# Adaptive Fuzzy Command Filtered Tracking Control for Flexible Robotic Arm with Input Dead-Zone
## 4 Design of Fuzzy Adaptive Controller and Stability Analysis
Step 3: Choose the virtual control function \(\alpha_{3}\) and the compensatory signal \(\dot{\omega}_{3}\) as follows:  
\[\alpha_{3}=-c_{3}z_{3}+\dot{\overline{\alpha}}_{2}-r_{2}-\frac{1}{2}r_{3}, \tag{42}\]  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]0  
where \(c_{3}>0\) is a designed parameter. Step 4: Choose the virtual control function \(\alpha_{4}\), the adaptive rate \(\dot{\theta}_{4}\), and the compensatory signal \(\dot{\omega}_{4}\) as follows:  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]1  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]2  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]3  
where \(c_{4}>0\) and \(\sigma_{4}>0\) are designed parameters. Step 5: Choose the Lyapunov function as follows:  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]4  
The derivative of \(V_{5}\) is as follows:  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]5As a result of the [PERSON]'s inequality,  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]6  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]7  
Through substituting Equations (45) and (46) into Equation (44), we can obtain the following:  
\[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]8 \[\varepsilon_{1}r_{1}\leq\frac{1}{2}r_{1}^{2}+\frac{1}{2}\varepsilon_{1}^{*2}, \tag{34}\]9  
As a result of the [PERSON]'s inequality,  
\[\dot{V}_{1}\leq\dot{V}_{0}+r_{1}\bigg{(}r_{2}+\omega_{2}+\frac{1}{2}r_{1}+ \overline{\alpha}_{1}-\dot{y}_{r}-\dot{\omega}_{1}\bigg{)}+\frac{1}{2} \varepsilon_{1}^{*2}, \tag{35}\]0  
where \(h^{*}\) is a constant representing the maximum value of \(h\). Through substituting Equation (17) into Equation (48), we can obtain the controller \(v\) as follows:  
\[\dot{V}_{1}\leq\dot{V}_{0}+r_{1}\bigg{(}r_{2}+\omega_{2}+\frac{1}{2}r_{1}+ \overline{\alpha}_{1}-\dot{y}_{r}-\dot{\omega}_{1}\bigg{)}+\frac{1}{2} \varepsilon_{1}^{*2}, \tag{35}\]1  
\[\dot{V}_{1}\leq\dot{V}_{0}+r_{1}\bigg{(}r_{2}+\omega_{2}+\frac{1}{2}r_{1}+ \overline{\alpha}_{1}-\dot{y}_{r}-\dot{\omega}_{1}\bigg{)}+\frac{1}{2} \varepsilon_{1}^{*2}, \tag{35}\]2  
The compensatory signal \(\dot{\omega}_{5}\) is given by the following:  
\[\dot{V}_{1}\leq\dot{V}_{0}+r_{1}\bigg{(}r_{2}+\omega_{2}+\frac{1}{2}r_{1}+ \overline{\alpha}_{1}-\dot{y}_{r}-\dot{\omega}_{1}\bigg{)}+\frac{1}{2} \varepsilon_{1}^{*2}, \tag{35}\]3  
The adaptive rate \(\dot{\theta}_{5}\) is given by the following:  
\[\dot{V}_{1}\leq\dot{V}_{0}+r_{1}\bigg{(}r_{2}+\omega_{2}+\frac{1}{2}r_{1}+ \overline{\alpha}_{1}-\dot{y}_{r}-\dot{\omega}_{1}\bigg{)}+\frac{1}{2} \varepsilon_{1}^{*2}, \tag{35}\]4  
where \(c_{5}>0\) and \(\sigma_{5}>0\) are designed parameters. **Theorem 1**. : _For the flexible robotic arm system (1) with unobservable states and dead-zone inputs, the state observer (21); virtual control functions (36), (39), (42), (44); parameter adaptation rates (40), (45), (56); and the controller (53) ensure bounded signals within the closed-loop system. Moreover, by selecting appropriate design parameters, the tracking error converges to a small neighborhood._  
**Proof of Theorem 1.