## 3 Problem Formulation  
### Gain-Scheduling Strategy  
Based on the knowledge from bench tests and NSGA-II optimization of the fixed-gain PID controller (see the next section), the \(I\) (integration) and \(D\) (derivation) gains are relatively small compared with the \(P\) (proportion) gain. A large \(I\) gain results in quick windup, while a large \(D\) gain leads to high sensitivity to noise. Normally, the proportional, integral, and derivative scheduled gains (functions) should be optimized simultaneously, which increases the search space dimensions and results in an extremely high number of iterations needing to converge. In this paper, we aim to find an optimization scheme that is feasible for implementation; thus, we propose searching for the optimal scheduled PID gains sequentially in the three steps shown in Figure 4, where \(e\) denotes the valve position error between the reference and feedback signals.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(J_{r}\) & 0.42 kg/m\({}^{2}\) & \(K_{0}\) & 0.698 Nms/rad \\ \(K_{i}\) & 205.3 Nm/duty cycle & \(K_{s}\) & 3.93 Nm/rad \\ \(\theta_{pre}\) & 0.873 rad & \(\theta_{s}\) & 0.0087 rad/s \\ \(T_{c}\) & 0.285 Nm & \(T_{r}\) & 57.01 Nm \\ \(m_{l}\) & 0.28 kg & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model parameters of EGR valve.  
Figure 3: Validation of the system model and its calibration.  
In Step 1, the algorithm searches for the optimal scheduling proportional gain function \(P(e)\), along with the optimal fixed \(I\) and \(D\) gains; in Step 2, the scheduling derivative gain \(D(e)\) is optimized using the scheduling proportional gain \(P(e)\) obtained in Step 1 and the fixed \(I\) gain; finally, in Step 3, the scheduling integral gain \(I(e)\) is optimized using the proportional gain \(P(e)\) obtained from Step 1 and derivative gain \(D(e)\) obtained from Step 2.  
Subject to the high static friction characteristics, the tracking error \(e\) of the valve position is selected as the scheduling parameter, where high gains are expected in the region with a small tracking error to overcome the increased friction force and reach the final destination. Considering the computational cost, the number of variables in the scheduled gain function should be low, even though complex scheduling functions could improve the closed-loop system performance further. After studying multiple candidate scheduling functions, \(f(e)=a/|e|^{b}+c\) is selected for the gain functions of \(P\), \(I\), and \(D\) for \(|e|\geq 0.1\), where \(e\) is the position error and \(a\), \(b\), and \(c\) are the calibration parameters. Note that a saturation function is used here to avoid the gains going to infinity as \(|e|\) goes to zero; see (9).  
\[\begin{cases}f(e)=\dfrac{a}{|e|^{b}}+c&\forall|e|\in[0.1,100]\\ f(e)=\dfrac{a}{0.1^{b}}+c&\forall|e|\in[0,0.1)\end{cases} \tag{9}\]  
Figure 5 shows the scheduled-gain function \(f(e)\) with \(a=1\), \(c=0\), and \(b\) between 0.1 and 1. It can be seen that as \(b\) increases the scheduled gain magnitude increases significantly with a small position error, resulting in a large gain difference between small and large \(e\) compared to a relatively small \(b\). With the help of the \(a\) (ramp-gain) and \(c\) (fixed-gain) parameters, the optimized scheduled gain is able to adapt to the nonlinear system dynamics.  
The gain-scheduled PID controller structure is shown in Figure 6, where \(e\) and \(u\) represent the valve position error and control, respectively, while \(P(e)\), \(I(e)\), and \(D(e)\) are  
Figure 4: Steps in sequential search for optimal gain-scheduled PID controller.  
Figure 5: Sample scheduled gains as a function of \(|e|\) with a range of \(b\).  
the scheduled proportional, integral, and derivative gains in the form of \(f(e)\), respectively. In another words, the optimization aims to search for the best shape of \(P(e)\), \(I(e)\), and \(D(e)\) in the form of \(f(e)\).