## 2 New Adaptive Fuzzy PID Controller
### Adaptive Fuzzy PID Control  
In this study, a general form of the \(m\)th-order nonlinear system is used as follows:  
\[\dot{\textbf{x}}_{ns}=\textbf{f}_{n}(\textbf{x}_{ns})+\textbf{g}_{n}(\textbf{ x}_{ns})u(t)+\textbf{d}_{n}(t) \tag{3}\]  
In Equation (3), the functions \(\textbf{f}_{n}(\textbf{x}_{ns})\in R^{m}\) and \(\textbf{g}_{n}(\textbf{x}_{ns})\in R^{m}\) are two unknown vectors; \(u(t)\in R^{1}\) is the input control; \(\textbf{d}_{n}(t)\in R^{m}\) is the disturbance (internal and external disturbances); and \(\left|\textbf{d}_{n}(t)\right|\leq\delta\textbf{d}_{n}\), where \(\delta\textbf{d}_{n}\in R^{m}\) is boundary of \(\textbf{d}_{n}(t)\) and \(\textbf{x}_{ns}=\left[x_{n1},x_{n2},\ldots,x_{nm}\right]=\left[x_{n1},\dot{x}_ {n1},\ldots,x_{n1}{}^{(m-1)}\right]^{T}\in R^{m}\) is the state variables. The system in Equation (3) can be rewritten as follows. \[\dot{\textbf{x}}_{ns}=\textbf{f}_{n0}(\textbf{x}_{ns})+\textbf{g}_{n0}(\textbf {x}_{ns})u(t)+\textbf{D}_{n}(t) \tag{4}\]  
where \(\textbf{D}_{n}(t)=\delta\textbf{f}_{n}+\delta\textbf{g}_{n}u(t)+\textbf{d}_{ n}(t)\) denotes the disturbances given by  
\[\textbf{D}_{n}=\left[0,0,\ldots,D_{n0}\right]^{T}\text{, }\textbf{f}_{n0}(\textbf{x}_{ns})=\left[x_{n2},\ldots,x_{nm},f_{n0}\right]^{T} \text{, }\textbf{g}_{n0}(\textbf{x}_{ns})=\left[0,\ldots,0,g_{n0}\right]^{T}\text{,}\] \[\delta\textbf{f}_{n}=\left[0,0,\ldots,\delta f_{n0}\right]^{T} \text{, }\delta\textbf{g}_{n}=\left[0,0,\ldots,\delta g_{n0}\right]^{T}\]  
Based on Equation (2), the relationship between the system in Equation (4) and IT2 FNN is determined as follows:  
\[\textbf{f}_{n00}(\textbf{x}_{ns})=\textbf{f}_{n0}(\textbf{x}_{ns})g_{f}= \theta_{nf}\xi_{nf} \tag{5}\]  
\[\textbf{g}_{n00}(\textbf{x}_{ns})=\textbf{g}_{n0}(\textbf{x}_{ns})g_{nf}= \theta_{ng}\xi_{ng} \tag{6}\]  
where \(\Theta_{nf}=\left[\Theta_{nf}^{l},\Theta_{nf}^{u}\right]^{T}\), \(\Theta_{g}=\left[\Theta_{g}^{l},\Theta_{g}^{u}\right]^{T}\) and \(\xi_{nf}=\xi_{ng}=\left[\xi_{nl},\xi_{nu}\right]\). Note that \(\Theta_{nf},\Theta_{ng}\) are the centroid of consequent vectors. \(\xi_{nf}\) and \(\xi_{ng}\) are consequent membership vectors of \(\textbf{f}_{n}\), \(\textbf{g}_{n}\), respectively. The sliding surface \(s_{\Delta}\) of sliding mode control is defined as follows. \[s_{\Delta}=k_{1}x_{n1}+k_{2}x_{n2}+k_{3}x_{n3}+\ldots+k_{m}x_{m}=\sum_{i=1}^{m}k _{i}x_{i} \tag{7}\]where \(\mathbf{K}_{m}=[k_{m},k_{m-1},k_{m-2},\ldots,k_{1}]\). It is noted that \(k_{m}=1\) in this analysis. The sliding surface in Equation (7) is rewritten as follows:  
\[x_{nm}=-k_{1}x_{n1}-k_{2}x_{n2}-k_{3}x_{n3}-\ldots-k_{m-1}x_{nm-1}+s_{\Delta} \tag{8}\]  
A new vector \(\mathbf{\widetilde{x}}_{s}\) is defined by \(\mathbf{\widetilde{x}}_{ns}=[x_{n1}\ x_{n2}\ x_{n3}\ \ldots\ x_{nm-1}]^{T}\), and thus the system in Equation (4) is rewritten as:  
\[\dot{\mathbf{\widetilde{x}}}_{ns}=\mathbf{S}_{n1}\mathbf{\widetilde{x}}_{ns}+ \mathbf{S}_{n2}^{T}\mathbf{s}_{\Delta} \tag{9}\]  
where  
\[\mathbf{S}_{n1}=\left[ \begin{array}{cccc}0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\.&.&.&\ldots&.\\ -k_{1}&-k_{2}&-k_{3}&\ldots&-k_{m-1}\end{array} \right]\ \mathbf{S}_{n2}=\left[ \begin{array}{c}0\\ 0\\.\\ 1\end{array} \right]\]  
The derivative of Equation (7) is obtained as follows. \[\dot{s}_{\Delta}=k_{1}\dot{x}_{n1}+k_{2}\dot{x}_{n2}+k_{3}\dot{x}_{n3}+\ldots+ k_{m-1}\dot{x}_{nm-1}+f_{n0}(x_{ns})+g_{s0}(x_{ns})u(t)+D_{n}(t) \tag{10}\]  
The lumped uncertainty is expressed by  
\[w_{nl}=\widetilde{\gamma}_{nf}\xi_{nf}+\widetilde{\gamma}_{ng}\xi_{ng}u+D_{n} \tag{11}\]  
where \(\widetilde{\gamma}_{nf}=f_{n0}(x_{ns})-f_{n00}(x_{ns})\) and \(\widetilde{\gamma}_{ng}=g_{n0}(x_{ns})-g_{n00}^{*}(x_{ns})\). Substituting Equation (11) into Equation (10) yields the following equation.