The predictions obtained from the model or its blocks can be found by averaging over the posterior. In our case, the Wiener model parameters of the linear part and the GP model of the nonlinear part are presented as the first and second moments, i.e. the mean values and covariances that match the first and second moments of the Gaussian mixtures of posterior realisations for each of the linear model parameters and the GP model. The predictions for the models with the parameters and the GP model obtained with the moment matching can be seen in Figs. 3.4 and 3.5 . 0 100 200 300 400 500 600 700 -1 -0.5 0 0.5 1 k u 0 100 200 300 400 500 600 700 -1 -0.5 0 0.5 1 k y Fig. 3.3 Input and output signals used for the identification of the Wiener model
112 3 Incorporation of Prior Knowledge -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 1.5 ξ y ± 2 σ predicted mean static nonlinearity Fig. 3.4 The static-nonlinear part of the Wiener model scheme obtained from Gaussian mixtures for every input point with the matching of moments 10 -2 10 -1 10 0 10 -2 10 0 Frequency response of the linear part Amplitude 10 -2 10 -1 10 0 -500 -400 -300 -200 -100 0 Phase [degrees] Frequency [rad/s] Fig. 3.5 Frequency response of the identified linear part with its mean ( full line ) and a 95% confidence interval ( grey band ) together with the frequency response of the original linear part ( dashed line ) The identified nonlinear part of the Wiener model with a good matching to the original nonlinearity is shown in Fig. 3.4 . The Bode plot of the identified linear part, together with the original linear part, is depicted in Fig. 3.5 . As is clear from Figs. 3.4 and 3.5 , the obtained model shows a satisfactory performance.
3.2 Wiener and Hammerstein GP Models 113 In the case of a validation with the simulation response of the model with an input signal that is different from the one used for the identification, we have various options. We can use the matched mean values and covariances for the linear model parameters and the matched posterior mean functions and covariance functions of the GP model to be used in the dynamic system simulation, taking account of the uncertainties as described in Sect. 2.6 or, alternatively, use the simulation with the Monte Carlo method. 3.2.2 GP Modelling Used in the Hammerstein Model The Hammerstein structure consists of a nonlinear static block followed by a linear dynamic block, as illustrated in Fig. 3.6 . It is a frequently applied, nonlinear dynamic systems modelling approach. This kind of model can be used where the actuator dominates the system’s behaviour with its nonlinear static characteristic. The structure of the Hammerstein model can be linearly parameterised, which can be reflected in the choice of regressors when modelling with the linear model. The idea behind this approach is to represent a static nonlinearity with a polynomial approximation and, in this case, the overall input–output relationship is linear in the parameters [ 18 ]. In the case of a GP model identification with a linear covari- ance function, this approach requires the manual or automated selection of poly-
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