# Adaptive Fuzzy Command Filtered Tracking Control for Flexible Robotic Arm with Input Dead-Zone
## 2 Preliminaries
### Fuzzy Logic System  
A fuzzy logic system (FLS) [40] consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. The knowledge base for the FLS comprises a collection of fuzzy if-then rules of the following form:  
\(R^{l}\): If \(x_{1}\) is \(F^{l}_{1}\) and \(x_{2}\) is \(F^{l}_{2}\) and \(\ldots\) and \(x_{n}\) is \(F^{l}_{n}\),  
then, \(y\) is \(G^{l},l=1,2,\ldots,N\).  
Here, \(x=x_{1},\ldots,x_{n}{}^{T}\) and \(y\) are the input and output of the fuzzy logic system. \(F^{l}_{i}\) and \(G^{l}\) are fuzzy sets. Associated membership functions \(\mu_{F^{l}_{i}}(x_{i})\) and \(\mu_{G^{l}}(y)\) are defined. \(N\) represents the number of rules. The fuzzy logic system can be expressed as follows:  
\[y(x)=\frac{\sum_{l=1}^{N}\overline{y_{l}}\prod_{i=1}^{n}\mu_{F^{l}_{i}}(x_{i}) }{\sum_{l=1}^{N}\left[\prod_{i=1}^{n}\mu_{F^{l}_{i}}(x_{i})\right]}, \tag{10}\]  
where \(\overline{y_{l}}=\underset{y\in R}{max}\mu_{G^{l}}(y)\).  
Define the fuzzy basis function:  
\[\phi_{l}=\frac{\prod_{i=1}^{n}\mu_{F^{l}_{i}}(x_{i})}{\sum_{l=1}^{N}\left( \prod_{i=1}^{n}\mu_{F^{l}_{i}}(x_{i})\right)}, \tag{11}\]  
where \(\theta{=[\overline{y}_{1},\overline{y}_{2},\ldots,\overline{y}_{N}]^{T}{=[ \theta_{1},\theta_{2},\ldots,\theta_{N}]^{T}}}\) and \(\phi^{T}(x)=[\phi_{1}(x),\cdots,\phi_{N}(x)]\).  
The general form of a fuzzy logic system is as follows:  
\[y(x)=\theta^{T}\phi(x), \tag{12}\]  
**Lemma 3** ([41]).: _The function \(f(x)\) is smooth on the compact set \(\Omega\subseteq R^{N}\). For a given constant \(\varepsilon>0\), there exists \(y(x)=\theta^{T}\phi(x)\) so that:_  
\[\underset{x\in\Omega}{sup}\Big{|}f(x)-\theta^{T}\phi(x)\Big{|}\leq\varepsilon, \tag{13}\]  
According to [42; 43], we define:  
\[\theta=\underset{\theta\in R^{N}}{arg\,min}\Bigg{\{}\underset{x\in\Omega}{sup} \Big{|}f(x)-\theta^{T}\phi(x)\Big{|}\Bigg{\}}, \tag{14}\]  
Then,  
\[f(x)=\hat{\theta}^{T}\phi(x)+\varepsilon, \tag{15}\]  
where \(|\varepsilon|\leq\varepsilon^{*}\), \(\varepsilon^{*}\) is an unknown constant.  
**Lemma 4** ([44]): _The Young's inequality is defined as follows: for any variable \(x,y\in R\), the following inequality must hold:_  
\[\mathrm{xy}\leq\frac{o^{p}}{p}|x|^{p}+\frac{1}{qo^{q}}|y|^{q}, \tag{16}\]  
_where \(o>0\), \(p>1\), \(q>1\), \((p-1)(q-1)=1\)._  
Before proceeding with the fuzzy state observer design, we first express the dead-zone [36] as a slowing time-varying input dependent function with a bounded disturbance:  
\[u=mv+h, \tag{17}\]  
The dead zone function is shown in Figure 2, where \(b_{i}>0\), \(b_{j}<0\). \(m_{i}(v)\) is the unknown smooth function.  
According to Equations (4) and (17), we can express the dead zone as follows:  
\[u=m(v-sat(t)), \tag{18}\]  
\[u=\begin{cases}m_{i}(v_{i}-b_{i})&v_{i}>b_{i}\\ 0&-b_{i}\leq v_{i}\leq b_{i}\\ m_{i}(v_{i}+b_{i})&v_{i}<b_{i}\end{cases}, \tag{19}\]  
where the asymmetric saturation function is defined as follows:  
\[sat(t)=\begin{cases}\begin{array}{cc}b_{i}&v_{i}>b_{i}\\ v_{i}&-b_{i}\leq v_{i}\leq b_{i}\\ b_{i}&v_{i}<b_{i}\end{array},\end{cases} \tag{20}\]