Spectral Determination of Nonlinear System Parameters

In this paper we propose an identification method of nonlinear system. This later can be structured by Wiener models. The determination of nonlinear system parameters can be done using spectral analysis. The system nonlinearity is allowed to be noninvertible general shape nonlinearity but it must be approximated by a polynomial function. The polynomial degree n can vary from one interval to another. The linear dynamic element is not-necessarily parametric but BIBO stable. In this work, a spectral method is developed allowing the estimates of the complex frequency gain as well as the estimates of nonlinear block parameters the identification method is built using one stage.


Introduction
This nonlinear system identification has been an active research area, especially over the last two decade [1]- [3]. A Wiener model is one of most popular models. It consists of linear element followed by nonlinearity block (Fig. 1). This model is more difficult than Hammerstein model (nonlinearity element followed by linear dynamic element). The usefulness of this nonlinear model has practically been confirmed in various domains fields [4] - [6].
In the present work, Wiener system identification is addressed in the case where the linear dynamic block is nonparametric and of unknown structure. This later is assumed to be BIBO stable. This assumption is commonly used in open loop identification case. On the other hand, it is assumed that the system nonlinearity f(.) can be approximated by a polynomial function. The polynomial degree n can vary from one interval to the author's knowledge; the identification approach is performed using only one stage unlike most of other methods [19].
The suggested identification approach involves periodic input signals. Then, in the steady state, the undisturbed system output w(t) is also periodic signal of the same period of u(t) (Fig. 1). Accordingly, the internal signal w(t) is decomposable of Fourier series. The nonlinear system parameters (i.e. the nonlinearity block parameters as well as the linear subsystem parameters) can be determined using u(t) and the system output.
The remaining parts of paper are organized as follows: the identification problem is formulated in Section 2; the identification method of linear and nonlinear elements is presented in Section 3; the identification method performances are illustrated by simulation example presented in Section 4. Except of this assumption, the nonlinear element can be of general shape and noninvertible.
Presently, a spectral analysis method is developed allowing an accurate estimates of complex frequency gain ) ( ω j G To the author's knowledge, the identification approach is performed using only one stage unlike most of other methods [19]. The suggested identification approach involves periodic input signals. Then, in the steady state, the undisturbed system output w(t) is also periodic signal of the same period of u(t) (Fig. 1). Accordingly, the internal signal w(t) is decomposable of Fourier series. The nonlinear system parameters (i.e. the nonlinearity block parameters as well as the linear subsystem parameters) can be determined using u(t) and the system output.
The remaining parts of paper are organized as follows: the identification problem is formulated in Section 2; the identification method of linear and nonlinear elements is presented in Section 3; the identification method performances are illustrated by simulation example presented in Section 4.

Identification Problem Statement
Use Wiener models consist of series connection of linear block ) (s G and nonlinear static element f(.). Presently, the linear dynamic block can be nonparametric and of unknown structure. Furthermore, the identification schema is built up in open loop. Then, the linear element is only supposed to be BIBO stable. The system nonlinearity is allowed to be noninvertible but can be approximated by polynomial function. The polynomial degree n can varies from one interval to another. One other hand, the nonlinear Wiener system (Fig. 1) under study can be analytically described by the following equations: where * refers to the convolution operator and: The notation L -1 denotes Laplace transform-inverse.
The internal signals ( ) v t and ( ) w t are related by the following equation: Finally, the system output ( ) y t can be expressed as follows: Using the fact that the system nonlinearity f(.)can be modelled within any interval by polynomial function of degree n. Then, one immediately has: where [ ] This question will be dealt in the next section.

Frequencies -Domain Identification
The frequency identification of the this nonlinear system (Wiener model), characterized by (1) Accordingly, one immediately gets using (3) and (6): The unknown parameters in (7)  Then, it is readily seen using (7) and (8a-b) that, the undisturbed output signal ( ) w t can be rewritten as: ϕ ω . Accordingly, one immediately gets using (3) and (9): This results shows that, the undisturbed output signal ( ) w t (not accessible to measurement) is periodic of the same period 2 / π ω of input ( ) u t . Likewise, the spectrum of inner signal ( ) w t is characterized by n components of frequency kω ( 1, , k n =  ) and a DC component (see (9)). It follows using the measurement of ( ) w t that, an accurate estimate of the frequency component amplitudes k S ( 0, , k n =  ) as well as of the phases (.) k λ . In this respect, it is interesting to emphasize that the undisturbed output ( ) w t is not accessible to measurement and the output signal ( ) y t (measurable) is ( ) w t mixed up to noise ( ) t ξ .
Fortunately, such an accurate estimation can be available thanks to the ergodicity property of the noise signal ( ) t ξ and the steady-state periodic nature of the inner signal ( ) w t . The ergodicity allows the substitution of arithmetic averages to probabilistic means, making simpler forthcoming developments. These remarks suggest that an accurate estimate ˆ( ) w t of ( ) w t can be obtained using the following periodical averaging: where N is any sufficiently large integer. It is readily shown that, the suggested estimator (11a-b) is consistent, i.e. the inner signal estimate ˆ( ) w t converges with probability 1 to the true signal ( ) w t . Indeed, one immediately gets using (3) and (11a-b): (12) Then, this latter becomes using the periodic nature of ( ) w t : Finally, the point is that the last term in (13) boils down to zero (w.p.1) using the ergodicity property of the noise ( ) t ξ . One gets thus from (13) the following convergence: On the other hand, it is important to point that, the determination of spectrum of the signal ( ) w t using (11a- Finally, an accurate estimate of nonlinear system parameters can be determined Equations Presently, the problem of nonlinear system identification is dealt. The nonlinear system is described by Wiener model. This identification problem is coped in the frequency-domain using the identification solution described in Section 3. Then, the identification algorithm is built up using a simple sine signals. It is interesting to point out that, the parameters of linear and nonlinear elements are determined using only one stage.