## 3 Adaptive Sliding Mode Controller Design via Backstepping
### Stability Analysis  
**Theorem 1**. : _For the reaching law of sliding mode control as Equation (13), the state of system \(s\) must converge to the equilibrium point in fixed time._  
Proof of Theorem 1.:
1. According to Equation (13), we get:\[ \begin{array}{rl}\dot{s\dot{s}}=&s\big{[}-D_{1}|s|^{\gamma}sgn(s)-D_{2}|s|^{\beta }sgn(s)-D_{3}s\big{]}\\ &=-D_{1}|s|^{\gamma+1}-D_{2}|s|^{\beta+1}-D_{3}s^{2}\leq 0\end{array}  \tag{58}\]  
Only if \(s=0\) can \(\dot{s\dot{s}}=0\); therefore, the system state \(s\) can reach the equilibrium point \(s=0\) under the approach law Equation (13). 2. Fixed-time convergence  
The initial state of the system is proposed as \(s_{0}>1\), the process of system convergence is divided into two stages, as follows:  
(1) \(s_{0}\to s=1\)  
Because \(\gamma>1\) and \(0<\beta<1\), then the first and third terms play a major role in the reaching law at this stage; the reaching law Equation (13) can be expressed as:  
\[\dot{s}=-D_{1}|s|^{\gamma}sgn(s)-D_{3}s \tag{59}\]  
Integrating Equation (59), we get:  
\[s^{1-\gamma}={s_{0}}^{1-\gamma}\frac{D_{1}}{D_{3}}e^{-(1-\gamma)D_{3}t}-\frac {D_{1}}{D_{3}} \tag{60}\]  
Then, the convergence time in this stage can be expressed as follows:  
\[t_{1}=\frac{1}{(1-\gamma)D_{3}}\bigg{[}ln\bigg{(}s^{1-\gamma}+\frac{D_{1}}{D_ {3}}\bigg{)}-ln\bigg{(}{s_{0}}^{1-\gamma}+\frac{D_{1}}{D_{3}}\bigg{)}\bigg{]} \tag{61}\]  
(2) \(s=1\)\(\rightarrow\)\(s=0\)  
In this stage, the second and third terms play a major role in Equation (13), and the reaching law Equation (13) can be expressed as:  
\[\dot{s}=-D_{2}|s|^{\beta}sgn(s)-D_{3}s \tag{62}\]  
Taking the integrate of Equation (62), we have:  
\[s^{1-\beta}=\bigg{(}1+\frac{D_{2}}{D_{3}}\bigg{)}e^{-(1-\beta)D_{3}t}-\frac{D _{2}}{D_{3}} \tag{63}\]  
Then, the convergence time in this stage can be expressed as follows:  
\[t_{2}=\frac{1}{(\beta-1)D_{3}}\bigg{[}ln\bigg{(}s^{1-\beta}+\frac{D_{2}}{D_{3} }\bigg{)}-ln\bigg{(}1+\frac{D_{2}}{D_{3}}\bigg{)}\bigg{]} \tag{64}\]  
From Equations (61) and (64), it is obvious that an increase in \(D_{3}\) can decrease \(t_{1}\) and \(t_{2}\), which can lead to a better convergence speed. Based on the above analysis, for the reaching law of sliding mode control as Equation (13), the convergence time satisfies the formula:  
\[T\leq t_{1}+t_{2} \tag{65}\]  
Thus, the proof is completed. \(\square\)  
**Theorem 2**. : _For the nonlinear system given by Equation (14), the controller given by Equation (55), and the adaptive laws represented by Equation (56), one can guarantee the stability of the system._  
**Proof of Theorem 2. ** A Lyapunov function candidate is chosen as Equation (48). Substituting Equations (55) and (56) into Equation (54), we have:\[\dot{V}_{n} =-\sum\limits_{j=1}^{n-1}k_{j}\tau_{j}^{T}z_{j}-\tau_{n-1}^{T}(c_{1}z _{1}+c_{2}z_{2}+\cdots+c_{n-1}z_{n-1})-S^{T}u_{sw}\] \[=-\sum\limits_{j=1}^{n-1}k_{j}\tau_{j}^{T}z_{j}-\tau_{n-1}^{T}(c_{1 }z_{1}+c_{2}z_{2}+\cdots+c_{n-1}z_{n-1}) \tag{66}\] \[-\sum\limits_{m=1}^{2}(D_{m,1}|s_{m}|^{\gamma_{m}+1}+D_{m,2}|s_{m }|^{\beta_{m}+1}+D_{m,3}s_{m}^{2})\]  
We define the matrix as follows:  
\[\mathbf{Q}= \begin{bmatrix}k_{1}&0&\cdots&0&0\\ 0&k_{2}&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&k_{n-2}&0\\ c_{1}&c_{2}&\cdots&c_{n-2}&k_{n-1}+c_{n-1}\end{bmatrix}  \tag{67}\]  
and  
\[z= \begin{bmatrix}z_{1}&z_{2}&\cdots&z_{n-1}\end{bmatrix} ^{T} \tag{68}\]  
Then, we have:  
\[\mathbf{z}^{T}\mathbf{Q}\mathbf{z}=\sum\limits_{j=1}^{n-1}k_{j}\tau_{j}^{T}z_{j}+\tau_{n-1 }^{T}(c_{1}\tau_{1}+c_{2}z_{2}+\cdots+c_{n-1}z_{n-1}) \tag{69}\]  
Substituting Equation (69) into Equation (66), we have:  
\[\dot{V}_{n}=-\mathbf{z}^{T}\mathbf{Q}\mathbf{z}-\sum\limits_{m=1}^{2}(D_{m,1}|s_{m}|^{ \gamma_{m}+1}+D_{m,2}|s_{m}|^{\beta_{m}+1}+D_{m,3}s_{m}^{2}) \tag{70}\]  
The principal minors in matrix \(\mathbf{Q}\) are positive, and \(\mathbf{Q}\) is a positive definite matrix.