Hammerstein-Wiener system estimator initialization

In nonlinear system identification, the system is often represented as a series of blocks linked together. Such block-oriented models are built with static nonlinear subsystems and linear dynamic systems. This paper deals with the identification of the Hammerstein-Wiener model, which is a block-oriented model where a linear dynamic system is surrounded by two static nonlinearities at its input and output. The proposed identification scheme is iterative and will be demonstrated on measurements. It will be proven that on noiseless data and in absence of modeling errors, the optimization procedure converges to the true system locally.     

Identification Scheme for Fractional Hammerstein Models With the Delayed Haar Wavelet

The parameter identification of a nonlinear Hammerstein-type process is likely to be complex and challenging due to the existence of significant nonlinearity at the input side. In this paper, a new parameter identification strategy for a block-oriented Hammerstein process is proposed using the Haar wavelet operational matrix(HWOM). To determine all the parameters in the Hammerstein model, a special input excitation is utilized to separate the identification problem of the linear subsystem from the complete nonlinear process. During the first test period, a simple step response data is utilized to estimate the linear subsystem dynamics. Then, the overall system response to sinusoidal input is used to estimate nonlinearity in the process. A single-pole fractional order transfer function with time delay is used to model the linear subsystem. In order to reduce the mathematical complexity resulting from the fractional derivatives of signals, a HWOM based algebraic approach is developed. The proposed method is proven to be simple and robust in the presence of measurement noises. The numerical study illustrates the efficiency of the proposed modeling technique through four different nonlinear processes and results are compared with existing methods.     

风洞系统非线性块状结构模型

本文将非线性块状模型的建模思想引入风洞系统模型的建立过程中,针对主排气阀和栅指电液伺服机构具有死区非线性特性,分别用含有死区输入的Hammerstein块状模型描述其动态特性,将主排气阀和栅指机构的输出作为风洞流场的输入,建立两输入两输出多变量耦合动态模型.两个独立的Hammerstein子模型与线性动态耦合的风洞流场模型串联构成一个非线性多变量块状模型.采用自适应加权递推辨识算法在线辨识Hammerstein子模型参数,采用带有遗忘因子的递推最小二乘法辨识风洞流场模型参数.仿真与风洞现场测试结果验证了本文方法的有效性.     

Fixed point iteration in identifying bilinear models

Inspired by fixed point theory, an iterative algorithm is proposed to identify bilinear models recursively in this paper. It is shown that the resulting iteration is a contraction mapping on a metric space when the number of input–output data points approaches infinity. This ensures the existence and uniqueness of a fixed point of the iterated function sequence and therefore the convergence of the iteration. As an application, one class of block-oriented systems represented by a cascade of a dynamic linear (L), a static nonlinear (N) and a dynamic linear (L) subsystems is illustrated. This gives a solution to the long-standing convergence problem of iteratively identifying LNL (Winer–Hammerstein) models. In addition, we extend the static nonlinear function (N) to a nonparametric model represented by using kernel machine.     

Identification of block-oriented nonlinear systems using orthonormal bases

In this paper, new noniterative algorithms for the identification of (multivariable) block-oriented nonlinear models consisting of the interconnection of linear time invariant systems and static nonlinearities are presented. The proposed algorithms are numerically robust, since they are based only on least squares estimation and singular value decomposition. Two different block-oriented nonlinear models are considered in this paper, viz., the Hammerstein model, and the Wiener model. For the Hammerstein model, the proposed algorithm provides consistent estimates even in the presence of colored output noise, under weak assumptions on the persistency of excitation of the inputs. For the Wiener model, consistency of the estimates can only be guaranteed in the noise free case. Key in the derivation of the results is the use of basis functions for the representation of the linear and nonlinear parts of the models. The performance of the proposed identification algorithms is illustrated through simulation examples of two benchmark problems drawn from the process control literature, viz., a binary distillation column and a pH neutralization process.     

Application of Wiener model predictive control (WMPC) to an industrial C2-splitter

Dual composition control of a high-purity distillation column is recognized as an industrially important, yet notoriously difficult control problem. It is proposed, however, that Wiener models, consisting of a linear dynamic element followed in series by a static nonlinear element, are ideal for representing this and several other nonlinear processes. They are relatively simple models requiring little more effort in development than a standard linear step response model, yet offer superior characterization of systems with highly nonlinear gains. Wiener models may be incorporated into MPC schemes in a unique way that effectively removes the nonlinearity from the control problem, preserving many of the favorable properties of linear MPC, especially in the analysis of stability. In this paper, Wiener model predictive control is applied to an industrial C2-splitter at the Orica Olefines plant with promising results.     

Modeling of pH process using wavenet based Hammerstein model

This paper outlines an approach for developing a Hammerstein model for nonlinear dynamic systems. The nonlinearity is sought to be captured through functional approximation using wavelets cast in a wavenet structure. Nonlinear block of wavenet at input side is cascaded with a linear dynamic block described by a state space model. A sequential approach is used for development of static nonlinear and linear dynamic parts of the model. Configuration and parameters of the nonlinear wavenet structure are determined from near steady state data extracted from dynamic test data while the state space model parameters of the linear dynamic part are obtained using a subspace identification approach. This approach has been applied for modeling a strongly nonlinear pH process operated over a wide range of operating conditions.     

辨识Hammerstein模型的两步法

本文利用稳态和动态信息提出了一种辨识Ｈａｍｍｅｒｓｔｅｉｎ模型的新方法－两步法，该方法利用稳态信息获取非线性增益的强一致性估计，利用动态信息获取线性子系统未知参数的强一致估计，该方法具有计算简单和辨识精度高等优点，最后的仿真结果说明了该方法的有效性和实用性。     

System identification by operatorial cancellation of nonlinear terms and application to a class of Volterra models

In this paper, a method is proposed for the identification of some SISO nonlinear models with two ill‐known components of different nature: a linear (possibly dynamic) part and a static nonlinear one. This method is well adapted when no a priori information is available about the nonlinear component to be identified. It is based on a difference operator, which enables to cancel the nonlinear term when applied to the model. Only the ill‐known linear part remains in the transformed model; it can therefore be identified independently of the nonlinear term. Based on the identified linear component, we have access to a pseudograph of the nonlinear term, whose shape can give precious information for the parameterization of the unknown nonlinear part and its identification. The identification model under consideration is defined in an abstract framework, with very weak hypotheses, so that the proposed approach has a large scope. To highlight the method, a class of dynamic Volterra models including some hybrid models such as dynamic inclusions is considered for application. Copyright © 2016 John Wiley & Sons, Ltd.     

Steady-state identification for large-scale industrial process by means of dynamic models

This paper investigates the steady-state identification of the large-scale industrial processes. Under mild conditions, the estimate of the steady-state model is formed from the estimated parameters of the approximate linear dynamic models of subsystems. To a class of nonlinear slow time-varying large-scale processes, which have many subsystems interconnected with one another, a parallel two-stage identification algorithm is put forward. The consistency of the estimate and the convergence of the parallel iteration are also proved. Simulation examples have shown that this new identification approach is efficient and reliable for the establishment of the steady-state model of the large-scale industrial process.     

Adaptive on-line steady-state optimization of slow dynamic processes

A software package OLIOPT was developed for the on-line optimization of the steady-state behaviour of slow dynamic processes in a relatively short time period. In the starting phase, the independently variable inputs are changed according to a special test signal. A nonlinear dynamic process model is identified on-line. Based on the static part of the model and the known inputs, the gradients of the performance index are calculated. An optimization algorithm changes the inputs towards their optimal values. On-line identification of the nonlinear model continues and the prediction of the optimum improves. In the last phase, the inputs take their optimal values and the process follows, feedforward controlled, to its optimal steady-state. The method is suited for industrial processes with one or more variable inputs, where a small gain in efficiency turns out to give a relatively large financial return. Results are shown for the on-line optimization of a thermal pilot process.     

A dynamic operability analysis approach for nonlinear processes

Current process operability indicators are mostly restricted to linear approximations of the process dynamics. Other operability analysis approaches that have the capability to include full nonlinear process models rely on mixed integer dynamic optimisation techniques which, in general, require large amount of computations. In this paper we propose a dynamic operability analysis approach for stable nonlinear processes that can be readily applied during process design and can be solved efficiently using a limited amount of computations. The process nonlinear dynamics are approximated by a series interconnection of static nonlinearities and linear dynamics, represented by the so-called Hammerstein–Wiener models. These type of models can often be obtained during process design where detailed steady-state nonlinear models are available, combined with some (usually limited) information on the process dynamics. Using an extended internal model control (IMC) framework, we investigate the interaction between the static nonlinearities and linear dynamics on the operability of the process. The framework extends the well-known equivalence between operability and invertibility of linear processes to nonlinear systems. In particular, by exploiting some results from the theory of passive systems we provide conditions that guarantee the existence of the inverse of the static nonlinearities. We show that the inverse can be attained inside a specific input/output region. This region imposes a constraint on the maximum magnitude of the signals that appear in the closed-loop and represents the effect of the static nonlinearities on the operability of the overall process. Dynamic operability is then quantified using a linear matrix inequality (LMI) optimisation approach that minimises a given performance criterion subject to the constraint imposed by the static nonlinearities.     

Robust model predictive control of Wiener systems

Block-oriented models (BOMs) have shown to be appealing and efficient as nonlinear representations for many applications. They are at the same time valid and simple models in a more extensive region than time-invariant linear models. In this work, Wiener models are considered. They are one of the most diffused BOMs, and their structure consists in a linear dynamics in cascade with a nonlinear static block. Particularly, the problem of control of these systems in the presence of uncertainty is treated. The proposed methodology makes use of a robust identification procedure in order to obtain a robust model to represent the uncertain system. This model is then employed to design a model predictive controller. The mathematical problem involved in the controller design is formulated in the context of the existing linear matrix inequalities (LMI) theory. The main feature of this approach is that it takes advantage of the static nature of the nonlinearity, which allows to solve the control problem by focusing only in the linear dynamics. This formulation results in a simplified design procedure, because the original nonlinear model predictive control (MPC) problem turns into a linear one.     

Hybrid fuzzy convolution modelling and identification of chemical process systems

This paper looks at a new method of modelling nonlinear dynamic processes, using grid-type Takagi-Sugeno fuzzy models and a priori knowledge. The proposed hybrid fuzzy convolution dynamic model consists of a non-linear fuzzy steady-state static and a gainindependent impulse response model-based dynamic part. The modelling of nonlinear pH processes is chosen as a realistic case study for demonstration of the proposed modelling approach. The off-line identified hybrid fuzzy convolution model is shown to be capable of modelling the nonlinear process and providing better multiple-step prediction than the conventional grid-type Takagi-Sugeno fuzzy model.     

Adaptive discrete-time sliding-mode control of nonlinear systems described by Wiener models

In this paper, we propose an adaptive control scheme that can be applied to nonlinear systems with unknown parameters. The considered class of nonlinear systems is described by the block-oriented models, specifically, the Wiener models. These models consist of dynamic linear blocks in series with static nonlinear blocks. The proposed adaptive control method is based on the inverse of the nonlinear function block and on the discrete-time sliding-mode controller. The parameters adaptation are performed using a new recursive parametric estimation algorithm. This algorithm is developed using the adjustable model method and the least squares technique. A recursive least squares (RLS) algorithm is used to estimate the inverse nonlinear function. A time-varying gain is proposed, in the discrete-time sliding mode controller, to reduce the chattering problem. The stability of the closed-loop nonlinear system, with the proposed adaptive control scheme, has been proved. An application to a pH neutralisation process has been carried out and the simulation results clearly show the effectiveness of the proposed adaptive control scheme.     

Modeling and parameter identification of systems with multisegmentpiecewise-linear characteristics

This paper deals with the modeling and parameter identification of nonlinear systems having multi-segment piecewise-linear characteristics. The decomposition of the corresponding mapping provides a new form of multi-segment nonlinearity representation, leading to an output equation where all the parameters to be estimated are separated. Hence, an iterative method with internal variable estimation can be applied for parameter identification using input/output data records. The only required a-priori knowledge of the nonlinear characteristic represents the limits for the domain partition. The proposed model of given static nonlinearity is also incorporated into the Hammerstein model. Examples of parameter identification for static and dynamic systems with multi-segment piecewise-linear characteristics are presented     

An integrating linearization method for Hammerstein models

This paper considers a method for linearization of models containing static input nonlinearities in series with a linear model, so called Hammerstein models. The method linearizes the system exactly, and is performed by differentiating the nonlinearity with respect to the input signal. Using this approach, an integration is added in the loop gain of the linearized system via the internal feedback. The method presented here is in essential senses an equivalent alternative to the standard method that utilizes the inverse of the static nonlinearity complemented with an integration. Particularly, in cases when analytic inversion is difficult, the presented method provides an attractive alternative.     

非线性动态系统的Wiener神经网络辨识法

提出了一种新的Wiener神经网络结构并将其应用于非线性动态系统辨识问题.首先,用Wiener模型对非线性动态系统进行描述,将其分解成线性动态子环节串接非线性静态增益的形式.其次,设计一种新型的神经网络结构,使网络权值对应于相应的Wiener模型参数;并推导了基于反向传播的网络权值调整方法.最后,通过网络迭代训练,可同时得到线性动态子环节和非线性静态增益的模型参数.通过一个Wiener模型的数值仿真来验证方法的有效性,仿真结果表明所提辨识方法切实可行.