Hammerstein model

S Lakshminarayanan, SL Shah, and K Nandakumar. Identification of Hammerstein models using multivariate statistical tools. Chem. Engg. Science, 50(22) 3599—3613, 1995. 

Transformation of Nonlinear Models Wiener-Hammerstein Models... 

Instead of fitting a fully nonlinear model, another approach to nonlinear system identification is to partition the nonUnearities from the linear component A common application of this approach is the Wiener-Hammerstein model. A Wiener-Hammerstein model is a generalisation of the Hammerstein model, where non-linearities are assumed cmly to be in the input and the Wiener model, where nonlinearities are assumed only to be in the output, which allows nonlinearities to be present in both the input and output The process model is assumed to be linear. Thus, the general form of the model can be written as... 

Weiner-Hammerstein models are useful with the actuators or sensors have significant nonlinearities in their behaviour. A common application is when the valve, used as an actuator, is not behaving normally and has significant nonlinearities, such as stiction. By removing the nonlinearities fi om the modelling process, it becomes possible to convert the initially nonlinear problem into a linear one. 

It is reasonable to assume that the actuator has many nonlinear characteristics and one of the most influential factors is the dynamics of load since the bending characteristics of the film highly depend on the load. In order to include the effect, static nonlinear block can be inserted in front of the linear dynamics of P2 s), i.e., Hammerstein model for P2 s) (see Fig. 11). 

Fig. 11 Structure of the actuator model using Hammerstein model (Reproduced from Development of an artificial muscle linear actuator using ionic polymer-metal composites. Yamakita M, Kamamichi N, Kaneda Y, Asaka K, Luo ZW, Advanced Robotics, 2004, Taylor Francis Ltd, reprinted by permission of the publisher Taylor Francis Ltd, www.tandfonline.com)...

It is therefore necessary to develop control-relevant techniques for characterizing nonlinearity. Through use of the Optimal Control Structure (OCS) approach [5], Stack and Doyle have shown that measures, such as Eq. (1), may still be applied but to a controlrelevant system structure. In the OCS approach, the necessary conditions for an optimal control trajectory given a process and performance objective are analyzed as an independent system. The nonlinearity of these equations determine the control-relevant nonlinearity. The OCS has been used to determine the control-relevance of certain commonly-exhibited nonlinear behaviors [6]. Using nonlinear internal model control (IMC) structures, similar analysis has been performed on Hammerstein and Wiener systems with polynomial nonlinearities to examine the role of performance objectives on the controlrelevant nonlinearity [7]. Though not applied to the examples in section 5, these controlrelevant analysis techniques have been shown to be beneficial and remain an active research area. 

Moreover, the nonlinearity measure methodology can be generalized to so-called suitability measures [21]. These suitability measures quantify the achievable model quality of an arbitrary model class instead of the linear model class. In Ref. 22 it has been demonstrated that an example distillation column can best be modeled by a Wiener model structure and that linear, Hammerstein and diagonal Volterra models of the same complexity achieve much worse results. Suitability measures thus allow nonlinear model structure identification. 

In this section, the application of the OCL nonlinearity measures to a simple scalar example system with saturating input nonlinearity will be illustrated. This system belongs to the class of Hammerstein systems that are commonly met in the modeling of process systems. The state equation is given by... 

Patwardhan, R.S., Lakshminarayanan, S. and Shah, S.L. (1998) Constrained non-Unear MPC using Hammerstein and Wiener models PLS framework. American Institute of Chemical Engineers Journal, 44 (7), 1611-22. 

Estimation of Muscle Force with EMG Signals Using Hammerstein-Wiener Model... 

Abstract— Estimation of muscie force is needed for monitoring or control purposes in many studies and applications that include direct human invoivement such as control of prosthetic arms and human-robot interaction. A new model is introduced to estimate the force of muscie from the EMG signals. Estimation is based on Hammerstein-Wiener Model which consists of three biocks. These biocks are used to describe the nonlinearity of input and output and iinear behavior of the model. The nonlinear network is designed base on the sigmoid network. The introduced modei is trained by some data sets which are recorded from different peopie and tested by some other data sets. The simuiation resuits show iow error rate between measured force and estimated force. 

In this paper we will introduce a non-parametric model based on Hammerstein-Wiener model with use of sigmoid network in nonlinear block of model which can be classified in neural network model category. Section II will discuss on structure of Hammerstein-Wiener model. In section III and TV, data acquisition and simulation procedure will be described and finally a conclusion remark will be represented. 

Figure 1 illustrates the block diagram of a Hammerstein-Wiener model structure [11] ...

The Hammerstein-Wiener model calculates the output y in three stages ... 

The input-output relationship will be decomposed into two or more interconnected elements, when the output of a system depends nonlinearly on its inputs. So, we can describe the relationship by a linear transfer function and a nonlinear function of inputs. The Hammerstein-Wiener model uses this configuration as a series connection of static nonlinear blocks with a dynamic linear block. 

Applications of Hammerstein-Wiener model are in wide areas, for example we can mention modelling electromechanical system and radio frequency components, audio and speech processing and predictive control of chemical processes. These models have a useful block representation, transparent relationship to linear systems, and are easier to implement than heavy-duty nonlinear. So, they are very usefiil. 

The Hammerstein-Wiener model can be used as a blackbox model structure since it prepares a flexible parameterization for nonlinear models. It is possible to estimate a linear model and try to improve its quality by adding an input or output nonlinearity to this model. 

Also, we can use Hammerstein-Wiener model as a grey box structure to take physical knowledge about process characteristics. For instance, the input nonlinearity might represent typical physical transformations in actuators and the output nonlinearity might describe common sensor characteristics [13]. 

A new method is introduced to estimate the force of the wrist by the EMG signal of the elbow muscle. The estimation is based on Hammerstein-Wiener model. The nonlinear block in this model is based on sigmoid network which can smoothly estimate the output values. Simulations results shows that this model have low error rate. 

J. Wingerden, M, Verhaegen, Closed-loop subspace identification of Hammerstein-Wiener models, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, P.R. China, December 16-18, 2009. 

Hunter, I. and Korenburg, M., The identification of nonlinear biological systems Wiener and Hammerstein cascade models. Biol. Cybern. 55 135-144,1986. 

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