## 3 The PID-CTOPC Approach
### Controller Design  
In Formula (10), differentiating \(\rho\) times for \(s^{[n-1]}\) relative to \(t\) yields:  
\[\left\{ \begin{array}{l}s^{[n]}=\phi_{1}\big{(}u^{2},u;x\big{)}+\overline{g} (x)\cdot\dot{u}\\ s^{[n+1]}=\phi_{2}\big{(}u^{3},u^{2},u,\dot{u};x\big{)}+\overline{g}(x)\cdot \dddot{u}\\ \vdots\\ s^{[n-1+\rho]}=\phi_{p}\Big{(}u^{p+1},\ldots,u,u^{[\rho-1]},\ldots,\dddot{u};x \Big{)}+\overline{g}(x)\cdot u^{[\rho]}\end{array} \right. \tag{11}\]  
where, \(\phi_{i}\Big{(}u^{i+1},\ldots,u,u^{[i-1]},\ldots,\dddot{u};x\Big{)}\), \(i=1,\ldots,\rho\) are nonlinear functions. \(\rho\) represents the control order. Let:  
\[\left\{ \begin{array}{l}\overline{s}=\big{[}H_{0}^{T},H^{T}\big{]}^{T}\\ H_{0}=\Big{[}s,\dot{s},\ldots,s^{[n-2]}\Big{]}^{T}\\ H=\Big{[}s^{[n-1]},s^{[n]},\ldots,s^{[n-1+\rho]}\Big{]}^{T}\end{array} \right. \tag{12}\]Then, prediction of \(s(t)\) within the predictive period \(t_{M}\) can be written as:  
\[s(t+\tau)=\Gamma_{M}\overline{S} \tag{13}\]  
where, \(0<\tau\leq t_{M}\) and \(\Gamma_{M}=\left[1,\tau,\ldots,\frac{\tau^{n+\rho-1}}{(n+\sigma-1)^{1}}\right] \in\mathbb{R}^{1\times(n+\rho)}\). The RHPI to be minimized is selected as:  
\[J_{M}=\frac{1}{2}\int_{0}^{t_{M}}s^{T}(t+\tau)\cdot s(t+\tau)d\tau \tag{14}\]  
One of the main results as well as the PID-CTOPC approach for the system (1) is given in Theorem 1. **Theorem 1. **_Consider the nonlinear affine system (1) satisfying (A1)-(A3). By introducing a new PID-structuralized state variable shown in Formula (8), the prescribed reference tracking problem of the system can be transformed into a stabilization one given by (9). Then, for a given control order \(\rho\in\mathbb{Z}^{+}\), the new state variable \(s(t)\) is predicted by Formula (13). The optimal predictive control law minimizing the RHPI (14) is given by:_  
\[u(t)=-[\overline{g}(x)]^{-1}\Big{[}\overline{f}(x)+K_{M}H_{0}\Big{]} \tag{15}\]  
\(K_{M}\in\mathbb{R}^{1\times(n-1)}\) _is the first row of matrix \(B_{\rho}^{-1}B_{1}\). The matrices \(B_{1}\) and\(B_{\rho}\) are given by:_  
\[\left\{ \begin{array}{l}B=\int_{0}^{t_{M}}\Gamma_{M}^{T}\Gamma_{M}d\tau= \left[\begin{array}{cc}B_{r}&B_{1}^{T}\\ B_{1}&B_{\rho}\end{array}\right]\in\mathbb{R}^{(n+\rho)\times(n+\rho)}\\ B_{r}\in\mathbb{R}^{(n-1)\times(n-1)},\;\;B_{\rho}\in\mathbb{R}^{(\rho+1) \times(\rho+1)}\end{array} \right. \tag{16}\]  
**Proof of Theorem 1. ** Denote \(\overline{u}=\left[u,\dot{u},\ldots,u^{[\rho]}\right]^{T}\), then the partial derivatives of \(H_{0}\) and \(H\) with respect to \(\overline{u}\) can be given by:  
\[\frac{\partial H_{0}}{\partial\overline{u}}=O^{(n-1)\times(\rho+1)},\;\;\frac {\partial H}{\partial\overline{u}}= \begin{bmatrix}\overline{g}(x)&0&\ldots&0 \\ *&\overline{g}(x)&\ldots&0\\ *&*&\ddots&\vdots\\ *&*&*&\overline{g}(x)\end{bmatrix} \in\mathbb{R}^{(\rho+1)\times(\rho+1)} \tag{17}\]  
where, \(\partial H/\partial\overline{u}\) is a lower triangular matrix with the diagonal elements all \(\overline{g}(x)\), and the symbol '\(\star\)' represents the non-zero elements. It is obvious that the matrix \(\partial H/\partial\overline{u}\) is full rank. By considering Formula (13), differentiating \(J_{M}\) with respect to \(\overline{u}\) yields:  
\[\frac{\partial J_{M}}{\partial\overline{u}}=\left[\left(\frac{\partial H_{0}} {\partial\overline{u}}\right)^{T},\left(\frac{\partial H}{\partial\overline{u }}\right)^{T}\right]\cdot B\cdot \begin{bmatrix}H_{0}\\ H\end{bmatrix}  \tag{18}\]  
where, \(B=\int_{0}^{t_{M}}\Gamma_{M}^{T}\Gamma_{M}d\tau\). Partition the matrix \(B\) by following Formula (16).