# Adaptive Fuzzy Command Filtered Tracking Control for Flexible Robotic Arm with Input Dead-Zone
## 3 Fuzzy Adaptive Observer  
The state \(\xi_{i}(i=2,3,4,5)\) of the flexible robotic arm system are unobservable. Therefore, it is necessary to establish a suitable state observer to monitor the system's state. The primary purpose of the observer is to provide a real-time estimate \(\xi(t)\) for the state \(\xi(t)\) in the aforementioned model (3).  
To start with, a fuzzy logic system is employed to approximate the unknown nonlinear functions within the robotic arm system. The nonlinear functions are represented as  
Figure 2: Dead zone model.  
follows: \(-\frac{mgd}{\hbar}cosy-\frac{F_{1}}{\hbar}\xi_{2}-\frac{K}{\hbar}\left(\xi_{1}- \frac{\xi_{3}}{N}\right)-\xi_{3}\), \(\frac{K}{2N}\left(\xi_{1}-\frac{\xi_{3}}{N}\right)-\frac{F_{2}}{\hbar}\xi_{4}- \frac{K_{4}}{\hbar}\xi_{5}-\xi_{5}\), \(-\frac{R}{\hbar}\xi_{5}-\frac{K_{6}}{\hbar}\xi_{4}-\frac{1}{\hbar}u-u\).  
The model of the fuzzy adaptive observer is depicted as follows:  
\[\left\{\begin{array}{l}\dot{\xi}_{1}=\dot{\xi}_{2}+k_{1}(y-\dot{\xi}_{1})\\ \dot{\xi}_{2}=\theta_{2}^{T}\phi_{2}(\xi_{1},\dot{\xi}_{2},\dot{\xi}_{3,f})+ \dot{\xi}_{3}+k_{2}(y-\dot{\xi}_{1})\\ \dot{\xi}_{3}=\dot{\xi}_{4}+k_{3}(y-\dot{\xi}_{1})\\ \dot{\xi}_{4}=\theta_{4}^{T}\phi_{4}(\xi_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{ \xi}_{4},\dot{\xi}_{5,f})+\dot{\xi}_{5}+k_{4}(y-\dot{\xi}_{1})\\ \dot{\xi}_{5}=\theta_{5}^{T}\phi_{5}(\xi_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{ \xi}_{4},\dot{\xi}_{5},u_{f})+u+k_{5}(y-\dot{\xi}_{1})\\ y=\xi_{1}\end{array}\right., \tag{21}\]  
Formula (21) can be rewritten as follows:  
\[\left\{\begin{array}{l}\dot{\xi}=A\dot{\xi}+Ky+F+C_{1}u\\ y=C_{2}\dot{\xi}\end{array}\right., \tag{22}\]  
where \(F=\left[0,\theta_{2}^{T}\phi_{2}\left(\xi_{1},\dot{\xi}_{2},\dot{\xi}_{3,f} \right),\;0,\;\theta_{4}^{T}\phi_{4}\left(\xi_{1},\dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4},\dot{\xi}_{5,f}\right),\;\theta_{5}^{T}\phi_{5}\left(\xi_{1}, \dot{\xi}_{2},\dot{\xi}_{3},\dot{\xi}_{4},\dot{\xi}_{5},u_{f}\right)\right]^{ T}\), \(\dot{\xi}=\left[\dot{\xi}_{1},\dot{\xi}_{2}\cdots,\dot{\xi}_{5}\right]^{T}\), \(A=\left[\begin{matrix}-k_{1}\\ \vdots&I_{4}\\ -k_{5}\cdots 0\end{matrix}\right],K=\left[k_{1},\cdots,k_{5}\right]^{T},C_{1}^{T}= \left[0,0,0,0,1\right]\), \(C_{2}=\left[1,\)0,0,0,0\right]\).  
By selecting appropriate parameters, \(k_{i}(i=1,\cdots,5)\), ensure that the polynomial \(p(s)=s^{5}+k_{1}s^{4}+\cdots+k_{5}\) satisfies the Hurwitz matrix condition. In other words, given a matrix of \(Q^{T}=Q>0\), there exists a positive definite symmetric matrix, \(P^{T}=P>0\), meaning that  
\[A^{T}P+PA=-Q, \tag{23}\]  
Define the observer error as \(e=\xi-\dot{\xi}=\left[e_{1},\cdots,e_{5}\right]^{T}\); then,  
\[\dot{e}=Ae+\varepsilon+\widetilde{\theta}, \tag{24}\]  
where \(\varepsilon=\left[0,\varepsilon_{2},0,\varepsilon_{4},\varepsilon_{5}\right]^ {T}\), \(\widetilde{\theta}_{i}=\hat{\theta}_{i}-\theta_{i}\), \(i=2,4,5\).  
Choose the Lyapunov function \(V_{0}\) as  
\[V_{0}=e^{T}Pe, \tag{25}\]  
\[\dot{V}_{0}=-e^{T}Qe+2e^{T}P\left(\varepsilon+\widetilde{\theta}\right), \tag{26}\]  
By using the [PERSON]'s inequality, we can obtain the following:  
\[\dot{V}_{0}\leq-e^{T}Qe+3e^{2}+P^{2}\left(\varepsilon_{2}^{*^{2}}+\varepsilon_ {4}^{*^{2}}+\varepsilon_{5}^{*^{2}}\right)+P^{2}\left(\widetilde{\theta}_{2} ^{T\sim}\widetilde{\theta}_{2}+\widetilde{\theta}_{4}^{T\sim}\widetilde{ \theta}_{4}+\widetilde{\theta}_{5}^{T\sim}\widetilde{\theta}_{5}\right), \tag{27}\]