Instrumental variables for nonlinearity recovering in block-oriented systems driven by correlated signals

The goal of the paper is to identify the Hammerstein-type systems excited and disturbed by correlated random processes. The problem is semi-parametric in the sense that the nonlinear static characteristic is recovered without prior knowledge about the linear dynamic block, i.e. when its order is unknown. The method is based on the instrumental variables technique, with the instruments generated by input inverse filtering. It is proved that, in contrast to the least-squares-based approach, the proposed algorithm leads to an asymptotically unbiased, strongly consistent estimate. Constructive procedures of instrumental variables generation are given for some popular cases.     

Iterative identification of Hammerstein systems

Convergences of iterative algorithms have been established for identification of Hammerstein systems in the case that the unknown nonlinearities are odd. Then, the results are further extended to nonsmooth nonlinearities.     

一种多输入单输出Hammerstein系统的集成辨识方法

针对多输入单输出(MISO)Hammerstein系统提出了一种稳态与动态辨识相结合的集成辨识方法.该方法利用稳态信息获取稳态模型的强一致性估计,并通过稳态模型以神经网络获得其非线性逼近函数,再利用动态信息辨识获取多输入单输出(MISO)Hammerstein系统的线性子系统未知参数的一致性估计.仿真结果表明了该方法的有效性和实用性.     

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.     

On identification of Hammerstein and Wiener model with application to virtualised software system

This paper proposes a system identification method for estimating virtualised software system dynamics within the framework of a Hammerstein–Wiener model. Building on the authors’ previous work in identification and control of the software systems, the approach utilises frequency sampling filter structure to describe the linear dynamics and B-spline curve functions for the inverse static output nonlinearity. Furthermore, the issue on parameter selection for B-spline model approximation of scatter data is addressed by using a data clustering method. An experimental test-bed of virtualised software system is established to generate real observational data which are used to confirm the performance of the proposed approach. The identification results have shown that the model efficacy is increased with the proposed approach because the dimension of the nonlinear model can be significantly reduced while maintaining the desired accuracy.     

Indirect closed-loop identification by optimal instrumental variable method

Recently, a new bias-compensating least-squares (BCLS) method was proposed for the identification of a closed-loop system with high-order controller. The major feature of this method is that it can achieve consistent parameter estimation without modelling the coloured noises acting on the system. This paper studies the connection between the BCLS method and the instrumental variable (IV) family. It is shown that the BCLS method is a kind of weighted instrumental variable (WIV) method whose results do not depend upon the order of the controller.     

Recursive identification of Hammerstein systems with dead-zone nonlinearity in the presence of bounded noise

The existing identification algorithms for Hammerstein systems with dead-zone nonlinearity are restricted by the noise-free condition or the stochastic noise assumption. Inspired by the practical bounded noise assumption, an improved recursive identification algorithm for Hammerstein systems with dead-zone nonlinearity is proposed. Based on the system parametric model, the algorithm is derived by minimising the feasible parameter membership set. The convergence conditions are analysed, and the adaptive weighting factor and the adaptive covariance matrix are introduced to improve the convergence. The validity of this algorithm is demonstrated by two numerical examples, including a practical DC motor case.     

基于非参数回归的路段速度估算方法*

城市道路交通运行状况的及时获取对交通管理和出行有着重要的作用,而准确判断交通状况需要利用准确的交通流速度。针对城市道路配备有一定数量的固定检测器,但是浮动车覆盖率不足的情况,提出了基于非参数回归的路段速度估算方法。该方法有效结合固定检测器数据和浮动车数据,对不同时段、路段、天气对行驶速度的影响分别进行分析,建立状态模式库,利用模式匹配估算出道路交通流速度。通过算例分析比较,可知该算法具有较高的准确度。     

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.     

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.     

Parameter learning for the nonlinear system described by Hammerstein model with output disturbance

A novel parameter learning scheme using multi-signal processing is developed that aims at estimating parameters of the Hammerstein nonlinear model with output disturbance in this paper. The Hammerstein nonlinear model consists of a static nonlinear block and a dynamic linear block, and the multi-signals are devised to estimate separately the nonlinear block parameters and the linear block parameters; the parameter estimation procedure is greatly simplified. Firstly, in view of the input–output data of separable signals, the linear block parameters are computed through correlation analysis method, thereby the influence of output noise is effectively handled. In addition, model error probability density function technology is employed to estimate the nonlinear block parameters with the help of measurable input–output data of random signals, which not only controls the space state distribution of model error but also makes error distribution tends to normal distribution. The simulation results demonstrate that the developed approach obtains high learning accuracy and small modeling error, which verifies the effectiveness of the developed approach.     

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.     

一类偶型Hammerstein模型辨识的相关算法

在介绍了一类输出只与其输入的平方有关的偶型H模型之后,分析了现有相关算法无法估计此类模型参数的原因,证明了若采用非过零逆重复M序例做激励信号,则仍然可以用相关方法估计其模型参数,并导出了这一模型参数估计的新相关算法。此算法还可用于估计一般的H模型参数。     

Variable selection in identification of a high dimensional nonlinear non-parametric system

The problem of variable selection in system identification of a high dimensional nonlinear non-parametric system is described. The inherent difficulty, the curse of dimensionality, is introduced. Then ...     

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.     

基于Hammerstein模型描述的非线性系统辨识新方法

Hammerstein模型常用来描述pH值或具有幂函数、死区、开关等特性的过程,本文提出了一种辨识此类对象模型结构和参数的新方法,首先将非线性静态部分和线性动态部分分别用非线性基和Laguerre级数表示,然后通过最小二乘法、矩阵特征值分解和矩阵扩维,辨识出两部分参数．并证明了该方法在输出端存在白噪声情况下误差的收敛性．此方法仅需假设输入为持续激励,适用范围广,计算简单,辨识精度高．最后通过pH中和滴定实验验证了以上结论．     

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.     

Decoupling the linear and nonlinear parts in Hammerstein model identification

The main result of this paper is to show that the linear part can be made decoupled from the nonlinear part in Hammerstein model identification. Therefore, identification of the linear part for a Hammerstein model becomes a linear problem and accordingly enjoys the same convergence and consistency results as if the unknown nonlinearity is absent.     

原子力显微镜中微悬臂梁分布参数系统的Hammerstein模型

为提高原子力显微镜(atomic force microscope,AFM)中微悬臂梁分布参数模型的精度,本文提出了包含非线性时空特性的改进模型,在此基础上简化控制器的结构.首先加入非线性补偿项修正传统分布参数模型;然后采用Karhunen-Loève(K–L)方法提取系统主导空间基函数,实现系统输出的时空变量分离.利用求解得到的时间系数和系统激励,建立系统时域Hammerstein模型,使系统无限维偏微分方程模型转化为时域有限维常微分方程形式,控制器的设计无需考虑空间耦合的影响;最后,利用最小二乘支持向量机结合奇异值分解法辨识模型中的参数.与传统分布参数模型进行仿真和实验结果比较,验证了方法的有效性.     

Impulse response constrained LS-SVM modelling for MIMO Hammerstein system identification

In this paper, a new methodology for identifying multiple inputs multiple outputs Hammerstein systems is presented. The proposed method aims at incorporating the impulse response of the system into a least-squares support vector machine (LS-SVM) formulation and therefore the regularisation capabilities of LS-SVM are applied to the system as a whole. One of the main advantages of this method comes from the fact that it is flexible concerning the class of problems it can model and that no previous knowledge about the underlying non-linearities is required except for very mild assumptions. Also, it naturally adapts to handle different numbers of inputs/outputs and performs well in the presence of white Gaussian noise. Finally, the method incorporates information about the structure of the system but still the solution of the model follows from a linear system of equations. The performance of the proposed methodology is shown through three simulation examples and compared with other methods in the literature.