Identificeren van nietlineaire systemen.

Identification of nonlinear systems

Philip Crama

The goal of this research is to develop identification methods for nonlinear systems. The models have to be usable in a design environment. Whenever this is necessary, the precision of the model may be lowered for the sake of simplicity. In a first attempt, we focus on Wiener and Hammerstein systems.
Wiener Systems
Wiener systems consist of two blocks taken in series: a linear dynamic block whose output is fed to a static nonlinear block. It is possible to identify both blocks using a special class of excitation signals. These excitation signals are random phase multisines. Among other properties, they allow to perform a consistent estimation of the linear part of the system.
The estimation of the linear part with random phase multisines looks very noisy because of the stochastic nonlinear contributions. There are two ways to reduce this noise: make more measurements with different phases for the random phase multisine or fit a parametric model through the non parametric estimation of the linear part. The reduction of parameters from one parameter for each frequency point to (usually) less than twenty parameters reduces the variability of the model sufficiently to limit the noise-like contributions.
Once the linear part is known, the input signal of the static nonlinearity can be guessed using the input signal and the non-parametric model of the linear dynamics. The knowledge of the input signals of the static nonlinearity and its output yield a cloud of points. Any approximation method can then be used to model the static nonlinearity starting from this input-output data.
Hammerstein Systems
Hammerstein systems contain the same building blocks as the Wiener system, but connected differently: the input signal goes first to a static nonlinear block. The nonlinear block's output is then applied to a linear dynamic block. Using random phase multisines, the linear part of the Hammerstein system is estimated easily. The properties of the random phase multisines for Wiener systems also apply to Hammerstein systems.
It is impossible to recover the output of the nonlinear static block without inverting the estimated model of the linear dynamic part of the Hammerstein system. This inversion is however a bad idea: if the value of the frequency response function is small, we amplify the noise. Another point against this method is that we have to extrapolate our model: the identification is only possible for frequencies where we put power in the input signal. However, the static nonlinear block causes new frequencies outside of the excitation band to appear in the output spectrum, that would also have to be inverted to recover correctly the output of the nonlinear block.
To avoid the inversion, the static nonlinear block is represented as the weighted sum of basis functions. The choice of basis functions is entirely free. For each basis function the output of a fictional Hammerstein system formed with the basis function as the static nonlinearity and the estimated frequency response function as linear dynamic block is computed for the measurement excitation signal. If we compute the Fourier transforms of each of these simulated system responses, we may compare them with the Fourier transform of the real output signal. This is a linear estimator for the weighting coefficients of the basis functions.