## 3 Control Design and Stability Analysis
### Controller Design
\tag{30}\]The main properties of \(V_{z2}\) are  
\[V_{z2}=0\text{ for }z_{2}\in[-C_{b},\ C_{b}]\] \[V_{z2}>0\text{ for }z_{2}\
otin[-C_{b},\ C_{b}]\] \[V_{z2}\text{ is continuous with respect to }z_{2},\text{ and it is bounded for }z_{2}\text{ bounded}\]  
The above properties and a stable dynamics of \(V_{z2}\) imply the convergence of \(z_{2}\) to \(\Omega_{z2}\), \(\Omega_{z2}=\{z_{2}:|z_{2}|\leq C_{b}\}\), as shown in the convergence theorem in Section 3.2. Differentiating \(V_{z2}\) with respect to time yields  
\[\dot{V}_{z2}=f_{z2}\frac{dz_{2}}{dt}, \tag{31}\] \[f_{z2}=\frac{dV_{z2}}{dz_{2}}=\left\{ \begin{array}{ll}z_{2}-C_{ b}\text{ for }z_{2}\geq C_{b}\\ 0\text{ for }z_{2}\in(-C_{b},C_{b})\\ z_{2}+C_{b}\text{ for }z_{2}\leq-C_{b}\end{array} \right. \tag{32}\]  
Incorporating the expression for \(z_{2}\) (26) into Equation (31) and arranging yields  
\[\dot{V}_{z2}=-k_{2}f_{z2}^{2}+f_{z2}\bigg{(}a_{1}a_{3}\frac{x_{1}- x_{2}}{x_{3}}+a_{1}a_{4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e} \bigg{)}\] \[+f_{z2}\big{(}a_{1}bv+(-1)W_{\psi}\big{)} \tag{33}\]  
Differentiating \(V_{z}\) in (29) with respect to time yields \(\dot{V}_{z}=\dot{V}_{z1}+\dot{V}_{z2}\). Substituting the expressions for \(\dot{V}_{z1}\) (21) and \(\dot{V}_{z2}\) (33) yields  
\[\dot{V}_{z}\leq -C_{b}|f_{z1}|+f_{z1}z_{2}-k_{1}f_{z1}^{2}-k_{2}f_{z2}^{2}-\varphi _{1}^{\top}\bar{\theta}_{1}f_{z1}\] \[+f_{z2}\bigg{(}a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_{4} \frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\] \[+f_{z2}\big{(}a_{1}bv+(-1)W_{\psi}\big{)} \tag{34}\]  
To obtain the required right hand side of \(dV_{z}/dt\), the terms  
\[f_{z1}z_{2}+f_{z2}\bigg{(}a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_{4}\frac{ x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]  
should be rewritten and tackled by means of proper definition of \(W_{\psi}\) and the control law. In the \(f_{z1}z_{2}\) term, the \(z_{2}\) signal must be expressed in terms of \(f_{z2}\), because the controller design is based on the dead zone functions \(f_{z1}\) and \(f_{z2}\) rather than \(z_{1}\) and \(z_{2}\). From definition (32), it follows that \(f_{z2}=z_{2}+\delta\),  
\[\delta=\left\{ \begin{array}{ll}-C_{b}\text{ for }z_{2}\geq C_{b}\\ 0\text{ for }z_{2}\in(-C_{b},C_{b})\\ C_{b}\text{ for }z_{2}\leq-C_{b}\end{array} \right.\]  
therefore, \(z_{2}=f_{z2}-\delta\) and \(|\delta|\leq C_{b}\). Therefore, \(f_{z1}z_{2}=f_{z1}f_{z2}-\delta f_{z1}\). Substituting into Equation (34) yields  
\[\dot{V}_{z}\leq -C_{b}|f_{z1}|-\delta f_{z1}-k_{1}f_{z1}^{2}-k_{2}f_{z2}^{2}- \varphi_{1}^{\top}\bar{\theta}_{1}f_{z1}\] \[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\] \[+f_{z2}\big{(}a_{1}bv+(-1)W_{\psi}\big{)} \tag{35}\]so that the error term \(\delta\) leads to the undesired uncertainty term \(-\delta f_{z1}\) in Equation (35). The property \(|\delta|\leq C_{b}\) implies that \(-\delta f_{z1}\leq C_{b}|f_{z1}|\), which is canceled by the already existing term \(-C_{b}|f_{z1}|\), so that \(-C_{b}|f_{z1}|-\delta f_{z1}\leq 0\), and Equation (35) yields  
\[\dot{V}_{z} \leq -k_{1}f_{z1}^{2}-k_{2}f_{z2}^{2}-\varphi_{1}^{\top}\bar{\theta}_{1 }f_{z1} \tag{36}\] \[+f_{z2}\Big{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_{4 }\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\Big{)}\] \[+f_{z2}\big{(}a_{1}bv+(-1)W_{\psi}\big{)}\]  
Therefore, the term \(-C_{b}|f_{z1}|\) incorporated in Equation (12) is necessary for counteracting the effect of the error term \(\delta\) resulting from \(f_{z2}=z_{2}+\delta\). In order to facilitate the design of the control law for \(u\), the constrained input signal \(v\) is expressed in terms of the unconstrained input signal \(u\) and the input error \(\Delta u\), and the effect of \(\Delta u\) is later canceled by the input of the auxiliary system, \(W_{\psi}\).