# Adaptive Fuzzy Command Filtered Tracking Control for Flexible Robotic Arm with Input Dead-Zone
## 4 Design of Fuzzy Adaptive Controller and Stability Analysis  
A block diagram of the control system, implementing tracking control of the flexible robotic arm using adaptive fuzzy control and employing an observer to estimate unobservable states, is shown in Figure 3.  
Define the error variables \(z_{i}(i=1,\cdots,5)\), where \(y_{r}\) is the system's reference signal. To address the issue of computational explosion in backstepping control, command filtering is introduced at this stage. \(\overline{\dot{\pi}}_{i}(i=1,\cdots,4)\) represents the filtered controller signals; \(\alpha_{i}(i=1,\cdots,4)\) represents the virtual control functions. \(\xi_{i}\) is the system's state value (obtained as \(\dot{\xi}_{i}\) through a state observer), which serves as the state estimate. \[\left\{ \begin{array}{l}z_{1}=y-y_{r}\\ z_{2}=\hat{\xi}_{2}-\overline{\alpha}_{1}\\ z_{3}=\hat{\xi}_{3}-\overline{\alpha}_{2}\\ z_{4}=\hat{\xi}_{4}-\overline{\alpha}_{3}\\ z_{5}=\hat{\xi}_{4}-\overline{\alpha}_{4}\end{array} \right., \tag{28}\]  
The defined compensation error signal is \(r_{i}=z_{i}-\omega_{i}(i=1,2,\cdots,5)\). The command filter is defined as:  
\[\dot{\kappa}_{1}=\omega\kappa_{2}, \tag{29}\]  
\[\dot{\kappa}_{2}=-2\zeta\omega\kappa_{2}-\omega(\kappa_{1}-\alpha_{i}), \tag{30}\]  
where \(\omega>0\) and \(\zeta\in(0,1]\) are the designed parameters, \(\overline{\alpha}_{i}(t)=\kappa_{1}(t)\) is the output of each filtering signal, and \(\alpha_{i-1}\) is the input to the command filter. The initial condition are \(\kappa_{1}(0)=\alpha_{i}(0)\) and \(\kappa_{2}(0)=0\). Step 1: The derivative of \(r_{1}\) is as follows:  
\[ \begin{array}{l}\dot{r}_{1} =\dot{z}_{1}-\dot{\omega}_{1}\\ =\dot{\xi}_{2}+k_{1}\big{(}y-\dot{\xi}_{1}\big{)}-\dot{y}_{r}-\dot{\omega}_{1 }\end{array} , \tag{31}\]  
Choose the Lyapunov function as follows:  
\[V_{1}=V_{0}+\frac{1}{2}r_{1}^{2}, \tag{32}\]  
The derivative of \(V_{1}\) is as follows:  
\[ \begin{array}{l}\dot{V}_{1} =\dot{V}_{0}+r_{1}\dot{r}_{1}\\ =\dot{V}_{0}+r_{1}\big{(}r_{2}+\omega_{2}+\varepsilon_{1}+\overline{\alpha}_{ 1}-\dot{y}_{r}-\dot{\omega}_{1}\big{)}\end{array} , \tag{33}\]  
where \(\varepsilon_{i}=k_{i}\big{(}y-\dot{\xi}_{1}\big{)}(i=1\cdots 5);\varepsilon_{i} {}^{*}\) is an unknown constant. Figure 3: Control system. As a result of the Young's inequality,  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]  
Through substituting Equation (31) into Equation (30), we can obtain the following:  
\[\dot{V}_{1}\leq\dot{V}_{0}+r_{1}\bigg{(}r_{2}+\omega_{2}+\frac{1}{2}r_{1}+ \overline{\alpha}_{1}-\dot{y}_{r}-\dot{\omega}_{1}\bigg{)}+\frac{1}{2} \varepsilon_{1}^{*2}, \tag{35}\]  
Choose the virtual control function \(\alpha_{1}\) as:  
\[\alpha_{1}=-c_{1}z_{1}+\dot{y}_{r}-\frac{1}{2}r_{1}, \tag{36}\]  
The compensatory signal \(\dot{\omega}_{1}\) is given by the following:  
\[\dot{\omega}_{1}=-c_{1}\omega_{1}+\omega_{2}+(\overline{\alpha}_{1}-\alpha_{1 }), \tag{37}\]  
where \(c_{1}>0\) is a designed parameter. Through substituting Equations (32) and (33) into Equation (31), we can obtain the following:  
\[\dot{V}_{1}\leq\dot{V}_{0}-c_{1}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}+r_{ 1}r_{2}, \tag{38}\]  
Step 2: Choose the virtual control function \(\alpha_{2}\), the adaptive rate \(\dot{\theta}_{2}\), and the compensatory signal \(\dot{\omega}_{2}\) as follows:  
\[\alpha_{2}=-c_{2}z_{2}-\theta_{2}^{T}\phi_{2}\bigg{(}\xi_{1},\dot{\xi}_{2}, \dot{\xi}_{3,f}\bigg{)}+\dot{\overline{\alpha}}_{1}-r_{1}-r_{2}, \tag{39}\]  
\[\dot{\theta}_{2}=r_{2}\phi_{2}\bigg{(}\xi_{1},\dot{\xi}_{2},\dot{\xi}_{3,f} \bigg{)}-\sigma_{2}\theta_{2}, \tag{40}\]  
\[\dot{\omega}_{2}=-c_{2}\omega_{2}+\omega_{3}+(\overline{\alpha}_{2}-\alpha_{2 }), \tag{41}\]  
where \(\sigma_{2}>0\) and \(c_{2}>0\) are designed parameters.