## 2 Valve System Model  
The nonlinear valve system architecture is shown in Figure 2, where the angular displacement \(\theta\), driven by a DC motor, is converted to the linear valve displacement \(y\) using a constrained slide rail-rod structure. Based on the valve structure in Figure 2, the relationship between \(\theta\) and \(y\) can be expressed by (1):  
\[y=53.19\cdot(1-cos(2\theta))-3.2 \tag{1}\]  
where \(\theta\) and \(y\) are the radian degree and valve opening percentage (%), respectively. The slide rod angular position \(\theta=0\) is at the upright vertical position and the initial slide rod position is at \(\theta=0.175\) rad (10 degrees), corresponding to the fully closed valve position (\(y=0\%\)), meaning that when the valve is fully open (\(y=100\%\)), the angle \(\theta\) is 1.4 rad (80 degrees); see the dashed-line in Figure 2.  
Figure 1: Controller design procedures categorized by offline optimization (black), online optimization (red), and the proposed model-assisted online optimization method (blue). The system moment of inertia \(J\) is calculated in (2), where \(J_{r}\) is the total rotational assembly moment of inertia, \(J_{m}\) is the equivalent moment of inertia of all the linear moving components at angular position \(\theta\), \(m\) is the total mass of linear moving part, and \(r\) is the length of the slide rod connection arm. \[J(\theta)=J_{r}+J_{m}=J_{r}+mr^{2}sin^{2}(\theta) \tag{2}\]  
The valve dynamics can be described by (3) using [PERSON]'s law. \[J\ddot{\theta}=T_{e}-T_{s}-T_{f}-T_{n} \tag{3}\]  
where \(T_{e}\) and \(T_{s}\) represent the electric motor output torque and spring load torque, stated in (4) and (5), respectively; \(K_{m}\) and \(K_{b}\) are the motor torque and back EMF (electromagnetic fields) coefficients, respectively; \(V_{c}\), \(R\), and \(\theta_{pre}\) represent the armature voltage, motor winding resistance, and spring preload position, respectively; \(T_{f}\) is the nonlinear friction torque; and \(T_{n}\) denotes the nonlinear torque created by the constraint of sliding rail-rod structure based on (6). \[T_{e}=K_{m}i=\frac{K_{m}}{R}(V_{c}-K_{b}\dot{\theta}) \tag{4}\]  
\[T_{s}=K_{s}(\theta+\theta_{pre}) \tag{5}\]  
\[T_{n}=mr^{2}\theta^{2}cos(\theta)sin(\theta) \tag{6}\]  
For the prominent nonlinear friction, the Stribeck friction model [34; 35] is used, which calculates the friction force based on the relative speed between two contacted surfaces. Based on the Stribeck friction model, the friction torque \(T_{f}\) can be represented by (7):  
\[T_{f}(\dot{\theta})=T_{c}+\frac{T_{r}-T_{c}}{1+(\dot{\theta}/\theta_{s})^{2}} +K_{v}\dot{\theta} \tag{7}\]  
where \(T_{c}\), \(T_{r}\), and \(K_{v}\) are the Coulomb friction torque, static friction torque, and viscous friction torque coefficients, respectively, and \(\theta_{s}\) is the characteristic velocity. To simplify the system identification process, the back EMF torque \(T_{b}=K_{m}K_{b}\dot{\theta}/R\) is lumped into the term \(K_{v}\dot{\theta}\) in \(T_{f}\), resulting in motor torque \(T_{e}=K_{i}u=K_{m}/R\), where \(u\) is the control input. Note that the scaled PWM (Pulse Width Modulation) range of \(u\) has been defined as \(u\in[-4000,4000]\) based on the software interface used for system test bench, presenting the full duty cycle range of the H-bridge output in the control unit. With \(mr^{2}=m_{l}\), the system dynamic equation is shown in (8). \[ \begin{split}&(J_{r}+m_{l}sin(\theta)^{2})\ddot{\theta}=K_{i}u-K_{s}( \theta+\theta_{pre})\\ &-(T_{c}+\frac{T_{r}-T_{c}}{1+(\dot{\theta}/\theta_{s})^{2}}+K_{ v}\dot{\theta})sign(\dot{\theta})-m_{l}\dot{\theta}^{2}cos(\theta)sin(\theta) \end{split}  \tag{8}\]  
Model coefficient identification is challenging in nonlinear systems. Multiple tests were conducted to isolate the effects of different parameters. First, the controller gain \(K_{i}\) was obtained from valve specification data, then the spring stiffness \(K_{s}\), spring preload  
Figure 2: Valve architecture.