Identification Of Hammerstein Systems Research Articles

The system identification and control of Hammerstein system using non-uniform rational B-spline neural network and particle swarm optimization

In this paper a new system identification algorithm is introduced for Hammerstein systems based on observational input/output data. The nonlinear static function in the Hammerstein system is modelled using a non-uniform rational B-spline (NURB) neural network. The proposed system identification algorithm for this NURB network based Hammerstein system consists of two successive stages. First the shaping parameters in NURB network are estimated using a particle swarm optimization (PSO) procedure. Then the remaining parameters are estimated by the method of the singular value decomposition (SVD). Numerical examples including a model based controller are utilized to demonstrate the efficacy of the proposed approach. The controller consists of computing the inverse of the nonlinear static function approximated by NURB network, followed by a linear pole assignment controller.

Convergence of normalized iterative identification of Hammerstein systems

An iterative identification algorithm of Hammerstein systems needs a proper initial condition to guarantee its convergence. In this paper, we propose a new algorithm by fixing the norm of the parameter estimates. The normalized algorithm ensures the convergence property under arbitrary nonzero initial conditions. The proofs of the property also give a geometrical explanation on why the normalization guarantees the convergence. An additional contribution is that the static function in the Hammerstein system is extended to square-integrable functions.

Parameter identification of Hammerstein systems containing backlash operators with arbitrary-shape parametric borders

A two-stage parameter identification method is developed for Hammerstein systems containing backlash nonlinearities bordered by parametric arbitrary-shape lines. In the first stage, a persistently exciting input is designed so that the linear subsystem can be made decoupled from the nonlinear element. Therefore, linear subsystem identification is coped with using a least squares estimator enjoying consistency, due to input persistent excitation. Then, the backlash parameters are estimated using appropriate periodic exciting signals and consistent parameter estimators.

Making parametric Hammerstein system identification a linear problem

In this paper, we study the identification of parametric Hammerstein systems with FIR linear parts. By a proper normalization and a clever characterization, it is shown that the average squared error cost function for identification can be expressed in terms of the inner product between the true but unknown parameter vector and its estimate. Further, the cost function is concave in the inner product and linear in the inner product square. Therefore, the identification of parametric Hammerstein systems with FIR linear parts is a globally convergent problem and has one and only one (local and global) minimum. This implies that the identification of such systems is a linear problem in terms of the inner product square and any local search based identification algorithm converges globally.

Making parametric Hammerstein system identification a linear problem

AbstractIn this paper, we study identification of parametric Hammerstein systems with FIR linear parts. By a proper normalization and a clever characterization, it is shown that the average squared error cost function for identification can be expressed in terms of the inner product between the true but unknown parameter vector and its estimate. Further the cost function is concave in the inner product and linear in the inner product square. Therefore, identification of parametric Hammerstein systems with FIR linear parts is a globally convergent problem and has one and only one (local and global) minimum. This implies that identification of such systems is a linear problem in terms of the inner product square and any local search based identification algorithm converges globally.

Markov Chain Approach to Identification of Wiener, Hammerstein and NARX Systems

AbstractA Markov chain approach to identification of the Wiener, Hammerstein, and nonlinear ARX (NARX) systems is presented. The motivation of this approach comes from the fact that these classes of nonlinear systems are connected with Markov chains, and hence their asymptotical properties, such as ergodicity, stationarity, and invariant probability distribution, can be derived from the corresponding chains. The estimates are given by the stochastic approximation algorithms with expanding truncations (SAAWET). All estimates proposed in the paper are strongly consistent. Numerical examples support the validity of this approach.

Closed-loop Identification of Hammerstein Systems Using Iterative Instrumental Variables

AbstractA feedback connection of a Hammerstein system and a linear controller provides a good approach to modeling systems with actuator nonlinearity or input saturation during closed-loop experiments. This paper discusses closed-loop identification of Hammerstein systems where both input and output signals are correlated with noise, and time domain signals are related via a nonlinear map. The static nonlinearity and the linear dynamic system are parametrized independently by using triangle basis functions and rational functions respectively. The independent parametrization method causes over-parametrization of the Hammerstein system, but parameter separation is conducted using a singular value decomposition. An iterative Instrumental Variables (IV) method minimizing an (filtered) output error (OE) is proposed to identify system parameters. The proposed iterative IV identification method is applied to the experimental closed-loop time domain data from the servo actuator in a Quantum LTO-3 tape drive and the estimation result shows the effectiveness of the proposed algorithm.

Frequency identification of Hammerstein systems with switch memory nonlinearities

AbstractThe problem of identifying fully nonparametric Hammerstein systems with input nonlinearities of the switch type is addressed. The nonlinearity borders assume no particular shape and so are allowed to be non smooth and noninvertible. A frequency identification method is developed that determines a set of points of the nonlinearity borders and estimates the linear subsystem frequency gain at a number of frequencies. The identification method relies on Fourier series decomposition and involves relatively simple excitation signals.

Computational burden reduction in Set-membership Hammerstein system identification

AbstractHammerstein system identification from measurements affected by bounded noise is considered in the paper. First, we show that computation of tight parameter bounds requires the solution to nonconvex optimization problems where the number of decision variables increases with the length of the experimental data sequence. Then, in order to reduce the computational burden of the identification problem, we propose a procedure to relax the previously formulated problem to a set of polynomial optimization problems where the number of variables does not depend on the size of the measurements sequence. Advantages of the presented approach with respect to previously published results are discussed.

Recursive Identification for Hammerstein Systems with State-space Model

摘要: 一般来说, 单输入单输出情形下的Hammerstein系统可以由转移函数来表示, 而对于多输入多输出情形下的系统其输入与输出间的关系难以表示. 本文基于Hammerstein系统的状态空间模型, 研究了其子空间辨识方法. 在开环情形, 对Hammerstein系统给出了在其非线性函数可以由有限基函数线性表示的情形的子空间辨识方法, 及其递推实现. 并且初步分析了这些方法的渐近性质. 针对这一方法, 我们给出了一个数据模拟实例分析方法的优劣. 关键词: 子空间方法 / Hammerstein系统 / 递推方法 / 主成分分析 / 多输入多输出系统 / 随机逼近

Subspace identification methods for Hammerstein systems: rank constraint and dimension problem

This article studies the subspace identification methods (SIMs) for Hammerstein systems with major focus on a rank constraint and the related dimension problem. We analyse the effects of the rank constraint on the three steps of a unifying framework for SIMs: the rank constraint has no effect on the first two steps, but does so on the third step. If the rank constraint is ignored, as in the existing over-parametrised method (OPM) for Hammerstein system identification, the optimality of the resulting estimate can still be established. Even so, the OPM may suffer from the dimension problem resulting in a low numerical efficiency. To resolve the dimension problem, we propose a new subspace-based method, named as the least-parametrised method (LPM), for identification of Hammerstein systems with non-coupling input nonlinearities. Simulation results are provided to demonstrate the effectiveness of the LPM, and show the necessity of considering the rank constraint to improve the numerical efficiency.

Recursive Identification for Hammerstein Systems with State-space Model

Abstract In this paper, the subspace methods for multi-input multi-output (MIMO) Hammerstein systems in state-space form are considered, for the reason that relation of input and output in MIMO case is appropriate to indicate in this form. In open-loop case, a subspace identification method is given for Hammerstein systems, in which the nonlinear static function can be represented as a linear combination of finite basis functions, and the recursive version of this method is also given. We show that, in mild conditions, the method given in this paper is consistent in some sense. A numerical example is given to show the performance of this method.

Convergence of the iterative algorithm for a general Hammerstein system identification

The convergence of the iterative identification algorithm for a general Hammerstein system has been an open problem for a long time. In this paper, it is shown that the convergence can be achieved by incorporating a regularization procedure on the nonlinearity in addition to a normalization step on the parameters.

An affine projection-based algorithm for identification of nonlinear Hammerstein systems

There are parametric and non-parametric methods for adaptive Hammerstein system identification. The most commonly used method is the non-parametric. In reality, the linear subsystem of a Hammerstein system is not of finite impulse response and non-parametric adaptive algorithms require large matrices and therefore increase computational complexity. The objectives of this paper are to identify the Hammerstein system adaptively based on the affine projection criterion using a parametric algorithm. We also develop a bound for control of step size of the proposed algorithm and derive an expression for its mean square error performance. The error surface of the nonlinear Hammerstein filter was determined by examining the non-quadratic nature and the global and local minima of the mean square error cost function. A bound was determined for the adaptive step size and an expression was derived for the mean square error convergence based on energy conservation theory. Simulations of system identification applications showed that convergence speed of the proposed algorithm was faster and the convergence was superior to previously existing Hammerstein algorithms. Applying the new algorithm to the identification of the human muscles stretch reflex dynamics showed good convergence results. The proposed algorithm is of practical value in real life situations.

Identification of Hammerstein Systems with Quantized Observations

This work is concerned with identification of Hammerstein systems whose outputs are measured by quantized sensors. The system consists of a memoryless nonlinearity that is polynomial and possibly noninvertible, followed by a linear subsystem. The parameters of linear and nonlinear parts are unknown but have known orders. Input design, identification algorithms, and their essential properties are presented under the assumptions that the distribution function of the noise is known and the quantization thresholds are known. The concept of strongly scaled full rank signals is introduced to capture the essential conditions under which the Hammerstein system can be identified with set-valued observations. Under strongly scaled full rank conditions, a strongly convergent algorithm is constructed. Asymptotic consistency and efficiency of the algorithm are investigated.

Adaptive tracking and recursive identification for Hammerstein systems

A weighted least squares (WLS) based adaptive tracker is designed for a class of Hammerstein systems. It is proved that the tracking error is asymptotically minimized. Incorporating with the diminishing excitation technique, the minimality of the tracking error and strong consistency of the estimates for parameters of the system are simultaneously achieved. Numerical examples are given and the simulation results are consistent with the theoretical analysis.

Revisiting Hammerstein system identification through the Two-Stage Algorithm for bilinear parameter estimation

The Two-Stage Algorithm (TSA) has been extensively used and adapted for the identification of Hammerstein systems. It is essentially based on a particular formulation of Hammerstein systems in the form of bilinearly parameterized linear regressions. This paper has been motivated by a somewhat contradictory fact: though the optimality of the TSA has been established by Bai in 1998 only in the case of some special weighting matrices, the unweighted TSA is usually used in practice. It is shown in this paper that the unweighted TSA indeed gives the optimal solution of the weighted nonlinear least squares problem formulated with a particular weighting matrix. This provides a theoretical justification of the unweighted TSA, and also leads to a generalization of the obtained result to the case of colored noise with noise whitening. Numerical examples of identification of Hammerstein systems are presented to validate the theoretical analysis.

Hammerstein Systems Identification in Presence of Hard Nonlinearities of Preload and Dead-Zone Type

Hammerstein system identification is considered in presence of preload and dead zone nonlinearities. The discontinuous feature of these nonlinearities makes it difficult to get a single system parameterization involving linearly all unknown parameters (those of the linear subsystem and those of the nonlinearity). Therefore, system identification has generally been dealt with using multiple stage schemes including different parametrizations and several data acquisition experiences. However, the consistency issue has only been solved under restrictive assumptions regarding the identified system. In this paper, a new identification scheme is designed and shown to be consistent under mild assumptions.

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.

Identification of Hammerstein systems without explicit parameterisation of non-linearity

This article proposes a new approach to identification of Hammerstein systems, where a non-linearity precedes a linear dynamic system, driven by piece-wise constant inputs. The proposed approach does not require an explicit parameterisation of the non-linearity. Moreover, the non-linearity does not have to be static, but could be the one with finite memories like backlash. By exploiting input's piecewise constant property, the denominator of the linear system described by an ARX model is consistently identified from the information of the output only; next, a subspace direct equalisation method estimates the unmeasurable inner signal based on the resulted denominator estimate and output measurements. Contrary to the existing blind approaches, the numerator of the linear system is not required, which leads to a significant improvement of removing an error propagation. On the basis of the estimated inner signal, the measured input and output, the non-linearity and linear system are obtained separately. The proposed approach is validated and compared with two existing blind approaches through numerical and experimental examples.