## 2 Preliminaries  
In this section, some useful preliminaries such as symbols, lemmas, and existing optimal predictive approaches are introduced first.  
Denote \(\mathbb{Z}^{+}\) as the set of positive integers, \(\mathbb{Z}=\{0,\mathbb{Z}^{+}\}\), \(\mathbb{R}\) as the set of real numbers, \(\mathbb{R}^{p\times q}\) as the set of \(p\times q\) real matrices, \(\mathbb{R}^{s}\) as the set of column matrices with dimension \(s\), \(O^{m\times n}\) as the set of \(m\times n\) zero matrices, \(s^{[j]}\) as the \(j\)th derivative of \(s\) relative to time, \(sl\) as the factorial of \(s\in\mathbb{Z}\).  
This paper considers the following single-input-single-output (SISO) nonlinear affine systems:  
\[\left\{\begin{array}{l}\dot{x}_{i}(t)=x_{i+1}(t),\;i=1,\ldots,n-1\\ \dot{x}_{n}(t)=f(x(t))+g(x(t))\cdot u(t)\\ y(t)=x_{1}(t)\end{array}\right. \tag{1}\]  
where, \(x(t)=[x_{1}(t),\ldots,x_{n}(t)]\in\mathbb{R}^{n}\) represents the state vector, \(n\geq 2\) represents the system dimension. \(y(t)\in\mathbb{R}\) and \(u(t)\in\mathbb{R}\) represent output and input, respectively. \(f(x(t))\in\mathbb{R}\) and \(g(x(t))\in\mathbb{R}\) are nonlinear mappings with \(g(x)\
eq 0\) for any \(x(t)\). In next,the variable \(\vartheta(t)\) is denoted as \(\vartheta\) while the prediction of \(\vartheta\) within the predictive period \(\tau>0\) would be expressed by \(\vartheta(t+\tau)\), \(\vartheta\) is a scalar or vector. The following assumptions are imposed on system (1):  
1. The zero dynamics are stable.
2. All the states \(x_{i}\), \(i=1,\ldots,n\) are available.
3. The system output \(y\) and the prescribed reference \(y_{d}\) are sufficiently many times continuously differentiable relative to time \(t\).  
Secondly, the PCTOPC approach is reviewed.  
Differentiating both sides of the last state differential equation in Formula (1) \(\sigma\) times relative to \(t\) yields:  
\[\left\{\begin{array}{l}y^{[n+1]}=p_{1}\big{(}u^{2},u;x\big{)}+g(x)\cdot\dot{u }\\ y^{[n+2]}=p_{2}\Big{(}u^{3},u^{2},u,\dot{u};x\Big{)}+g(x)\cdot\dot{u}\\ \vdots\\ y^{[n+\sigma]}=p_{\sigma}\Big{(}u^{\sigma+1},\ldots,u,u^{[\sigma-1]},\ldots,\dot{u};x\Big{)}+g(x)\cdot u^{[\sigma]}\end{array}\right. \tag{2}\]  
where, \(\sigma\in\mathbb{Z}^{+}\) is called control order. Details of the nonlinear terms \(p_{i}\), \(i=1,\ldots,\sigma\) can be seen in [8].  
By using the Taylor series expansion, predictions of \(y\) and \(y_{d}\) within the predictive period \(t_{C}\) can be expressed by:  
\[\left\{\begin{array}{l}y(t+\tau)=\Gamma_{c}\cdot Y_{c}\\ y_{d}(t+\tau)=\Gamma_{c}\cdot Y_{cd}\end{array}\right. \tag{3}\]  
where, \(0<\tau\leq t_{C}\), \(\Gamma_{c}=\left[1,\tau,\ldots,\frac{\tau^{n+\sigma}}{(t+\sigma)!}\right]\), \(Y_{c}(t)=\left[y,\dot{y},\ldots,y^{[n+\sigma]}\right]^{T}\), and \(Y_{cd}(t)=\left[y_{d},\dot{y}_{d},\ldots,y_{d}^{[n+\sigma]}\right]^{T}\).  
By employing following RHPI:  
\[J_{C}=\frac{1}{2}\int_{0}^{t_{C}}[y(t+\tau)-y_{d}(t+\tau)]^{T}[y(t+\tau)-y_{d }(t+\tau)]d\tau \tag{4}\]  
the optimal predictive control law for the system (1) is summarized in following lemma.  
Lemma 1: [8]_. Consider the nonlinear affine system (1) satisfying (A1)-(A3). Then, for a given control order \(\sigma\in\mathbb{Z}^{+}\), the system output \(y\) and the prescribed reference \(y_{d}\) are predicted by Formula (3). The optimal predictive control law minimizing the RHPI (4) is given by:_  
\[u(t)=[g(x)]^{-1}\Big{[}y_{d}^{[n]}-f(x)-K_{c}M_{r}\Big{]} \tag{5}\]  
_where, \(M_{r}\) is given by:_  
\[M_{r}=\Big{[}x_{1}-y_{d},x_{2}-\dot{y}_{d},\ldots,x_{n}-y_{d}^{[n-1]}\Big{]}^{T} \tag{6}\]  
\(K_{c}\in\mathbb{R}^{1\times(\sigma+1)}\)_is the first row of matrix \(A_{\sigma}^{-1}A_{1}\). The matrices \(A_{1}\) and \(A_{\sigma}\) are given by:_  
\[\left\{\begin{array}{l}A=\int_{0}^{t_{1}}\Gamma_{c}^{T}\Gamma_{c}d\tau= \left[\begin{array}{cc}A_{r}&A_{1}^{T}\\ A_{1}&A_{\sigma}\\ \end{array}\right]\in\mathbb{R}^{(n+\sigma+1)\times(n+\sigma+1)}\\ A_{r}\in\mathbb{R}^{n\times n},\;\;A_{\sigma}\in\mathbb{R}^{(\sigma+1)\times( \sigma+1)}\end{array}\right. \tag{7}\]