## 3 Controller Design
### Stability Analysis  
**Theorem 2**. : _Consider system (28) with the control laws (44), (45), and the nonlinear first-order filter (32) under Assumption 1. Then, the closed-loop system is fixed-time stable, and the tracking errors can be stabilized in an arbitrarily small neighborhood of the origin._  
Before proving Theorem 2, the following necessary lemma that contributes to the derivation process needs to be stated:  
**Lemma 4** ([33]). : _If \(\eta_{1},\eta_{2},\ldots\eta_{n}\geq 0\) and \(p>\)0, then_  
\[\max\left(n^{p-1},1\right)\left(\eta_{1}^{p}+\eta_{2}^{p}+\cdots+\eta_{n}^{p} \right)\geq(\eta_{1}+\eta_{2}+\cdots+\eta_{n})^{p} \tag{46}\]  
**Proof of Theorem 2. ** Define a Lyapunov function candidate as follows:  
\[V_{1}=\frac{1}{2}{s_{1}}^{2}+\frac{1}{2}{s_{2}}^{2}+\frac{1}{2}{y_{1}}^{2}+ \frac{1}{2}{e_{2}}^{2}+\frac{1}{2}{e_{4}}^{2} \tag{47}\]  
Taking the time derivative of \(V_{1}\) gives  
\[\dot{V}_{1}=s_{1}\dot{s}_{1}+s_{2}\dot{s}_{2}+y_{1}\dot{y}_{1}+{e_{2}}\dot{e}_{ 2}+{e_{4}}\dot{e}_{4} \tag{48}\]  
Since the disturbance estimation error dynamics are unrelated to control inputs, the estimation errors \(e_{2}\) and \(e_{4}\) will converge to the neighborhood of the origin in fixed time \(T_{1}\) and \(T_{2}\), respectively, according to Theorem 1. When \(t\geq t_{1}=\max(T_{1},T_{2})\), we have \(e_{2}=0\), \(e_{4}=0\), \(V_{1}=\frac{1}{2}{s_{1}}^{2}+\frac{1}{2}{s_{2}}^{2}+\frac{1}{2}{y_{1}}^{2}\). Then, substituting Equations (32), (34), (36), (44) and (45) into Equation (48), we have  
\[ \begin{array}{l}\dot{V}_{1}=s_{1}\dot{s}_{1}+s_{2}\dot{s}_{2}+y_{1}\dot{y}_ {1}+{e_{2}}\dot{e}_{2}+{e_{4}}\dot{e}_{4}\\ \dot{V}_{1}=s_{1}\big{(}s_{2}+y_{1}+\omega_{zc}+f_{\alpha}+d_{\alpha}-\dot{ \alpha}_{c}\big{)}+s_{2}\big{(}g_{1}\delta+f_{\omega_{z}}+d_{\omega_{z}}- \dot{\omega}_{zd}\big{)}+y_{1}\big{(}\dot{\omega}_{zd}-\dot{\omega}_{zc}\big{)} \\ =s_{1}\Big{[}s_{2}+y_{1}-a_{1}\text{sig}(s_{1})^{\tau_{1}}-c_{1}\text{sig}(s_{ 1})^{\tau_{2}}+d_{\alpha_{z}}-\hat{d}_{\alpha}-b_{1}\text{sign}(s_{1})\Big{]} \\ +s_{2}\Big{[}-a_{2}\text{sig}(s_{2})^{\tau_{1}}-c_{2}\text{sig}(s_{2})^{\tau_{ 2}}+d_{\omega_{z}}-\hat{d}_{\omega_{z}}-b_{2}\text{sign}(s_{2})-s_{1}\Big{]} \\ +y_{1}\Big{(}-\frac{1}{\tau}\text{sig}(y_{1})^{\tau_{1}}-\frac{1}{\tau}\text{ sig}(y_{1})^{\tau_{2}}+\xi_{\omega_{z}}\Big{)}\\ \leq s_{1}y_{1}-a_{1}\big{(}{s_{1}}^{2}\big{)}^{\frac{\tau_{1}+1}{2}}-c_{1} \big{(}{s_{1}}^{2}\big{)}^{\frac{\tau_{2}+1}{2}}-a_{2}\big{(}s_{2}^{2}\big{)} ^{\frac{\tau_{1}+1}{2}}-c_{2}\big{(}{s_{2}}^{2}\big{)}^{\frac{\tau_{2}+1}{2}} \\ -\frac{1}{\tau}\big{(}y_{1}\big{)}^{\frac{\tau_{1}+1}{2}}-\frac{1}{\tau}({y_{1} }^{2})^{\frac{\tau_{1}+1}{2}}+y_{1}\xi_{\omega_{z}}\\ \end{array}  \tag{49}\]  
where \(\xi_{\omega_{z}}\) is the upper bound of \(\omega_{zc}\). Using [PERSON]'s inequality for \(s_{1}y_{1}\) and \(y_{1}\xi_{\omega_{z}}\) in the above equation, we can obtain  
\[ \begin{array}{l}\dot{V}_{1}\leq-a_{1}\big{(}{s_{1}}^{2}\big{)}^{\frac{\tau_{1 }+1}{2}}-c_{1}\big{(}{s_{1}}^{2}\big{)}^{\frac{\tau_{1}+1}{2}}-a_{2}\big{(}{s_{ 2}}^{2}\big{)}^{\frac{\tau_{1}+1}{2}}-c_{2}\big{(}{s_{2}}^{2}\big{)}^{\frac{ \tau_{2}+1}{2}}\\ -\frac{1}{\tau}\big{(}y_{1}\big{)}^{\frac{\tau_{1}+1}{2}}-\frac{1}{\tau}({y_{1 }}^{2})^{\frac{\tau_{2}+1}{2}}+\frac{1}{2}{y_{1}}^{2}+\frac{1}{2}{\varepsilon \omega_{z}}^{2}+\frac{1}{2}{s_{1}}^{2}+\frac{1}{2}{y_{1}}^{2}\\ \end{array}  \tag{50}\]  
Consider the following inequality:  
\[p\leqslant p^{q_{1}}+p^{q_{2}} \tag{51}\]  
where \(p\geq 0\), \(0<q_{1}<1\), and \(q_{2}>1\).