## 3 Control Design and Stability Analysis
### Controller Design
Substituting this into Equation (16) yields  
\[\dot{V}_{z1} \leq -k_{1b}|f_{z1}|-k_{1}f_{z1}^{2}+f_{z1}a_{1}x_{2} \tag{20}\] \[+f_{z1}\left(\varphi_{1}^{\top}\bar{\theta}_{1}+k_{1}f_{z1}-\dot {y}_{d}-\dot{\psi}_{1}+k_{1b}sat_{z1}\right)-\varphi_{1}^{\top}\bar{\theta}_{1 }f_{z1}\]  
The dynamics of \(V_{z1}\) is affected by the following terms: (i) the term \(-k_{1b}|f_{z1}|-k_{1}f_{z1}^{2}\) is negative and it provides stability; (ii) the term \(-\varphi_{1}^{\top}\bar{\theta}_{1}f_{z1}\), which is later tackled by defining a quadratic form for \(\bar{\theta}_{1}\), as can be noticed in the boundedness and convergence theorems in Section 3.2; and (iii) the term \(f_{z1}a_{1}x_{2}+f_{z1}\left(\varphi_{1}^{\top}\bar{\theta}_{1}+k_{1}f_{z1}- \dot{y}_{d}-\dot{\psi}_{1}+k_{1b}sat_{z1}\right)\), which is later tackled by adequate definition of the new state \(z_{2}\), so that this term equals \(f_{z1}z_{2}\) and (20) yields  
\[\dot{V}_{z1}\leq-k_{1b}|f_{z1}|-k_{1}f_{z1}^{2}+f_{z1}z_{2}-\varphi_ {1}^{\top}\hat{\theta}_{1}f_{z1}\] (21) where \[z_{2}=a_{1}x_{2}+(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1})+k_{1b} sat_{z1}+\varphi_{1}^{\top}\hat{\theta}_{1} \tag{22}\]  
The effect of the term \(f_{z1}z_{2}\) can be tackled by adequate dynamics of \(V_{z1}\) and \(V_{z2}\), such that \(\dot{V}_{z1}+\dot{V}_{z2}\) involves the term \(-kf_{z1}^{2}-kf_{z2}^{2}\), being \(f_{z2}\) a saturation function of \(z_{2}\) and \(V_{z2}\) a quadratic function of \(f_{z2}\). To this end, the required dynamics of \(z_{2}\) is generated next. Differentiating \(z_{2}\) (22) with respect to time yields  
\[\dot{z}_{2}=a_{1}\dot{x}_{2}+k_{1}\frac{df_{z1}}{dz_{1}}\dot{z}_{1}-\ddot{y}_{ d}-\ddot{\psi}_{1}+k_{1b}\frac{dsat_{z1}}{dz_{1}}\dot{z}_{1}+\varphi_{1}^{ \top}\hat{\theta}_{1}+\varphi_{1}^{\top}\hat{\theta}_{1} \tag{23}\]  
The auxiliary state \(\psi_{1}\) and its time derivatives are provided by the auxiliary system, whose general structure consists of a second order linear filter whose input \(W_{\psi}\) involves the input error \(\Delta u\):  
\[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]0 \[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]1  
where the term \(W_{\psi}\), which will be defined later, is used to cancel the effect of the \(\Delta u\) term, which is the difference between the saturated and non-saturated input signals. From (24), (25) it follows that \(\ddot{\psi}_{1}=K_{\psi 1}^{2}\psi_{1}-(K_{\psi 1}+K_{\psi 2}\psi_{2})+W_{\psi}\). Substituting this expression and the expressions for \(\dot{z}_{1}\) (7), \(\dot{x}_{1}\) (1) and \(\dot{x}_{2}\) (3) into Equation (23) and arranging yields  
\[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]2 \[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]3  
where  
\[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]4 \[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]5 \[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]6 (28) \[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]7 \[+f_{z1}(k_{1}f_{z1}-\dot{y}_{d}-\dot{\psi}_{1}-a_{2}x_{1}-r_{x_{1 }}+k_{1b}sat_{z1})\]8  
where  
\[\frac{df_{z1}}{dz_{1}}=\left\{ \begin{array}{l}2(z_{1}-C_{b})\;\;\text{for}\; \;z_{1}\geq C_{b}\\ 0\;\;\text{for}\;\;z_{1}\in(-C_{b},C_{b})\\ (-2)(z_{1}+C_{b})\;\;\text{for}\;\;z_{1}\leq-C_{b}\end{array} \right.\]  
and \(dsat_{z1}/dz_{1}\) is defined in Equation (15). Let  
\[V_{z}=V_{z1}+V_{z2} \tag{29}\] \[V_{z2}=\left\{ \begin{array}{l}(1/2)(z_{2}-C_{b})^{2}\;\;\text{ for}\;\;z_{2}\geq C_{b}\\ 0\;\;\text{for}\;\;z_{2}\in(-C_{b},C_{b})\\ (1/2)(z_{2}+C_{b})^{2}\;\;\text{for}\;\;z_{2}\leq-C_{b}\end{array} \right.