## 2 Problem Formulation and Preliminary
### Performance Function Design  
To ensure that the transient and steady performance of the tracking errors evolves within the reasonable boundaries, the description of the prescribed performance of the \(\mathbf{z}_{1}\) is expressed as  
\[-\rho_{i}(t)<\mathbf{z}_{1i}(t)<\rho_{i}(t), \tag{11}\]  
where \(i=1,\,2,\,3\), and \(\rho_{i}(t)\) denotes a novel PTPPF, which is described as follows:  
\[\rho_{i}(t)= \begin{cases}a_{1}+a_{2}\mathrm{arctan}\Big{(}\frac{t}{T_{f}} \Big{)}+a_{3}e^{-k\frac{t}{T_{f}}}+a_{4}e^{-k(\frac{t}{T_{f}})^{2}}&,\,\,0\leq t \leq T_{f},\\ \rho_{i\infty}&,\,\,t>T_{f}\end{cases}  \tag{12}\]  
\[ \begin{array}{l}a_{1}=\rho_{i0}+3a_{4}\,,\,\,\,\,a_{2}=-4 ke^{-k}a_{4}\\ a_{3}=-4a_{4}\,\,\,\,\,\,\,,\,\,\,\,a_{4}=\frac{\rho_{i\infty}-\rho_{i0}}{- \pi ke^{-k}-3e^{-k}}\end{array} , \tag{13}\]  
where \(\rho_{i0}\) is the maximum overshoot, and \(\rho_{i\infty}\) is the prescribed steady-state boundary. \(\rho_{i0}\) and \(\rho_{i\infty}\) can be flexibly adjusted according to the control requirements. \(k\) is a positive constant, \(0.5<k<3\), which determines the decreasing rate of \(\rho_{i}\) when \(0\leq t\leq T_{f}\). \(T_{f}\) denotes the prescribed terminal time that \(\rho_{i}\) reaches \(\rho_{i\infty}\). **Theorem 2**. : _The PTPPF \(\rho_{i}(t)\) satisfies \(\dot{\rho}_{i}(t)<0\) when \(0\leq t\leq T_{f}\) and \(\lim_{t\to T_{f}^{*}}\rho_{i}=\  
\[\ddot{\rho}_{i}=\frac{-2a_{2}t}{T_{f}\gamma^{3}\bigg{(}1+\left(t/T_{f}\right)^{2} \bigg{)}^{2}}-\frac{k^{2}a_{3}}{T_{f}}e^{-k\frac{t}{T_{f}}}-\frac{2 ka_{4}}{T_{f} ^{2}}e^{-k(\frac{t}{T_{f}})^{2}}+\frac{4k^{2}a_{4}t^{2}}{T_{f}4}e^{-k(\frac{t}{ T_{f}})^{2}}. \tag{15}\]  
According to Equation (13), it is obvious that \(\lim_{t\to T_{f}^{+}}\rho_{i}=\lim_{t\to T_{f}^{+}}\rho_{i}\) and \(\lim_{t\to T_{f}}\dot{\rho}_{i}=\lim_{t\geq T_{f}}\dot{\rho}_{i}=0\). The PTPPF and its first-order derivative are continuous with respect to time. Considering Equation (13) and the range of \(k\), it can be calculated that \(a_{4}<0\). To reduce computational cost, the proof of \(\dot{\rho}_{i}<0\) can be converted to \(y>0\) on \(\left[0,\ T_{f}\right]\), where \(y\) is  
\[y=-\frac{4 ke^{-k}}{T_{f}\bigg{(}1+\left(t/T_{f}\right)^{2}\bigg{)}}+\frac{4k} {T_{f}}e^{-k\frac{t}{T_{f}}}-\frac{2 kt}{T_{f}^{2}}e^{-k(\frac{t}{T_{f}})^{2}},\ \ 0\leq t<T_{f}. \tag{16}\]  
Let \(x=t/T_{f}\); Equation (16) can then be simplified as follows:  
\[y=-\frac{4 ke^{-k}}{T_{f}(1+x^{2})}+\frac{4k}{T_{f}}e^{-kx}-\frac{2 kx}{T_{f}}e ^{-kx^{2}}\,\ \ 0\leq x<1, \tag{17}\]  
where \(y(0)>0\) and \(y(1)=0\). Under the condition \(0\leq x<1\), the derivative of \(y\) is obtained as  
\[\dot{y}=\frac{8 ke^{-k}x}{T_{f}\big{(}1+x^{2}\big{)}^{2}}-\frac{4k^{2}}{T_{f}} e^{-kx}-\frac{2k}{T_{f}}e^{-kx^{2}}+\frac{4k^{2}x^{2}}{T_{f}}e^{-kx^{2}}, \tag{18}\]  
\[\ddot{y}=\frac{8 ke^{-k}}{T_{f}}\bigg{(}\frac{1}{\left(1+x^{2}\right)^{2}}+ \frac{4x^{2}}{\left(1+x^{2}\right)^{3}}\bigg{)}+\frac{4k^{2}}{T_{f}}\Big{[} \Big{(}ke^{-kx}-kx^{3}e^{-kx^{2}}\Big{)}+\Big{(}3 xe^{-kx}-kx^{3}e^{-kx^{2}} \Big{)}\Big{]}. \tag{19}\]  
We can get that \(\dot{y}>0\), \(\dot{y}(0)<0\), and \(\dot{y}(1)=0\) when \(0\leq x<1\). This means that \(\dot{y}\) is in a trend of monotone increasing, and \(\dot{y}\) will not be greater than zero because of \(\dot{y}(1)=0\). As a result, \(y\) is monotonically decreasing and greater than zero in \(0\leq x<1\). Specifically, it can be concluded that \(\dot{\rho}_{i}<0\) for all \(0\leq t\leq T_{f}\). Thus, the proof of Theorem 2 is complete. \(\square\)  
Some studies have been conducted to affirm the property of the proposed PTPPF. The parameters are given as \(\rho_{i0}=10\) and \(\rho_{i0\infty}=0.01\). In Figure 2, we can see that the variate \(\rho_{i}\) converges to the prescribed steady-state error range within a predefined time \(T_{f}\). Figure 3 shows that \(\rho_{i}(t)\) satisfies \(\dot{\rho}_{i}(t)<0\) when \(0\leq t\leq T_{f}\) and \(\lim_{t\to T_{f}^{-}}\rho_{i}=\lim_{t\to T_{f}^{+}}\rho_{i}=\rho_{i\infty}\).