## 2 PTOS Control Scheme  
In this section, we simply introduce the control scheme of PTOS. As many typical servo systems can be modeled by double integral operators, the specific expression is depicted as  
\[\left\{\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx\end{array}\right. \tag{1}\]  
with  
\[x=\left(\begin{array}{c}y\\ v\end{array}\right),A=\left[\begin{array}{cc}0&1\\ 0&0\end{array}\right],B=\left[\begin{array}{cc}0\\ a\end{array}\right],C=\left[\begin{array}{cc}1&0\end{array}\right],\]  
where \(x\) is the system state and \(A,B,C\) are the system state matrices. State \(y\) is the system position output, and \(v\) is the system speed output. \(u\) is the control input, which is also the control law output, and \(a\) is the positive acceleration constant.  
To take the effects of control input saturation into account, \(u\) should be expressed as shown  
\[u=\text{sign}\left(u_{c}\right)\cdot\min\left\{u_{\max},\left|u_{c}\right| \right\}, \tag{2}\]  
where \(u_{c}\) is the original output of the control law.  
For the double integral operators, the goal of PTOS control scheme is to make the system output \(y\) track the target position signal \(r\) as quickly, accurately, and smoothly as possible, and the control law \(u_{p}\) is  
\[u_{p}=\text{sign}\left(u\right)\cdot\min\left\{u_{\max},\left|u_{out}\right| \right\}. \tag{3}\]  
\(u_{out}\) is given by  
\[u_{out}=k_{2}\left[f(e)-v\right] \tag{4}\]  
trade-off between the overshoot and response settling time. Second, due to the windup phenomenon (The so-called windup refers to the phenomenon that the closed-loop performance of the system is deteriorated due to the unequal control input and actuator output) [25; 26], the relative stability of the system may decrease. According to the classical linear control theory, the requirement of phase margin is usually higher than \(45^{\circ}\). As show in Figure 1, the solid black line represents a simple Bode diagram of the Type II system, where \(\omega_{c}\) is the system open-loop crossover frequency, \(\gamma\) is the system phase margin, and \(\omega_{a}\), \(\omega_{b}\) are the frequency values corresponding to a phase margin of \(45^{\circ}\) in the phase-frequency curve.  
When the system input signal is a SRS, the system error signal will encounter windup with a high probability. If so, the controller input is saturated, which equivalent to the system gain's (\(k\) Value in (11)) reduction. Correspondingly, the system amplitude-frequency curve will change from the solid black line to the blue dotted line or even the red dotted line in Figure 1. In the blue dotted line, the system phase margin is \(\gamma^{\prime}\). By comparing \(\gamma\) and \(\gamma^{\prime}\), we can see that the relative stability of the system is declined, but \(\gamma^{\prime}\) still meets the stability margin requirement in actual engineering. If the system gain \(k\) decreases so that the crossover frequency becomes \(\omega_{c}\)\({}^{\prime\prime}\), the phase margin \(\gamma^{\prime\prime}\) will be lower than \(45^{\circ}\). At this time, the dynamic performance of the system is poor and the adaptability to the control system parameters is weak. As can be seen from the above analyses of the frequency response, the windup phenomenon can be regarded as an uncertain gain in the open loop transfer function. Due to the consistency of the frequency and time domain, the output signal may encounter oscillations or divergence after actuator saturated when the reference is a step/constant signal.