## 3 Prescribed Performance Constraint Control of EHS
### Controller Design Based on PPC  
Together with (7), the system state errors are defined as follows  
\[\left\{ \begin{array}{l}z_{1}=\ln\left(\frac{e_{\max}(e_{\min}-e/\rho)}{e_{ \min}(e_{\max}-e/\rho)}\right)\\ z_{2}=x_{2}-\alpha_{1}\\ z_{3}=x_{3}-\alpha_{2}\end{array} \right., \tag{11}\]  
where \(e\) is the position tracking error defined in (4), \(\alpha_{i}(i=1,2)\) is the virtual control variable in controller design. To avoid the explosion of complexity caused by the repeatedly calculated differentiations of \(\dot{\alpha}_{i}(i=1,2)\) in the backstepping iteration, the dynamic surfaces of \(z_{i+1}(i=1,2)\) are given as follows  
\[\tau_{i}\dot{\alpha}_{i}+\alpha_{i}=\beta_{i},\quad\alpha_{i}(0)=\beta_{i}(0) \tag{12}\]  
where \(\beta_{i}(i=1,2)\) are the stabilizing functions to be designed, \(\tau_{i}(i=1,2)\) are the time constants of the dynamic surfaces. Thus, the output errors of two dynamic surfaces are defined as \(S_{i}=\alpha_{i}-\beta_{i}(i=1,2)\). Substituting \(S_{i}\) into (12), the virtual control derivatives \(\dot{\alpha}_{i}=-S_{i}/\tau_{i}(i=1,2)\) are obtained. Based on the system state errors (11) and the dynamic surface (12), the prescribed performance constraint controller \(u\) is designed as follow  
\[\left\{ \begin{array}{l}\beta_{1}=\dot{y}_{d}+\frac{\dot{\rho}}{\rho}e-k_{1} \frac{z_{1}}{r}\\ \beta_{2}=-\frac{1}{g_{2}}\left(k_{2}z_{2}+rz_{1}+\tilde{f}_{2}+\frac{S_{1}}{ \tau_{1}}\right)\\ \alpha_{i}=-\int\limits_{0}^{t}\frac{S_{i}}{\tau_{i}}dt,i=1,2\\ S_{i}=\alpha_{i}-\beta_{i},i=1,2\\ u=-\frac{1}{g_{3}}\left(k_{3}z_{3}+\tilde{f}_{3}+\tilde{g}_{2}z_{2}+\frac{S_{2 }}{\tau_{2}}\right)\end{array} \right., \tag{13}\]  
where the attenuated parameter \(r\) is  
\[r=\frac{\partial T^{-1}}{\partial(e/\rho)}\frac{1}{\rho}=\frac{e_{\max}-e_{ \min}}{(e_{\max}-e/\rho)(e/\rho-e_{\min})\rho}\geq\frac{e_{\max}-e_{\min}}{ \left(\frac{e_{\max}-e_{\min}}{2}\right)^{2}}=\frac{4}{e_{\max}-e_{\min}}>0. \tag{14}\]  
**Theorem 2**. : _Considering the stabilizing functions (13) together with their dynamic surfaces (12) for the EHS model (2) under Hypothesis 1 and Remarks 1 and 2, regardless of the system state errors \(z_{i}(t)(i=1,2,3)\) start from any initial values \(-\infty<z_{i}(0)<\infty\), the generalized error \(Z_{g}(t)\) including \(z_{i}(i=1,2,3)\) and \(S_{j}(j=1,2)\) is ultimate boundedness [42] and its convergence domain is an hypersphere \(H_{r}\),_  
\[H_{r}\in\left\{\sum_{i=1}^{3}z_{i}^{2}+\sum_{j=1}^{2}S_{j}^{2}=2V(0)e^{-ct_{f} }+2\delta/c\right\} \tag{15}\]  
_where \(\delta\) and \(c\) are positive constants, \(V(0)\) is the initial system state error, \(\forall t>t_{f}\) (\(t_{f}\) is a finite time)._  
Proof. : The candidate quadratic Lyapunov function of the EHS model (2) is given by  
\[V=\frac{1}{2}\sum_{i=1}^{3}z_{i}^{2}+\frac{1}{2}\sum_{j=1}^{2}S_{j}^{2}. \tag{16}\]  
For convenient proof, \(V\) is rewritten into the cascade elements for the convenient controller design as follows  
\[\left\{ \begin{array}{l}V_{1}=\frac{1}{2}z_{1}^{2}+\frac{1}{2}S_{1}^{2}\\ V_{2}=V_{1}+\frac{1}{2}z_{2}^{2}+\frac{1}{2}S_{2}^{2}\\ V_{3}=V_{2}+\frac{1}{2}z_{3}^{2}\end{array} \right., \tag{17}\]  
and the following inequalities are satisfied by [PERSON]'s inequality  
\[\left|z_{i}S_{i}\right|\leq\frac{z_{i}^{2}+S_{i}^{2}}{2}\ \ \ \ \left|S_{i}\dot{\beta}_{i}\right|\leq \frac{z_{i}^{2}\left|\dot{\beta}_{i}\right|^{2}_{\max}}{2\sigma_{i}}+\frac{ \sigma_{i}}{2}\ \ \ \ \left|z_{i+1}\Delta_{i+1}\right|\leq\frac{z_{i+1}^{2}+\Delta_{i+1}^{2} \max}{2}\ \, \tag{18}\]  
for \(i=1,2\), where \(\sigma_{i}(i=1,2)\) are positive constants, \(\left|\dot{\beta}_{i}\right|_{\max}(i=1,2)\) are the maximal boundaries of \(\dot{\beta}_{i}(i=1,2)\). If a constant \(c\) is defined as \(c=\min\{2\Gamma_{1},2\Gamma_{2},2\Omega_{1},2\Omega_{2},2k_{3}\}\), from the definitions of \(\tilde{V}_{1}\) and \(\tilde{V}_{2}\), (27) is rewritten as  
\[\dot{V}_{3}\leq-cV_{3}+\delta. \tag{28}\]  
Integrating (28), \(\dot{V}_{3}\) yields  
\[