## 2 New Adaptive Fuzzy PID Controller
### Adaptive Fuzzy PID Control
\[\dot{s}_{\Delta}=\sum\limits_{i=1}^{m-1}k_{i}x_{ni+1}+f_{n00}^{*}(x_{ns})+g_{ n00}^{*}(x_{ns})u+w_{nl} \tag{12}\]  
Then, Equation (12) can be written by substituting values of the fuzzy mode as  
\[\dot{s}_{\Delta}=\sum\limits_{i=1}^{m-1}k_{i}x_{ni+1}+\theta_{nf}^{*}\xi_{nf} +\theta_{ng}^{*}\xi_{ng}u+w_{nl} \tag{13}\]  
where  
\[\theta_{nf}^{*}=\text{argmin}_{\theta_{ng}\in\Delta\theta_{ng}} \Big{[}\sup_{x_{ns}\in\Delta x_{n}}\big{|}f_{n}(x_{ns})-f_{n00}(x_{ns})\big{|} \Big{|},\] \[\theta_{ng}^{*}=\text{argmin}_{\theta_{ng}\in\Delta\theta_{ng}} \Big{[}\sup_{x_{ns}\in\Delta x_{ns}}\big{|}g_{n}(x_{ns})-g_{n00}(x_{ns})\big{|} \Big{|},\ \Delta\theta_{nf}=\Big{\{}\theta_{nf}\in R^{m},\|\theta_{nf}\|\leq \varepsilon_{nf}\Big{\}},\] \[\Delta\theta_{ng}=\Big{\{}\theta_{ng}\in R^{m},\|\theta_{ng}\|\leq \varepsilon_{ng}\Big{\}},\ \Delta x_{n}=[x_{ns}\in R^{m},\|x_{ns}\|\leq \varepsilon_{nx}]\]  
In the above equations, \(\varepsilon_{nf}\,\varepsilon_{ng},\ \varepsilon_{nx}\) are constant boundaries. An equivalent control of the system is derived from Equation (13) based on assumption of \(w_{nl}\approx 0\):  
\[u_{n1}=-\frac{1}{\hat{\theta}_{ng}\xi_{ng}}\bigg{(}\sum\limits_{i=1}^{m-1}k_{i} x_{ni+1}+\hat{\theta}_{nf}\xi_{nf}\bigg{)} \tag{14}\]  
To guarantee the robustness and stability in control, a new robust control \(u_{n2}\) is suggested as follows:  
\[u_{n2}=\frac{1}{\hat{\theta}_{ng}\xi_{ng}}\Bigg{(} \begin{array}{c}-\sum \limits_{i=1}^{m-1}P_{(m-1)}x_{ni}-\frac{s_{\Delta}}{\beta}+\frac{1}{2}\Gamma_{n }s_{\Delta}\xi_{f}z_{\mathbf{\widetilde{x}}_{ns}}\mathbf{P}_{n}\mathbf{S}_{n2} \mathbf{S}_{n2}^{T}\mathbf{P}_{n}\mathbf{\widetilde{x}}_{ns}{}^{T}+K_{p} \mathbf{S}_{2}^{T}\mathbf{E}_{n}\mathbf{E}_{n}^{T}\mathbf{P}_{n}\mathbf{B}_{p}+ \\ +K_{f}\int\mathbf{S}_{2}^{T}\mathbf{E}_{n}\mathbf{E}_{n}^{T}\mathbf{P}_{n} \mathbf{B}_{p}dt+K_{D}\mathbf{S}_{2}^{T}\mathbf{E}_{n}\mathbf{E}_{n}^{T}\mathbf{ P}_{n}\mathbf{B}_{p}\end{array} \Bigg{)} \tag{15}\]where \(e_{n}=x_{nd}-x_{ns}\), \(\mathbf{E}_{n}=[\mathrm{e}_{n1},\mathrm{e}_{n2},\ldots,\mathrm{e}_{nm}]\) is tracking error and \(x_{d}\) is the desired value. The value of \(\mathbf{B}_{P}\) is determined as follows:  
\[\dot{\mathbf{x}}_{ns}=\mathbf{f}_{n0}(\mathbf{x}_{ns})+\mathbf{g}_{n0}( \mathbf{x}_{ns})u(t)+\mathbf{D}_{n}(t)=\mathbf{A}_{P}(x_{ns}(t),t)\mathbf{x}_{ ns}(t)+\mathbf{B}_{P}(x_{ns}(t),t)u(t)+\mathbf{D}_{n}(t) \tag{16}\]  
where \(\mathbf{A}_{P}\in R^{m\times m},\mathbf{B}_{P}\in R^{m\times m},u(t)\in R^{v}(v =1)\). The value \(\Gamma_{n}\) is the adaptive parameter related the modified Riccati-like equation, and its boundary is given by \(\Delta\Gamma_{n}=\left\{\Gamma_{n}\in R,\left\|\Gamma_{n}\right\|\leq\varepsilon _{n\Gamma},\Gamma_{n}\xi_{fz}\leq\rho\right\}\); \(\varepsilon_{n\Gamma}\) is constant boundary. The matrix \(\mathbf{P}_{n}=\mathbf{P}_{n}^{T}\geq 0\) in which its result is a solution of a new modified Riccati-like equation given by  
\[ \begin{array}{c}\mathbf{P}_{n}\mathbf{S}_{n1}+\mathbf{S}_{n1}^{\mathrm{T}} \mathbf{P}_{n}+\mathbf{Q}_{n}-s_{\Delta}\Gamma_{n}\xi_{f2}\mathbf{P}_{n} \mathbf{S}_{n2}\mathbf{S}_{n2}^{\mathrm{T}}\mathbf{P}_{n}+\rho\mathbf{P}_{n} \mathbf{S}_{n2}\mathbf{S}_{n2}^{\mathrm{T}}\mathbf{P}-K_{P}\xi_{fz}\mathbf{S }_{n2}^{\mathrm{T}}\mathbf{E}_{n}\mathbf{E}_{n}^{\mathrm{T}}\mathbf{P}_{n} \mathbf{B}_{P}-\\ -K_{I}\xi_{fz}\int\mathbf{S}_{n2}^{\mathrm{T}}\mathbf{E}_{n}\mathbf{E}_{n}^{ \mathrm{T}}\mathbf{P}_{n}\mathbf{B}_{P}dt-K_{D}\xi_{fz}\mathbf{S}_{n2}^{ \mathrm{T}}\dot{\mathbf{E}}_{n}\mathbf{E}_{n}^{\mathrm{T}}\mathbf{P}_{n} \mathbf{B}_{P}=0\end{array}  \tag{17}\]  
where \(\left\{s_{\Delta}\Gamma_{n}\xi_{fz},K_{P}\xi_{fz},K_{I}\xi_{fz},K_{D}\xi_{fz} \right\}\leq\rho\). The boundaries of \(K_{P}\), \(K_{I}\), and \(K_{D}\) are determined as \(\Delta K_{P}=\left\{K_{P}\in R,\left\|K_{P}\right\|\leq\varepsilon_{nK\Gamma} \right\}\), \(\Delta K_{I}=\left\{K_{I}\in R,\left\|K_{I}\right\|\leq\varepsilon_{nK\Gamma} \right\}\), and \(\Delta K_{D}=\left\{K_{D}\in R,\left\|K_{D}\right\|\leq\varepsilon_{nK\Gamma} \right\}\), respectively.