## 3 Control Design and Stability Analysis
### Controller Design
Let  
\[\Delta u=v-u \tag{37}\] \[\text{hence}\] \[v=u+\Delta u \tag{38}\]  
Substituting (38) into (36), yields  
\[\dot{V}_{z} \leq -k_{1}f_{z1}^{2}-k_{2}f_{z2}^{2}-\varphi_{1}^{\top}\bar{\theta}_{ 1}f_{z1} \tag{39}\] \[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]0 \[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]1  
The effect of the term \(a_{1}f_{z2}b\Delta u\) should be canceled by an adequate choice of \(W_{\psi}\), but this is hampered by the uncertainty on the control gain \(b\). To this end, \(b\) is expressed in terms of an updated parameter and a parameter updating error. Recall from assumption 5 that \(|b|\leq\bar{\mu}_{ub}|b_{m}|\). Therefore,  
\[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]2  
As the upper bound \(\bar{\mu}_{ub}\) is unknown, it is expressed in terms of updated parameter and parameter updating error. Let \(\bar{\theta}_{ub}=\hat{\theta}_{ub}-\bar{\mu}_{ub}\), where \(\hat{\theta}_{ub}\), \(\hat{\theta}_{ub}\) are a parameter estimation error and an updated parameter, and \(\hat{\theta}_{ub}\) is provided by an updating mechanism defined later. From the above definition it follows that \(\bar{\mu}_{ub}=\hat{\theta}_{ub}-\hat{\theta}_{ub}\). Substituting into Equation (40) and arranging, yields  
\[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]3  
Substituting into Equation (39), yields  
\[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]4 \[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]5 \[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]6  
Thus, the right hand side of \(dV_{z}/dt\) is affected by the following terms: (i) the term \(\hat{\theta}_{ub}a_{1}|f_{z2}b_{m}\Delta u|\), which is later tackled by properly choosing \(W_{\psi}\), and (ii) the term \(-a_{1}|f_{z2}b_{m}\Delta u|\bar{\theta}_{ub}\), which is later tackled by defining the quadratic form for \(\bar{\theta}_{ub}\). To this end, if \(|f_{z2}|\) is expressed as \(|f_{z2}|=f_{z2}sign(f_{z2})\), the resulting expression of \(W_{\psi}\) would contain the \(sign(f_{22})\) signal, so that chattering might occur. Therefore, we notice from definition (32) that  
\[+f_{z2}\bigg{(}f_{z1}+a_{1}a_{3}\frac{x_{1}-x_{2}}{x_{3}}+a_{1}a_ {4}\frac{x_{2}}{x_{3}}+B_{1d}(-a_{2}x_{1}-r_{x1})+B_{1e}\bigg{)}\]7 (43) where \[sat_{22}=\left\{ \begin{array}{l}+1\;\;\text{for}\;\;z_{2}\geq C_{b}\\ (1/C_{b})z_{2}\;\;\text{for}\;\;z_{2}\in(-C_{b},C_{b})\\ -1\;\;\text{for}\;\;z_{2}\leq-C_{b}\end{array} \right. \tag{44}\]  
Therefore, (41) leads to  
\[a_{1}f_{22}b\Delta u+f_{22}(-1)W_{\psi}\leq f_{22}\big{(}\hat{\theta}_{ub}sat _{22}a_{1}|b_{m}\Delta u|-W_{\psi}\big{)}-a_{1}|f_{22}b_{m}\Delta u|\tilde{ \theta}_{ub} \tag{45}\]  
we choose  
\[W_{\psi}=a_{1}\hat{\theta}_{ub}sat_{22}|b_{m}\Delta u| \tag{46}\]  
substituting into Equation (24), (25) gives the auxiliary system  
\[\dot{\psi}_{1}=-K_{\psi 1}\psi_{1}+\psi_{2} \tag{47a}\] \[\dot{\psi}_{2}=-K_{\psi 2}\psi_{2}+a_{1}\hat{\theta}_{ub}sat_{ 22}|b_{m}\Delta u|\] (47b) \[K_{\psi 1}>1/2,\;\;K_{\psi 2}>1/2 \tag{47c}\]  
where \(sat_{22}\) is defined in (44). Substituting (46) into Equation (45) yields \(a_{1}f_{22}b\Delta u+f_{22}(-1)W_{\psi}\leq-a_{1}|f_{22}b_{m}\Delta u|\tilde{ \theta}_{ub}\).