## 3 Controller Design Based on Legendre Polynomial FAT
### Stability Analysis of the Proposed Method  
For the adaptive control law and error-boundary control law in Equations (18)-(20), the Lyapunov theory is used for the stability analysis to prove the boundedness of control parameters and the uniformly convergence of the joint error. The following lemma is needed for the stability analysis. **Lemma 1**. : _[_26_]_ _If the differentiable function \(\Delta(t)\) has a finite limit as \(t\rightarrow+\infty\), and if \(\dot{\Delta}(t)\) is uniformly continuous with respect to \(t\) (a sufficient condition for a differentiable function to be uniformly continuous is that its derivative is bounded), then \(\dot{\Delta}(t)\to 0\) as \(t\rightarrow\infty\)._  
Consider the following Lyapunov function candidate:  
\[V=\frac{1}{2}s^{T}s+\frac{1}{2}E^{T}P_{0}E+\frac{1}{2\lambda_{1}}\overline{P}^ {T}\overline{P}+\frac{1}{2\lambda_{2}}\overline{\rho}^{T}\overline{\rho} \tag{21}\]  
Combined Equation (17), the time derivative of Equation (21) is  
\[ \begin{array}{ll}\dot{V}&=s^{T}\dot{s}+\frac{1}{2}\dot{E}^{T}P_{0}E+\frac{1}{ 2}E^{T}P_{0}\dot{E}+\frac{1}{\lambda_{1}}\overline{P}^{T}\dot{\overline{P}}+ \frac{1}{\lambda_{2}}\overline{\rho}^{T}\dot{\overline{\rho}}\\ &=s^{T}\dot{s}+\frac{1}{2}E^{T}A^{T}P_{0}E+\frac{1}{2}w^{T}B^{T}P_{0}E+\frac{ 1}{2}E^{T}P_{0}AE+\frac{1}{2}E^{T}P_{0}Bw-\frac{1}{\lambda_{1}}\overline{P}^ {T}\dot{\hat{p}}-\frac{1}{\lambda_{2}}\overline{\rho}^{T}\dot{\hat{p}}\\ &=s^{T}\dot{s}-\frac{1}{2}E^{T}QE+E^{T}P_{0}Bw-\frac{1}{\lambda_{1}}\overline{ P}^{T}\dot{\hat{p}}-\frac{1}{\lambda_{2}}\overline{\rho}^{T}\dot{\hat{p}}\end{array}  \tag{22}\]  
Substituting Equation (15) into \(s^{T}\dot{s}\) in the Equation (22), and then integrating the Equations (11) and (12), it can yield  
\[ \begin{array}{ll}s^{T}\dot{s}&=s^{T}k_{d}(H-\hat{H})-s^{T}[\mu s+\hat{\rho} sign(s)]-s^{T}k_{d}u_{c}\\ &=s^{T}k_{d}\overline{P}^{T}\varphi(x)-s^{T}hs-s^{T}\beta sign(s)-s^{T}k_{d}u_ {c}+s^{T}k_{d}\varepsilon_{m}\end{array}  \tag{23}\]  
Substituting Equations (17c), (11) and (12) into \(E^{T}P_{0}Bw\) in the Equation (22), one has  
\[E^{T}P_{0}Bw=E^{T}P_{0}B\overline{P}^{T}\varphi(x)-E^{T}P_{0}Bk_{d}{}^{-1}[hs+ \hat{\rho}sign(s)]-E^{T}P_{0}Bu_{c}+E^{T}P_{0}Be_{m} \tag{24}\]  
Then, in a similar way, substituting Equations (23) and (24) into Equation (25), we have  
\[\dot{V}=-\frac{1}{2}E^{T}QE+V_{1}+V_{2} \tag{25a}\]  
Figure 1: The control system block diagram of the proposed controller. \[V_{1}=\overline{P}^{T}\Big{[}s^{T}k_{d}\varphi(x)\ +E^{T}P_{0}B\varphi(x)\Big{]}- \frac{1}{\lambda_{1}}\overline{P}^{T}\hat{P}-\Big{[}s^{T}k_{d}+E^{T}P_{0}B\Big{]} \Big{[}u_{\epsilon}+k_{d}{}^{-1}hs\Big{]} \tag{25b}\] \[V_{2}=\Big{[}s^{T}k_{d}\varepsilon_{m}-s^{T}\hat{\rho}sign(s)\Big{]}+\Big{[}E^{T}P_{0 }B\varepsilon_{m}-E^{T}P_{0}Bk_{d}{}^{-1}\hat{\rho}sign(s)\Big{]}-\frac{1}{ \lambda_{2}}\overline{P}^{T}\hat{\rho} \tag{25c}\]  
On the basis of Equation (18b) and adaptive law Equation (19), we obtain \(V_{1}<0\). Due to \(\overline{\rho}=\rho-\hat{\rho}\), the Equation (25c) can be expressed as  
\[ \begin{array}{rl}V_{2}=&\Big{[}s^{T}k_{d}\varepsilon_{m}-s^{T}\beta sign(s) \Big{]}+\Big{[}E^{T}P_{0}B\varepsilon_{m}-E^{T}P_{0}Bk_{d}{}^{-1}\rho sign(s) \Big{]}\\ &+\overline{\rho}^{T}\Big{[}s+k_{d}{}^{-1}(E^{T}P_{0}B)^{T}\Big{]}sign(s)- \frac{1}{\lambda_{2}}\overline{P}^{T}\hat{\rho}\end{array}  \tag{26}\]  
Thus, according to the error-boundary control law Equation (20), yields  
\[V_{2}=\Big{[}s^{T}k_{d}\varepsilon_{m}-s^{T}\rho sign(s)\Big{]}+\Big{[}E^{T}P _{0}B\varepsilon_{m}-E^{T}P_{0}Bk_{d}{}^{-1}\rho sign(s)\Big{]} \tag{27}\]  
For (27), to prove the stability of the control system (i.e., \(\dot{V}\leq 0\)), the necessary and sufficient conditions are as follows. \[s^{T}k_{d}\varepsilon_{m}-s^{T}\rho sign(s)\leq 0 \tag{28a}\] \[E^{T}P_{0}B\varepsilon_{m}-E^{T}P_{0}Bk_{d}{}^{-1}\rho sign(s)\leq 0 \tag{28b}\]  
Therefore, to satisfy the above formula, the following analysis proves  
(a) Due to \(k_{d}\in\)[0,1], we have \(|\varepsilon_{m}|\leq\rho\leq\delta=\rho/k_{d}\), then  
\[