# A Novel Adaptive PID Controller with Application to Vibration Control of a Semi-Active Vehicle
## 2 New Adaptive Hybrid Controller
### Adaptive Hybrid Control
Substituting Equation (11) into Equation (10) yields the equation  
\[\dot{s}_{s}=\sum_{i=1}^{n-1}k_{i}x_{i+1}+f_{00}^{*}(x)+g_{00}^{*}(x)u+w_{l} \tag{12}\]  
Then, Equation (12) can be written by substituting values of OIT2 FNN as  
\[\dot{s}_{s}=\sum_{i=1}^{n-1}k_{i}x_{i+1}+\theta_{f}^{*}\xi_{f}+\theta_{g}^{*} \xi_{g}u+w_{l} \tag{13}\]  
where, \(\theta_{f}^{*}=\operatorname*{argmin}_{\theta_{f}\in\Delta\theta_{f}}\bigl{[} \sup_{x\in\Delta x}\bigl{|}f(x)-f_{00}(x)\bigr{|}\bigr{]}\), \(\theta_{g}^{*}=\operatorname*{argmin}_{\theta_{g}\in\Delta\theta_{g}}\bigl{[} \sup_{x\in\Delta x}\bigl{|}g(x)-g_{00}(x)\bigr{|}\bigr{]}\), \(\Delta\theta_{f}=\Bigl{\{}\theta_{f}\in R^{n},\;\|\theta_{f}\|\leq\varepsilon_ {f}\Bigr{\}}\), \(\Delta g=\bigl{\{}\theta_{g}\in R^{n},\;\|\theta_{g}\|\leq\varepsilon_{g}\bigr{\}}\), \(\Delta x=\{x\in R^{n},\;\|x\|\leq\varepsilon_{x}\}\). In the above, \(\varepsilon_{f}\), \(\varepsilon_{g}\), \(\varepsilon_{x}\) are constant boundaries. An equivalent control of the system is derived from Equation (13) based on assumption of \(w_{l}\approx 0\)  
\[u_{1}=-\frac{1}{\bar{\theta}_{g}\xi_{g}}\biggl{(}\sum_{i=1}^{n-1}k_{i}x_{i+1} +\hat{\theta}_{f}\xi_{f}\biggr{)} \tag{14}\]  
The equivalent control \(u_{1}\) cannot control the system because it cannot compensate the error from the fuzzy approximation. To guarantee the robustness and stability in control, a robust control part \(u_{2}\) should be added as  
\[u_{2}=\frac{1}{\bar{\theta}_{g}\xi_{g}}\biggl{(}-\sum_{i=1}^{n-1}P_{(n-1)i}x_ {i}-\frac{s_{s}}{\beta}+\frac{1}{2}\Gamma s_{s}\xi_{fz}\widetilde{x}\mathbf{P} \mathbf{S}_{2}\mathbf{S}_{2}^{T}\mathbf{P}\widetilde{x}^{T}+K_{p}{s_{s}}^{2}e+K_{l}{s_{s}} ^{2}\int edt+K_{D}{s_{s}}^{2}\frac{de}{dt}\biggr{)} \tag{15}\]  
where, \(e=x_{d}-x\) is tracking error and \(x_{d}\) is the desired value. The control \(u_{2}\) is the combination of the sliding surface of \(s_{s}\), H-infinity controller and PID controller. The value \(\Gamma\) is the adaptive parameter where its boundary is given by \(\Delta\Gamma=\Bigl{\{}\Gamma\in R,\;\|\Gamma\|\leq\varepsilon_{\Gamma},\; \Gamma_{\xi fz}\leq\rho\Bigr{\}}\), and \(\varepsilon_{\Gamma}\) is constant boundary. The matrix \(\mathbf{P}=\mathbf{P}^{T}\geq 0\) in which its result is a solution of Riccati-like equation given by  
\[\mathbf{P}\mathbf{S}_{1}+\mathbf{S}_{1}^{T}\mathbf{P}+\mathbf{Q}-s_{s}\Gamma_{\xi fz}^{*}P\mathbf{S}_{ 2}\mathbf{S}_{2}^{T}\mathbf{P}+\rho\mathbf{P}\mathbf{S}_{2}\mathbf{S}_{2}^{T}\mathbf{P}=0 \tag{16}\]  
Besides, the boundaries of \(K_{P}\), \(K_{I}\), and \(K_{D}\) are determined as \(\Delta K_{P}=\{K_{P}\in R,\;\|K_{P}\|\leq\varepsilon_{KP}\}\), \(\Delta K_{I}=\{K_{I}\in R,\;\|K_{I}\|\leq\varepsilon_{KI}\}\), \(\Delta K_{D}=\{K_{D}\in R,\;\|K_{D}\|\leq\varepsilon_{KD}\}\). In these boundaries, \(\varepsilon_{KP}\), \(\varepsilon_{KI}\) and \(\varepsilon_{KD}\) are constant. Finally, the control \(u\) of the system is determined as  
\[u=u_{1}+u_{2} \tag{17}\]  
**Remark 1**. : _To evaluate the stability and robustness of the system, Lyapunov function of the system (3) is suggested as_  
\[V=\frac{1}{2}{s_{s}}^{2}+\frac{1}{2}\widetilde{x}\mathbf{P}\widetilde{x}^{T}+ \frac{1}{2\eta_{1}}\widetilde{\gamma}_{f}^{2}+\frac{1}{2\eta_{2}}\widetilde{ \gamma}_{g}^{2}+\frac{1}{2\eta_{4}}K_{P}^{2}+\frac{1}{2\eta_{5}}K_{I}^{2}+ \frac{1}{2\eta_{6}}K_{D}^{2} \tag{18}\]Based on the evaluation, six adaptation laws are found as  
\[\dot{\widetilde{\gamma}}_{f}=-\eta_{1}s_{\mathrm{s}}\xi_{p} \tag{19}\]  
\[w_{l}=\widetilde{\gamma}_{f}\xi_{f}+\widetilde{\gamma}_{g}\xi_{g}u+D \tag{11}\]0  
\[w_{l}=\widetilde{\gamma}_{f}\xi_{f}+\widetilde{\gamma}_{g}\xi_{g}u+D \tag{11}\]1  
\[w_{l}=\widetilde{\gamma}_{f}\xi_{f}+\widetilde{\gamma}_{g}\xi_{g}u+D \tag{11}\]2  
\[w_{l}=\widetilde{\gamma}_{f}\xi_{f}+\widetilde{\gamma}_{g}\xi_{g}u+D \tag{11}\]3  
\[w_{l}=\widetilde{\gamma}_{f}\xi_{f}+\widetilde{\gamma}_{g}\xi_{g}u+D \tag{11}\]4  
Now, the adaptation laws of the proposed hybrid adaptive controller can be summarized in the following theorem. **Theorem 1**.