In the literature, various techniques have been reported on nonlinear process identification using integer-order (classical) models. The **Hammerstein** process was estimated using the special test signal in [3] and further extended for **Hammerstein**- Wiener processes in [4], [5]. The separate block-oriented non- iterative relay feedback (SRF) method was illustrated in [6]. Mehta and Majhi [7] presented the non-iterative relay feedback (NRF) method to determine the structure prior to the parameters of the **Hammerstein** model. The **Hammerstein** **system** with time delay has been accurately identified using a recursive least squares method in [8]. Recently, a separate block-oriented parameter identification method for **Hammerstein** systems using least squares was described in [9]. Furthermore, another special input based identification of **Hammerstein**-Wiener nonlinear **system** with noise was discussed in [10]. Even though some efficient integer-order techniques have been developed so far, a fractional domain approach is yet to be fully explored for nonlinear process identification.

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A practical orthogonal frequency-division multiplexing (OFDM) **system** can generally be modelled by the **Hammerstein** **system** that includes the nonlinear distortion effects of the high power amplifier (HPA) at transmitter. In this contribution, we advocate a novel nonlinear equalisation scheme for OFDM **Hammerstein** systems. We model the nonlinear HPA, which represents the static nonlinearity of the OFDM **Hammerstein** channel, by a B-spline neural network, and we develop a highly effective alternating least squares algorithm for estimating the parameters of the OFDM **Hammerstein** channel, including channel impulse response (CIR) coefficients and the parameters of the B-spline model. Equalisation of the OFDM **Hammerstein** channel can then be accomplished by the usual one-tap linear equalisation as well as the inversion of the estimated B-spline neural network model. We propose to use an efficient Gauss-Newton algorithm for the latter inversion task. The effectiveness of our nonlinear equalisation scheme for OFDM **Hammerstein** channels is demonstrated by simulation results.

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There exists a lot of works on nonlinear **system** identification which use block-oriented models. For example estimating the formal information of neurons[27], using ARMA model for dynamic linear block and a multilayer feed forward neural network to model the static nonlinear[13], least square and SVD for **Hammerstein** model[11],[8], recursive identification for **Hammerstein** **system** with state space model[33], eigensystem realization algorithm(ERA) for accurate parameter estimation and the **system** order determination[23], over parameterization and iterative methods[22], iterative approaches[20][29], frequency-domain method[24], subspace method[17],[18][25],[26], stochastic algorithm[6], blind approaches[4], magnetosphere identification[3], constructing a model for ionospheric dynamics[16], using Genetic algorithm for H-W identification[14], initializing parameters and order determination by Lipchitz[1], fully automated recurrent neural network[31], using the spectral magnitude matching method[32], using a new maximum- likelihood based method[33].

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The primary diﬃculty in the identification of **Hammerstein** nonlinear systems (a static memoryless nonlinear **system** in series with a dynamic linear **system**) is that the output of the nonlinear **system** (input to the linear **system**) is unknown. By employing the theory of aﬃne projection, we propose a gradient-based adaptive **Hammerstein** algorithm with variable step-size which estimates the **Hammerstein** nonlinear **system** parameters. The adaptive **Hammerstein** nonlinear **system** parameter estimation algorithm proposed is accomplished without linearizing the systems nonlinearity. To reduce the eﬀects of eigenvalue spread as a result of the **Hammerstein** **system** nonlinearity, a new criterion that provides a measure of how close the **Hammerstein** filter is to optimum performance was used to update the step-size. Experimental results are presented to validate our proposed variable step-size adaptive **Hammerstein** algorithm given a real life **system** and a hypothetical case.

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have been proposed in [18]. The Least Squares Sup- port Vector Machines (LS-SVMs) have been presented in [19,20]. A generalized **Hammerstein** model con- sisting of a static polynomial function in series with time-varying linear model is developed in order to model the **Hammerstein**-like multivariable processes whose linear dynamics vary over the operating space in [21]. In this work, we propose a new coupled struc- ture identification of MIMO model with separate non- linearities. It is organized as follows: SISO Hammer- stein **system** is presented in part 1 of section 2. A new coupled structure for MIMO **Hammerstein** **system** is developed in part 2 of section 2. Simulation results of a quadruple-tank process is given in section 3. Finally, a conclusion is made.

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In recent years, a growing amount of research has been done on nonlinear **system** identification [1, 2]. Nonlinear dynami- cal **system** models generally have a high number of param- eters although many problems can be su ﬃ ciently well ap- proximated by simplified block-based models consisting of a linear dynamic subsystem and a static nonlinearity. The model consisting of a cascade of a linear dynamic **system** and a memoryless nonlinearity is known as the Wiener **system**, while the reversed model (a static nonlinearity followed by a linear filter) is called the **Hammerstein** **system**. These systems are illustrated in Figures 1 and 2, respectively. Wiener sys- tems are frequently used in contexts such as digital satellite communications [3], digital magnetic recording [4], chemi- cal processes, and biomedical engineering. **Hammerstein** sys- tems are, for instance, encountered in electrical drives [5] and heat exchangers.

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approximation of solutions. First, the integration method is proposed to transform the considered boundary value problems into **Hammerstein** integral equations. Second, the existence of solutions for the obtained **Hammerstein** integral equations is analyzed by using the Schauder ﬁxed point theorem. The contraction mapping theorem in Banach spaces is further used to address the uniqueness of solutions. Third, the approximate solution of **Hammerstein** integral equations is constructed by using a new numerical method, and its convergence and error estimate are analyzed. Finally, some numerical examples are addressed to verify the given theorems and methods.

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be able to track channel variations simultaneously with data transmission. This problem is still not solved in this paper. However, when the symbol timing of the desired signal and the self-interference signal are synchronized, the problem does not arise because specific training symbols are unnecessary. Although this problem is very important, we treat it as a future work in this paper. The rest of this paper is organized as follows. In Section II, a detailed model of the self interference which includes nonlinearities of the IQ mixers and the power amplifier is provided. The proposed selection technique with the frequency domain **Hammerstein** self-interference canceller is presented in Section III. In Section IV, the performance of the proposed technique under different scenarios is analyzed with equivalent baseband signal simulations

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Abstract: Acceleration control of turbofan engines is conventionally designed through either schedule-based or acceleration-based approach. With the widespread acceptance of model-based design in aviation industry, it becomes necessary to investigate the issues associated with model-based design for acceleration control. In this paper, the challenges for implementing model-based acceleration control are explained; a novel **Hammerstein**-Wiener representation of engine models is introduced; based on the **Hammerstein**-Wiener model, a nonlinear generalized minimum variance type of optimal control law is derived; the feature of the proposed approach is that it does not require the inversion operation that usually upsets those nonlinear control techniques. The effectiveness of the proposed control design method is validated through a detailed numerical study.

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Abstract: The solid oxide fuel cell (SOFC) is widely acknowledged for clean distributed power generation use, but critical process problems frequently occur when the stand-alone fuel cell is directly linked with the electricity grid. To guarantee the optimal operation of the SOFC in a power **system**, it is essential, that its generation ramp rate and load following is fast enough to sustain power quality. In order to address these problems, a suitable and highly efficient control **system** will be required to control and track power load demands for complex SOFC power systems under grid connection. Therefore, novel nonlinear hybrid adaptive Fuzzy Neural Network (AFNN) is developed for control of grid connected SOFC. During peak power demand schedules from electric utility grid and large load perturbations, maintaining optimal power quality and load-following is a big challenge. Both the rapid power load following and safe SOFC operation requirement is taken into account in the design of the closed-loop control **system**. Simulation results showed that the proposed hybrid AFNN enhance the optimal power quality and load-following than conventional PI controller.

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The notion of osculating operators has been considered from diﬀerent points of view see 2, 3. In this note we reformulate the definition of osculating operators. Our setting is a locally convex topological vector space. Moreover, we present a new implicit function theorem and, as an example of application, we study the solutions of an **Hammerstein** equation containing a parameter.

Roughly, the iterative methods (e.g. [2]-[3]) necessitate a large amount of data; since computation time and memory usage drastically increase, and have local convergence properties which necessitates that a fairly accurate parameter estimates are available to initialize the search process. This prior knowledge is not required in stochastic methods but these are generally relied on specific assumption on the input signals (e.g. gaussianity, persistent excitation....) and on **system** model (e.g. MA linear subsystems, smooth nonlinearity). The frequency methods are generally applied to nonparametric systems under minimal assumptions and only require simple periodic excitations. But, they sometime necessitate several data generation experiments.

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be simple, accurate and general. This approximate description of the **system** can be constructed by **system** identification strategy, as the goal of **system** identification is to build a mathematical model of a dynamic **system** based on some initial information about the **system** and the measurement data collected from the **system**. According to [1], the process of **system** identification consists of designing and conducting the identification experiment in order to collect the measurement data, selecting the structure of the model and specifying the parameters to be identified and eventually fitting the model parameters to the obtained data [2]. Finally the quality of the obtained model is evaluated through model validation process. Generally **system** identification is an iterative process and if the quality of the obtained model is not satisfactory, some or all of the listed phases can be repeated in order to obtain one satisfied model.

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Using computer simulations, the properties of the lattice-**Hammerstein** filter derived in previous parts are inspected. The input is a colored Gaussian signal generated by an FIR filter defined by h=[0.9045 0.7 0.9045] whose input is zero-mean white Gaussian noise. The results are averaged over 100 independent trials. The number of input samples is 1000. In the first experiment, the noise variance is 0.0248, the degree of nonlinearity of **Hammerstein** is P=2, and the number of stages (memories) is M=10. To verify equations Eq. (31), Eq. (17), Eq. (18), and Eq. (19), power of forward and backward errors are depicted in Figs 4 (a), (b) and (c). In agreement with Eq. (19), Fig. 4 (a) shows that backward errors of different stages are orthogonal, and thus, nonzero at the same stages and zero elsewhere. Also, from Fig. 5 (b), it is seen that in accordance to Eq. (31), the power of forward and backward errors of similar stages are identical. Moreover, Fig. 5 (c) shows that forward/backward errors are orthogonal to the input signal as proved in Eq. (17) and Eq. (18).

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that the **Hammerstein** type equation u + KF u = 0 has a solution in H . It is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized **Hammerstein** type equation. The results obtained in this paper improve and extend known results in the literature.

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with the iterated collocation method. Guoqiang (1993) introduced and discussed the asymptotic error expansion of a collocation-type method for Volterra-**Hammerstein** integral equations. The methods in Kumar et al. (1987) and Guoqiang [1993] transform a given integral equation into a **system** of non-linear equations, which has to be solved with some kind of iterative method. In Kumar et al. (1987) the definite integrals involved in the solution may be evaluated analytically only in favorable cases, while in Guoqiang (1993) the integrals involved in the solution have to be evaluated at each time step of the iteration. Orthogonal functions, often used to represent arbitrary time functions, have received considerable attention in dealing with various problems of dynamic systems. The main characteristic of this technique is that it reduces these problems to those of solving a **system** of algebraic equations, thus greatly simplifying the problem. Orthogonal functions have also been proposed to solve linear integral equations. Runge –Kutta methods are being applied to determine numerical solutions for the problems, which are modeled as Initial Value Problems (IVP’s) involving differential equations that arise in the fields of Science and Engineering by Alexander and Coyle (1990), Murugesan et al. (1999; 2000; 2001; 2003), Shampine [1994] and Yaakub and Evans (1999). Runge-Kutta methods have both advantages and disadvantages. Runge-Kutta methods are stable and easy to adapt for variable stepsize and order. However, they have difficulties in achieving high accuracy at reasonable cost, which were discussed recently by Butcher (2003). Murugesan et al. (1999) have analyzed different second-order systems and multivariable linear systems via RK method based on centroidal mean. Park et al. (2004; 2005) have applied the RK- Butcher algorithm to optimal control of linear singular systems and observer design of singular systems (transistor circuits).Murugesan et al. (2004) and Sekar et al. (2004) applied the RK- Butcher algorithm to industrial robot arm control problem and second order IVP’s. In this paper, we are introducing here the STHW for finding the numerical solution of nonlinear Volterra-

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Abstract. In this study a numerical method is developed to solve the **Hammerstein** integral equations. To this end, the kernel has been approximated using the least-squares approximation schemes based on Legendre-Bernstein basis. The Legendre polynomials are orthogonal and this property improves the accuracy of the approx- imations. Also the nonlinear unknown function has been approxi- mated by using the Bernstein basis. The useful properties of Bern- stein polynomials help us to transform **Hammerstein** integral equa- tion to solve a **system** of nonlinear algebraic equations.

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This paper is organized as follows. In Section 2, the Bai's **system** for which we are developing the adaptive controler is presented. In Section 3 the proposed technique using the approximate inverse method is given. In Section 4 the proposed technique using the estimate of the plant noise is presented and Section 5 includes illustrative example and the simulation results for the developed technique.

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In this paper, we have extended the idea of the modified Taylor-series expansion method from one dimensional Ham- merstein equation to multivariate **Hammerstein** equation. Then we have applied it to find the numerical solution of multivariate **Hammerstein** equation. This new method finds an approximation of the solution pointwise and therefore lends itself to numerical computations which can be done in parallel. The method also computes the derivatives of the solution concurrently. The results of several numerical experiments have shown the efficiency and accuracy of this new method.

Abstract: **Hammerstein** and Wiener models are nonlinear representations of systems composed by the coupling of a static nonlinearity N and a linear **system** L in the form N-L and L-N respectively. These models can represent real processes which made them popular in the last decades. The problem of identifying the static nonlinearity and linear **system** is not a trivial task, and has attracted a lot of research interest. It has been studied in the available literature either for **Hammerstein** or Wiener systems, and either in a discrete-time or continuous-time setting. The objective of this paper is to present a unified framework for the identification of these systems that is valid for SISO and MIMO systems, discrete and continuous-time setting, and with the only a priori knowledge that the **system** is either Wiener or **Hammerstein**. Keywords: Nonlinear **system** identification, Wiener systems, **Hammerstein** systems.

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