Revisiting Hammerstein system identification through the Two-Stage Algorithm for bilinear parameter estimation (original) (raw)

Hammerstein System Identification by a Semi-Parametric Method

2000

A semi-parametric algorithm for identification of Hammerstein systems in the presence of correlated noise is proposed. The procedure is based on the non-parametric kernel regression estimator and the standard least squares. The advantages of the method in comparison with the standard non-parametric approach are discussed. Limit properties of the proposed estimator are studied, and the simulation results are presented.

New results for Hammerstein system identification

1995

Abstract A novel approach is presented for the analysis and design of identification algorithms for Hammerstein models, which consist of a static nonlinearity followed by an LTI system. The authors examine two identification problems. In the first problem, the system is excited with white noise and the LTI system is FIR, and they find a simple explicit solution for the optimal parameter estimate and show that for sufficiently large data lengths a standard iterative technique globally converges to this optimal value.

Combined parametric-nonparametric identification of Hammerstein systems

Ieee Transactions on Automatic Control, 2004

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.

A Decoupling Derivative-Based Approach for Hammerstein System Identification

Proceedings of the 17th IFAC World Congress, 2008, 2008

This paper proposes a non iterative algorithm for the identification of Hammerstein model, using the sampled output data obtained from the step response, giving a continuoustime model for the linear part and a point-wise estimation of the nonlinear one. Key in the derivation of the results is the algebraic derivative method in the frequency domain yielding exact formula in terms of multiple integrals of the output signal, when placed in the time domain. By investigating the connection between such integrals and parameters to be estimated, a set of three linear regression equations is proposed. The first equation is used to estimate the structure of poles in the linear part, the second to estimate a point of the nonlinearity, the third to estimate the structure of zeros in the linear part. No a priori knowledge of the structure of the nonlinearity is required. The proposed algorithm is numerically robust, since it is based only on least squares estimation. Simulation results validate the proposed algorithm.

Hammerstein system identification by non-parametric instrumental variables

International Journal of Control, 2009

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.

An adaptive approximation method for Hammerstein systems identification

2012 IEEE International Conference on Control Applications, 2012

This paper aims to describe a new identification method for Hammerstein systems relying on the framework of basis functions approximation in order to obtain an adequate model for the nonlinear static component. The specific coefficients of the basis functions approximation and also the parameters of the linear dynamic component are estimated using a nonlinear least squares method based on a modified version of Gauss-Newton algorithm. An algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity approximation of the nonlinear component built on a grid hierarchy using adaptive bases. Such bases provides a powerful means to detect local singularities and often lead to quite simple refinement strategies. Finally, we present some numerical results for our method that show its efficiency.

Hammerstein system identification by nonparametric instrumental variables,” Accepted for publication in

2008

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 numeri...

Hammerstein system identification through best linear approximation inversion and regularisation

International Journal of Control, 2017

Hammerstein systems are composed by the cascading of a static nonlinearity and a linear system. In this paper, a methodology for identifying such systems using a combination of least squares support vector machines (LS-SVM) and best linear approximation (BLA) techniques is proposed. To do this, a novel method for estimating the intermediate variable is presented allowing a clear separation of the identification steps. First, an approximation to the linear block is obtained through the BLA of the system. Then, an approximation to the intermediate variable is obtained using the inversion of the estimated linear block and the known output. Afterwards, a nonlinear model is calculated through LS-SVM using the estimated intermediate variable and the known input. To do this, the regularisation capabilities of LS-SVM play a crucial role. Finally, a parametric re-estimation of the linear block is made. The method was tested in three examples, two of them with hard nonlinearities, and was compared with four other methods showing very good performance in all cases. The obtained results demonstrate that also in the presence of noise, the method can effectively identify Hammerstein systems. The relevance of these findings lies in the fact that it is shown how the regularisation allows to bypass the usual problems associated with the noise backpropagation when the inversion of the estimated linear block is used to compute the intermediate variable.

Blind maximum likelihood identification of Hammerstein systems

Automatica, 2008

This paper is about the identification of discrete-time Hammerstein systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramér-Rao lower bound is calculated. In practice, the latter can be computed accurately without using the strong law of large numbers. A two-step procedure is described that allows to find high quality initial estimates to start up the iterative Gauss-Newton based optimization scheme. The paper includes the illustration of the method on a simulation example. A theoretical analysis demonstrates that additive output measurement noise introduces a bias that is proportional to the variance of that additive, unmodeled noise source. The simulations support this result, and show that this bias is insignificant beyond a certain Signal-to-Noise Ratio (40 dB in the example).

An identification algorithm for Hammerstein systems using subspace method

Proceedings of the 2011 American Control Conference, 2011

This paper describes a new algorithm for the identification of single-input single-output Hammerstein systems using the multivariable output error state space (MOESP) class of subspace identification algorithms. The algorithm consists of three main steps. First, the MOESP algorithm is used to determine the system order and estimate two of the state space model matrices. Second, a least squares problem is solved to minimize the prediction error. Finally, the global search optimization is needed to be used to estimate optimal values for the remaining parameters. Performance of the model was evaluated by simulating a model of ankle joint reflex stiffness, a well known Hammerstein system. The results demonstrate that the algorithm estimated the model parameters very accurately in the presence of additive, output noise.