# Complex-Order Models: A System Identification Point of View
## 5 Identification Results  
Model identification and validation was performed by data measured from an experimental test bench, which is a prototype of a commercial car [36]. In particular, for a tank pressure of 40 bar, the solenoid valve was commanded as shown in Figure 6. In this way, the control chamber pressure and, hence, the response in the common-rail pressure, determines the output shown in the same Figure 6.  
The behavior of the described process could be modeled as a two-inputs-one-output system. The first input is given by abrupt variations of the duty cycle of the control valve driving current, while the ramp variation of the engine speed, which affects the frequency of injections, is considered as the second input. The output response is the common-rail pressure. Figure 6 shows the experimental input and output signals. An integer-order discrete-time ARX model is considered as a term of comparison. It is characterized by a sufficiently high order such that a non-parametric identification can be made, independently of any fixed model structure, which is, in any case, unknown and can be determined by comparison only. Moreover, in this way, problems of sensitivity to noise and disturbances are prevented by fitting time-domain data with the ARX model of high order and regularizing identification. The discrete-time representation is:  
\[A(q)y(k)=B_{1}(q)u_{1}(k)+B_{2}(q)u_{2}(k)+e(k), \tag{36}\]  
where \(u_{1}\) is the input representing the duty cycle for the driving current, \(u_{2}\) is the second input representing the engine speed, \(y\) is the output, i.e., the common-rail pressure, \(e\) is the disturbance, \(A(q)\) and \(B_{i}(q)\), for \(i\in\{1,2\}\), are polynomials in the forward shift operator \(q\) of orders \(n\) and \(n-1\), respectively, and \(n\) is taken as sufficiently large for capturing the process dynamics. The parameters of the ARX models are estimated by the least-squares method [49] and \(n=10\) is empirically determined by numerical tests (this value was necessary to optimally represent the frequency response). Each of such discrete-time ARX models has 20 parameters in total, as indicated by the following polynomials:  
Figure 6: Inputs and output data used to fit the high-order ARX model (data are sampled every \(1\,\mathrm{ms}\)). \[A(z) = 1-1.028\,z^{-1}-0.1695\,z^{-2}+0.208\,z^{-3}-0.2639\,z^{-4}+0.3578\,z ^{-5}-0.3181\,z^{-6}\] \[- 0.01813\,z^{-7}-0.02773\,z^{-8}+0.3347\,z^{-9}-0.07473\,z^{-10}\] \[B_{1}(z) = 0.02052-0.005222\,z^{-1}-0.09316\,z^{-2}+0.1346\,z^{-3}-0.1157\,z^ {-4}+0.1004\,z^{-5}\] \[- 0.01834\,z^{-6}-0.07983\,z^{-7}+0.09431\,z^{-8}-0.03754\,z^{-9}\] \[B_{2}(z) = -6.924\cdot 10^{-5}+0.0007713\,z^{-1}-0.002213\,z^{-2}+0.002686\,z ^{-3}-0.001219\,z^{-4}\] \[- 0.0004619\,z^{-5}+0.0004533\,z^{-6}+0.0007231\,z^{-7}-0.001037\,z ^{-8}+0.000366\,z^{-9}\]  
Therefore, the obtained model is a multi-input system which is described by two transfer functions. Figure 7 shows the frequency response of the fitted ARX model, which considers the duty cycle as input and the common-rail pressure as output (the first transfer function), and the influence of the engine speed on the output (the second transfer function). Figure 8 shows the time-domain responses provided by the ARX model and by the experimental common-rail pressure obtained when the inputs are as indicated in Figure 6.0 A reasonable accuracy is obtained if one considers the remarkable process nonlinearities and uncertainties of the relation between the inputs and the common-rail pressure [50]. To obtain the unknown parameter values, the following cost function is used:  
Figure 6.1  
Figure 6.2 Comparison between experimental and simulated common-rail pressures. Figure 6.3 Frequency response of the fitted ARX model: the first transfer function (**left**) and the second transfer function (**right**).