. The asymptotic properties of the method (consistency and rate of convergence) remain the same as in the classical versions of nonparametric algorithms.

Generalized Kernel Regression Estimate for the Identification of Hammerstein Systems

Generalized Kernel Regression Estimate for the Identification of Hammerstein Systems A modified version of the classical kernel nonparametric identification algorithm for nonlinearity recovering in a Hammerstein system under the existence of random noise is proposed. The assumptions imposed on the unknown characteristic are weak. The generalized kernel method proposed in the paper provides more accurate results in comparison with the classical kernel nonparametric estimate, regardless of the number of measurements. The convergence in probability of the proposed estimate to the unknown characteristic is proved and the question of the convergence rate is discussed. Illustrative simulation examples are included.

Combined Parametric-Nonparametric Identification of Block-Oriented Systems

Hammerstein system -- Wiener system -- Wiener-Hammerstein (sandwich) system -- Large-scale interconnected systems -- Structure detection and model order selection -- Time-varying systems -- Simulation studies -- Summary.

Wiener-Hammerstein system identification with non-Gaussian input.

Abstract The paper addresses the problem of non-parametric estimation of the static characteristic in Wiener-Hammerstein (sandwich) system excited and disturbed by random processes. A new, kernel-like method is presented. The proposed estimate is consistent under small amount of a priori information. An IIR dynamics, non-invertible static non-linearity, and non-Gaussian excitations are admitted. The convergence of the estimate is proved for each continuity point of the static characteristic and the asymptotic rate of convergence is analysed. The results of computer simulation example are included to illustrate the behaviour of the estimate for moderate number of observations.

Kernel-based identification of Wiener–Hammerstein system

Abstract This paper addresses the problem of Wiener–Hammerstein (LNL) system identification. We present two estimates, which recover the static nonlinear characteristic and the linear dynamic blocks separately. Both algorithms are based on kernel preselection of data and application of local least squares and cross-correlation techniques. Formal proofs of consistency are derived under very mild a priori restrictions imposed on the input excitation and system characteristics. In particular, the input need not be Gausssian, and a wide (nonparametric) class of nonlinear characteristics is admitted. Finally, we propose a universal multi-stage identification strategy which allows to split the resulting linear model into two separate blocks. We also present a simple simulation example to illustrate the behavior of the method in practice.

Parametric Versus Nonparametric Approach to Wiener Systems Identification

The problem of nonlinear dynamic systems modelling by means of block-oriented models has been strongly elaborated for the last four decades, due to vast variety of applications. The concept of block-oriented models assumes that the real plant, as a whole, can be treated as a system of interconnected blocks, static nonlinearities (N) and linear dynamics (L), where the interaction signals cannot be measured.

Semiparametric Approach to Hammerstein System Identification

Abstract Two semiparametric algorithms to recover a nonlinear characteristic in a Hammerstein system are proposed. Both are obtained by incorporating a parametric component into the kernel nonparametric algorithm. For small number of observations, their identification errors are smaller than that of the purely nonparametric algorithm. The same idea is also proposed for identification of linear dynamic component. Parametric instrumental variables estimate is elastically substituted by the nonparametric correlation-based method, when the number of observations tends to infinity.