# Complex-Order Models: A System Identification Point of View
## 4 A Use Case in System Identification
However, the intrinsic compressibility of gas makes pressure regulation a very hard task, depending on complex and nonlinear fuel dynamics and parameter changes due to various working points associated with power and speed requests [38; 39; 40; 41; 42]. Therefore, it is necessary to develop suitable models for analyzing system performance, predicting correct and anomalous behaviors in the injection process, evaluating working conditions and system configurations, and designing controllers for common-rail pressure regulation [38; 43]. In any case, a compromise between simplicity and accuracy must be achieved to simplify control design and make the controller effective. Moreover, few approaches are available in the literature for modeling and controlling gas-injection systems [44; 42]. However, the available models do not show a good fitting to experimental data and are not so suitable for control. Herein, a model represented by a transfer function with aperiodic CO-RV elements is identified based on an integer-order ARX model. (ARX models are used as regularized representations of frequency responses. Although it would be, in principle, possible to determine the frequency characteristics of the system under consideration by computing the ratios of Fourier transforms of the output and input, such a procedure would be numerically ill-conditioned, and ill-suited for practical applications.) The models are identified by using data from a real, prototype injection system. The CO-RV model parameters are determined by an optimization procedure based on a cost function that considers deviations of the magnitude and phase characteristics of the considered model from the corresponding ones of the ARX model. The Injection System: General Description and Problem FormulationThe proposed modeling methodology is applied to the key part of the injection system, such that a good data fitting is obtained and the gas pressure dynamics can be easily predicted and controlled by the identified model. The considered injection system is based on six interacting main components in which gas flows: a gas tank, a mechanical pressure reducer, an electro-hydraulic valve, a common-rail volume, a set of injectors, and an electronic control unit (see Figure 5). Temperature is nearly constant and uniform in all parts of the system. The components are connected by pipes, in which transmission delays are neglected. Gas from the tank feeds the other elements, but its pressure inside the tank can be assumed as slowly varying (40-200 bars), with respect to pressure dynamics in the other parts. The task of the mechanical element, thanks to a valve-shutter coupling--in the upper part of the reducer--actuated by the valve, and to a piston-shutter coupling--in the lower part of the reducer, is to decrease pressure to lower values (5-20 bars) before gas reaches the common-rail volume. To this aim, the valve regulates the gas inflow into the \"control chamber\" of the mechanical reducer volume; then, this gas pressure acts on the upper surface of the piston separating the control chamber from a \"main chamber\". On the other hand, gas inflow into the main chamber is determined by the motion of the lower shutter, in its turn caused by the gas arriving from the tank, and acts on the lower surface of the piston. In more detail, the gas pressure in the control chamber pushes the piston and the lower shutter down; hence, this allows more gas into the main chamber. Moreover, if the gas pressure on the piston is decreased, the lower shutter closes and the pressure in the main chamber is reduced. To synthesize, the equilibrium between pressure forces determines the amount of gas that feeds the common-rail volume from the main chamber and from the control chamber. Consequently, gas flows through the injectors to the intake manifold, where gas is mixed with air. Finally, the control unit commands the valve to control the gas flow.