# A Control-Oriented Linear Parameter-Varying Model of a Commercial Vehicle Air Brake System
## 4 Linearization Through System Identification  
Since the developed system model is highly nonlinear with respect to control inputs and supply pressure and it is well-known that design a nonlinear controller with guaranteed performance is a challenge, a Linear Parameter-Varying (LPV) model for the system is developed so that a gain-scheduling controller can be designed for the air brake system in future. One of the approaches of developing an LPV model is to link a set of linearized system models to form a single LPV model and this method is adopted in this paper. For system linearization, \(q\)-Markov Cover (COVariance Equivalent Realization) [22] system identification approach is adopted in this paper. The main reasons for adopting the \(q\)-Markov Cover system identification approach for linearization is due to highly nonlinear characteristics of the supply and exhaust solenoid valves, where both valves operate in either on or off state. This makes conventional linearization approaches difficult to be applied. The \(q\)-Markov Cover utilizing the Pseudo-Random Binary Signal (PRBS) as the excitation. It is possible to conduct system identification using either pulse or white noise. However, for the air brake system, the pulse input may not generate a rich response for \begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline
**Reference Pressure/Bar** & **Supply Pressure/bar** & **Maximum Relative Error** & **Maximum Steady Error** & **Response Time under Simulation/Experiment in Charge Process/ms** & **Response Time under Simulation/Experiment in Discharge Process/ms** \\ \hline
2 & 5.8 & 15\% & 5.6\% & 302/293 & 223/215 \\
3 & 5.8 & 13\% & 3.3\% & 324/306 & 240/220 \\
4 & 5.8 & 11.5\% & 2.4\% & 359/330 & 290/265 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of experimental and simulation results under different reference pressure. Figure 6: Comparing between experimental and simulation results under different reference pressure levels with 4.5/5.8 bar supply pressure. (**a**) Comparison results with 4.5 bar supply pressure, (**b**) Comparison results with 5.8 bar supply pressure. system identification purposes; and white noise cannot be realized accurately due to signal saturation, leading to system identification error. Note that PRBS can be realized exactly during system identification. The \(q\)-Markov Cover uses the pseudo-random binary signal (PRBS) as the excitation signal mainly due to its generation accuracy and the model identified is a linearized model of the target nonlinear system. In general, there are two methods to obtain the linearized model through system identification using either simulated and experimental data. There are two reasons for using the simulation data for generating the linearized model. One is due to the long experimental time required to generate the steady-state PRBS response and the other is that system identification may operate the brake system outside its physical capability. The general structure of a linear discrete-time SISO system is shown in Figure 7, where \(R(z)\) is the reference signal, \(U(z)\) is the input of plant, \(Y(z)\) is the output pressure of plant, \(K(z)\) is closed-loop controller gain, and \(G(z)\) is the plant transfer function. The relationship between input and output signals, shown in Figure 7, can be expressed below in the form of closed-loop transfer function from \(R(z)\) to \(Y(z)\). \[G_{CL}(z)=\frac{Y(z)}{R(z)}=\frac{K(z)G(z)}{1+K(z)G(z)} \tag{21}\]  
Assuming that \(1-G_{CL}(z)\) is invertible, and the plant transfer function can be expressed by  
\[G_{P}(z)=\frac{G_{CL}(z)}{K(z)-K(z)G_{CL}(z)} \tag{22}\]  
To identify the system model, the PRBS signal is added as a disturbance signal to the input signal to identify the system at the operational condition at the current input level (or air chamber pressure).