## 3 Model-Free Robust Control Design
### Adaptive PD RBFNN Controller Design
#### 3.1.3 Lyapunov Stability Analysis  
By submitting the control law in Equation (14), the closed loop system is expressed as:  
\[\ddot{e}=-K^{T}E+\left\{\hat{f}(x)-f(x)\right\} \tag{19}\]  
Let:  
\[A_{mf}=\left[\begin{array}{cc}0&1\\ -k_{p}&-k_{d}\end{array}\right],B_{mf}=\left[\begin{array}{c}0\\ 1\end{array}\right]\]  
Equation (18) can be rewritten as:  
\[\dot{E}=A_{mf}E+B_{mf}\Big{\{}\hat{f}_{lpd}(x)-f_{lpd}(x)\Big{\}} \tag{20}\]  
Optimal weight value is \(W^{*}=arg\underset{W\in\Omega}{min}\{sup\Big{|}\hat{f}_{lpd}(x)-f_{lpd}(x) \Big{|}\}\).  
Modeling Error: \(\varepsilon=\hat{f}_{lpd}(x|\hat{W})-f_{lpd}(x|W^{*})\).  
Equation (19) becomes:  
\[\dot{E}=A_{mf}E+B_{mf}[\{\hat{f}_{lpd}(x|\hat{W})-f_{lpd}(x|W^{*})\}+\varepsilon] \tag{21}\]  
Using Equation (20):  
\[\dot{E}=A_{mf}E+B_{mf}\{{(\hat{W}-W^{*})}^{T}h(x)+\varepsilon\} \tag{22}\]  
Choosing the Lyapunov function as:  
\[V=\frac{1}{2}E^{T}PE+\frac{1}{2\gamma}{(\hat{W}-W^{*})}^{T}{(\hat{W}-W^{*})} \tag{23}\]  
where \(\gamma\) is a positive constant. \((\hat{W}-W^{*})\) denotes the parameter estimation error and the matrix \(P\) is symmetric and positive definite, and satisfies the following Lyapunov equation:  
\[A_{mf}^{T}P+PA_{mf}=-Q \tag{24}\]  
with \(Q\geq 0\). Finding \(\dot{V}\):  
\[\dot{V}=\dot{V}_{1}+\dot{V}_{2} \tag{25}\]  
\[V_{1}=\frac{1}{2}E^{T}PE\]  
Taking the derivative of \(V_{1}\):  
\[\dot{V}_{1}=\frac{1}{2}\dot{E}PE+\frac{1}{2}E^{T}p\dot{E}\]  
Rewriting Equation (20) as:  
\[\dot{V}_{1}=\frac{1}{2}\dot{E}PE+\frac{1}{2}E^{T}p\dot{E} \tag{26}\]  
\[\dot{E}=A_{mf}E+M \tag{27}\]  
where \(M=B_{mf}\{{(\hat{W}-W^{*})}^{T}h(x)+\varepsilon\}\).  
Submitting Equation (26) into Equation (25):  
\[\dot{V}_{1}=\frac{1}{2}E^{T}\Big{(}A_{mf}^{T}P+PA_{mf}\Big{)}E+\frac{1}{2}M^{ T}PE+\frac{1}{2}E^{T}PM\]Using Equation (24):  
\[\dot{V}_{1}=\frac{1}{2}E^{T}QE+E^{T}PM\]  
Submitting \(M\) into the above equation:  
\[\dot{V}_{1}=-\frac{1}{2}E^{T}QE+\left(\hat{W}-W^{*}\right)^{T}E^{T}PB_{mf}h(x)+E ^{T}PB_{mf}\varepsilon \tag{28}\]  
Taking the derivative of \(V_{2}\):  
\[\dot{V}_{2}=\frac{1}{\gamma}{\left(\hat{W}-W^{*}\right)^{T}}\hat{W} \tag{29}\]  
Substituting Equations (27) and (28) into Equation (24):  
\[\dot{V}=-\frac{1}{2}E^{T}QE+E^{T}PB_{mf}\varepsilon+\frac{1}{\gamma}{\left( \hat{W}-W^{*}\right)^{T}}\Big{\{}\hat{W}+\gamma E^{T}PB_{mf}h(x)\Big{\}} \tag{30}\]  
The adaptive law is chosen as:  
\[\dot{\hat{W}}=-\gamma E^{T}PB_{mf}h(x) \tag{31}\]  
Substituting Equation (30) into Equation (29):  
\[\dot{V}=-\frac{1}{2}E^{T}QE+E^{T}PB_{mf}\varepsilon \tag{32}\]  
Since \(-\frac{1}{2}E^{T}QE\leq 0\), considering the adaptive control system convergence analysis in [52], if the approximation error \(\varepsilon\) is made very small using the RBF, \(\dot{V}\leq 0\) can be achieved.  
In order to design an adaptive PD RBFNN, first, a Lyapunov-based adaptive PD control law was designed. The unknown lumped function is approximated using the RBFNN. The error with the desired position \(y_{d}\) is defined as \(E=\left[\begin{array}{cc}e&e^{T}\\ \end{array}\right]\) and the adaptive PD control law is formulated as:  
\[v^{*}=\frac{1}{\alpha}\Big{\{}-f(x)+\ddot{y}_{d}+K^{T}E\Big{\}} \tag{33}\]  
where \(K=\left[\begin{array}{cc}k_{P}&k_{D}\\ \end{array}\right]\).  
The adaptive law is defined as:  
\[\hat{W}=-\gamma E^{T}PB_{mf}h(x) \tag{34}\]