## 2 PTOS Control Scheme
### TP-Transition Process  
To improve the performance of the PTOS control scheme for set-point tracking of SRSs, that is, to increase the swiftness and smoothness while avoiding windup, the TP is introduced in detail, which causes the jumping input signal \(r\) to become a slowly varying rising signal \(r_{tp}\). After the system input becomes the TP signal \(r_{tp}\), the control law will always calculate the control input depending on small errors, which completely avoids the windup to reduce the overshoot. By performing TP operation on  
Figure 1: A simple Bode diagram of the Type II system. the SRS, \(r_{tp}\) is obtained. And the control law of the PTOS for the SRSs in Equation (10) is integrated into Figure 2, where the new system can be rewritten as  
\[\left\{ \begin{array}{l}\dot{x}=\left(A-BK\right)x+Bk_{1}r_{tp}\\ y=Cx\end{array} \right. \tag{12}\]  
with  
\[x=\left( \begin{array}{l}y\\ v\end{array} \right),A=\left[ \begin{array}{ll}0&1\\ 0&0\end{array} \right],B=\left[ \begin{array}{l}0\\ a\end{array} \right],\]  
\[C=\left[ \begin{array}{cc}1&0\end{array} \right],K=\left[ \begin{array}{cc}k _{1}&k_{2}\end{array} \right].\]  
The TP signal in Equation (12) is designed to track the SRSs fast and smoothly. The block diagram is shown in Figure 3, where \(r_{tp}\) tracks \(r\), \(\hat{r}_{tp}=r_{tp}^{\prime}\). The method of solving TP is the minimum principle in the optimal control, wherein the time-optimal tracking is the performance index and the state \(r_{tp}\) is obtained by solving \(u\). The TP portion in Figure 3 is  
\[\left\{ \begin{array}{l}\dot{z}=f\left(z,t\right)=A^{\prime}z+B^{\prime}u\\ y=C^{\prime}z\end{array} \right. \tag{13}\]  
with  
\[z=\left( \begin{array}{c}z_{1}\\ z_{2}\end{array} \right)=\left( \begin{array}{c}r_{tp}\\ r^{\prime}_{tp}\end{array} \right),A^{\prime}=\left[ \begin{array}{cc}0&1\\ 0&0\end{array} \right],B^{\prime}=\left[ \begin{array}{cc}0\\ 1\end{array} \right],C^{\prime}=\left[ \begin{array}{cc}1&0\end{array} \right].\]  
The initial conditions and terminal conditions are, resepectively,  
\[ \begin{array}{l}z_{1}\left(0\right)=z_{2}\left(0\right)=0,\\ z_{1}\left(t_{f}\right)=r_{A},z_{2}\left(t_{f}\right)=0,\end{array}  \tag{14}\]  
where \(r_{A}\) is the absolute amplitude of input signal. And the control constraint is  
\[\left|u\right|\leq a_{\max},0\leq t\leq t_{f}. \tag{15}\]  
The objective is to solve the optimal control that minimizes the following performance index,  
Figure 3: The control block for solving TP. Figure 2: The control block of tracking SRSs after TP added into the PTOS. \[J=\varphi\left(z\left(t_{f}\right),t_{f}\right)+\int_{t_{0}}^{t_{f}}F\left(z\left(t \right),u\left(t\right),t\right)dt. \tag{16}\]  
According the optimal control theory performance index design method in [27], we can choose \(F=1\) as a usually used, convenient and effective functional for time optimum design, and \(\varphi=0\) without considering the energy and fuel minimum. Then take the Hamilton function as  
\[H=F+\lambda^{\mathrm{T}}f\left(z,t\right)=1+\lambda_{1}z_{2}\left(t\right)+ \lambda_{2}u\left(t\right), \tag{17}\]  
where \(\lambda\) is the co-state vector and expressed as  
\[\lambda\left(t\right)=\left[ \begin{array}{cc}\lambda_{1}\left(t\right)& \lambda_{2}\left(t\right)\end{array} \right]^{T}.\]  
From the co-state equation  
\[\left\{ \begin{array}{c}\dot{\lambda}_{1}\left(t\right)=-\frac{\partial H}{ \partial z}=0\\ \dot{\lambda}_{2}\left(t\right)=-\frac{\partial H}{\partial z_{2}}=-\lambda_{ 1}\end{array} \right. \tag{18}\]  
we can obtain  
\[\left\{ \begin{array}{c}\lambda_{1}\left(t\right)=c_{1}\\ \lambda_{2}\left(t\right)=-c_{1}t+c_{2}\end{array} \right., \tag{19}\]  
where \(c_{1}\) and \(c_{2}\) is the integral constant.