# A Novel Adaptive PID Controller with Application to Vibration Control of a Semi-Active Vehicle
## 2 New Adaptive Hybrid Controller
### Adaptive Hybrid Control
: _The adjusted adaptation laws of the proposed hybrid controller given by Equations (19)-(24) are modified to satisfy Lyapunov stability as_  
\[\dot{\widetilde{\gamma}}_{f}=\left\{ \begin{array}{rl}-\eta_{1}s_{\mathrm{s}} \xi_{f}&if\parallel\widetilde{\gamma}_{t}\parallel<\varepsilon_{f}\text{ or }\left(\parallel\widetilde{\gamma}_{f}\parallel=\varepsilon_{f}\text{ and }s_{\mathrm{s}}\xi_{f}\widetilde{\gamma}_{f}\geq 0\right)\\ -\eta_{1}s_{\mathrm{s}}\xi_{f}+\eta_{1}\frac{\left(\parallel\widetilde{\gamma }_{f}\parallel^{2}-\varepsilon_{f}\right)s_{\mathrm{s}}\xi_{f}\widetilde{ \gamma}_{f}}{\delta_{1}\parallel\widetilde{\gamma}_{f}\parallel^{2}}&if \parallel\widetilde{\gamma}_{f}\parallel=\varepsilon_{f}\text{ and }s_{ \mathrm{s}}\xi_{f}\widetilde{\gamma}_{f}<0\end{array} \right. \tag{25}\]  
\[\dot{\widetilde{\gamma}}_{g}=\left\{ \begin{array}{rl}-\eta_{2}s_{\mathrm{s}} \xi_{g}u_{1}&if\parallel\widetilde{\gamma}_{g}\parallel<\varepsilon_{g}\text{ or }\left(\parallel\widetilde{\gamma}_{g}\parallel=\varepsilon_{g}\text{ and }s_{\mathrm{s}}\xi_{g}u_{1}\widetilde{\gamma}_{g}\geq 0\right)\\ -\eta_{2}s_{\mathrm{s}}\xi_{g}u_{1}+\eta_{2}\frac{\left(\parallel\widetilde{ \gamma}_{g}\parallel^{2}-\varepsilon_{g}\right)s_{\mathrm{s}}\xi_{g}u_{1} \widetilde{\gamma}_{g}}{\delta_{2}\parallel\widetilde{\gamma}_{g}\parallel^{2 }}&if\parallel\widetilde{\gamma}_{g}\parallel=\varepsilon_{g}\text{ and }s_{\mathrm{s}}\xi_{g}u_{1}\widetilde{\gamma}_{g}<0\end{array} \right. \tag{26}\]  
\[\dot{\Gamma}=\left\{ \begin{array}{rl}-\eta_{3}s_{\mathrm{s}}\xi_{fz} \widetilde{\mathbf{x}}P\mathbf{S}_{2}\mathbf{S}_{2}^{T}\mathbf{P}\widetilde{ \mathbf{x}}^{T}&if\parallel\Gamma\parallel<\varepsilon_{\Gamma}\text{ or }\\ -\eta_{4}s_{\mathrm{s}}^{2}\xi_{fz}+\eta_{4}\frac{\left(\parallel K_{P} \parallel^{2}-\varepsilon_{F}\right)s_{\mathrm{s}}^{2}\xi_{fz}\mathbf{X}_{P} }{\delta_{4}\parallel K_{P}\parallel^{2}}&if\parallel K_{P}\parallel= \varepsilon_{F}\text{ and }s_{\mathrm{s}}^{2}\xi_{fz}K_{P}<0\end{array} \right. \tag{27}\]  
\[\dot{K}_{I}=\left\{ \begin{array}{rl}-\eta_{5}s_{\mathrm{s}}^{2}\xi_{fz} \int edt&if\parallel K_{I}\parallel<\varepsilon_{KI}\text{ or }\left(\parallel K_{I}\parallel=\varepsilon_{KI}\text{ and }s_{\mathrm{s}}^{2}\xi_{fz}\int edtK_{I}\geq 0\right)\\ -\eta_{5}s_{\mathrm{s}}^{2}\xi_{fz}\int edt+\eta_{5}\frac{\left(\parallel K_{I} \parallel^{2}-\varepsilon_{KI}\right)s_{\mathrm{s}}^{2}\xi_{fz}\int edtK_{I} }{\delta_{5}\parallel K_{I}\parallel^{2}}&if\parallel K_{I}\parallel= \varepsilon_{KI}\text{ and }s_{\mathrm{s}}^{2}\xi_{fz}\int edtK_{I}<0\end{array} \right. \tag{29}\]  
\[\dot{K}_{D}=\left\{ \begin{array}{rl}-\eta_{6}s_{\mathrm{s}}^{2}\xi_{fz}\frac{de }{dt}&if\parallel K_{D}\parallel<\varepsilon_{KD}\text{ or }\left(\parallel K_{D}\parallel=\varepsilon_{KD}\text{ and }s_{\mathrm{s}}^{2}\xi_{fz}\frac{de}{dt}K_{D}\geq 0\right)\\ -\eta_{6}s_{\mathrm{s}}^{2}\xi_{fz}\frac{de}{dt}+\eta_{6}\frac{\left(\parallel K _{D}\parallel^{2}-\varepsilon_{KD}\right)s_{\mathrm{s}}^{2}\xi_{fz}\frac{de}{dt} K_{D}}{\delta_{6}\parallel K_{D}\parallel^{2}}&if\parallel K_{D} \parallel=\varepsilon_{KD}\text{ and }s_{\mathrm{s}}^{2}\xi_{fz}\frac{de}{dt}K_{D}<0\end{array} \right. \tag{30}\]  
where, \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\), \(\delta_{4}\), \(\delta_{5}\), \(\delta_{6}\) are choosing parameters related boundaries of \(f(x)\), \(g(x)\), \(\Gamma\), \(K_{P}\), \(K_{I}\), and \(K_{D}\)  
the Luenberger observer [31] is used in this work. The closed-loop block-diagram of the proposed control system is shown in Figure 1. In the first process of control, the granular clustering module will find the feature of the system based on the standard data. The data includes two main parameters which are correlative with the desired fuzzy models. Hence, the results of the first process are the input of OIT2 FNN module to find fuzzified value for calculating both adaptation laws and adaptive hybrid controller.