# A Novel Adaptive PID Controller with Application to Vibration Control of a Semi-Active Vehicle
## 2 New Adaptive Hybrid Controller
### Adaptive Hybrid Control  
In this study, the \(n\)th-order nonlinear system is used to define a new proposed controller. Its equation is expressed by  
\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})+\mathbf{g}(\mathbf{x})u(t)+\mathbf{d}(t) \tag{3}\]  
The above equation strictly follows the assumption given below. **Assumption 1**. : _In Equation (3), the functions \(f(\mathbf{x})\in R^{n}\) and \(\mathbf{g}(\mathbf{x})\in R^{n}\) are two unknown non-linear function vectors, \(u(t)\in R^{1}\) is the control function, \(\mathbf{d}(t)\in R^{n}\) is the external disturbance vector, \(|\mathbf{d}(t)|\leq\delta\mathbf{d}\) where \(\delta\mathbf{d}\in R^{n}\) is the upper bound of \(\mathbf{d}(t)\), and \(\mathbf{x}=\left[x_{1},\ x_{2},\ \ldots,\ x_{n}\right]=\left[x_{1},\ x_{1},\ \ldots,\ x_{1} {}^{(n-1)}\right]^{T}\in R^{n}\) is the state vector of the system._  
The system (3) can be rewritten as  
\[\dot{\mathbf{x}}=\mathbf{f_{0}}(\mathbf{x})+\mathbf{g_{0}}(\mathbf{x})u(t)+\mathbf{D}(t) \tag{4}\]  
where, \(\mathbf{D}(t)=\delta\mathbf{f}+\delta\mathbf{g}u(t)+\mathbf{d}(t)\) denotes the uncertain disturbances which can be redefined by \(\mathbf{D}=\left[0,0,\ldots,D_{0}\right]^{T}\), \(\mathbf{f_{0}}(\mathbf{x})=\left[x_{2},\ldots,x_{n},f_{0}\right]^{T}\), \(\delta\mathbf{g}=\left[0,0,\ldots,\delta g_{0}\right]^{T}\), \(\delta\mathbf{f}=\left[0,0,\ldots,\delta g_{0}\right]^{T}\). It is noted that \(\delta\mathbf{f}\) and \(\delta\mathbf{g}\) are two positive vectors. Based on Equation (2), the relationship between system (4) and OIT2 FNN is determined as  
\[\mathbf{f_{00}}(\mathbf{x})=\mathbf{f_{0}}(\mathbf{x})g_{f}=\mathbf{\theta_{f}}\mathbf{ \xi_{f}} \tag{5}\] \[\mathbf{g_{00}}(\mathbf{x})=\mathbf{g_{0}}(\mathbf{x})g_{f}=\mathbf{\theta_{g}}\mathbf{ \xi_{g}} \tag{6}\]  
In the above, \(\mathbf{\theta_{f}}=\left[\mathbf{\theta_{f}^{l}},\mathbf{\theta_{f}^{u}}\right]^{T}\), \(\mathbf{\theta_{g}}=\left[\mathbf{\theta_{g}^{l}},\mathbf{\theta_{g}^{u}}\right]^{T}\) and \(\mathbf{\xi_{f}}=\mathbf{\xi_{g}}=\left[\mathbf{\xi_{l}},\mathbf{\xi_{u}}\right]\). Note that \(\mathbf{\theta_{f}},\mathbf{\theta_{g}}\) are the centroid of consequent vectors. \(\mathbf{\xi_{f}}\), \(\mathbf{\xi_{g}}\) are consequent membership vectors of \(\mathbf{f}\), \(\mathbf{g}\), respectively. The sliding surface \(s_{s}\) of the sliding mode control is defined as  
\[s_{s}=k_{1}x_{1}+k_{2}x_{2}+k_{3}x_{3}+\ldots+k_{n}x_{n}=\sum\limits_{i=1}^{n}k _{i}x_{i} \tag{7}\]  
where, \(\mathbf{K}=\left[k_{n},k_{n-11},k_{n-2},\ldots,k_{1}\right]\)is defined as the coefficients such that all of the roots of the polynomial \(\sigma^{n}+k_{n-1}\sigma^{n-1}+k_{n-2}\sigma^{n-2}+\ldots+k_{1}\) are in the open left-half complex plane. It is noted that \(k_{n}=1\) in this analysis. The sliding surface in Equation (7) is rewritten using the state variables as  
\[x_{n}=-k_{1}x_{1}-k_{2}x_{2}-k_{3}x_{3}-\ldots-k_{n-1}x_{n-1}+s_{s} \tag{8}\]  
A new vector \(\widetilde{\mathbf{x}}\) is defined by \(\widetilde{\mathbf{x}}=\left[x_{1}\ x_{2}\ x_{3}\ \ldots\ x_{n-1}\right]^{T}\), and thus the system (4) is rewritten as  
\[\dot{\widetilde{\mathbf{x}}}=\mathbf{S_{1}}\widetilde{\mathbf{x}}+\mathbf{S_{2}}^{T}s_{s_{s}} \tag{9}\]  
where,  
\[\mathbf{S_{1}}=\left[ \begin{array}{cccc}0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \cdot&\cdot&\cdot&\ldots&\cdot\\ -k_{1}&-k_{2}&-k_{3}&\ldots&-k_{n-1}\end{array} \right],\ \mathbf{S_{2}}=\left[ \begin{array}{c}0\\ 0\\ \cdot\\ 1\end{array} \right]\]The derivative of Equation (7) yields the equation  
\[\dot{s}_{s}=k_{1}\dot{x}_{1}+k_{2}\dot{x}_{2}+k_{3}\dot{x}_{3}+\ldots+k_{n-1} \dot{x}_{n-1}+f_{0}(x)+g_{0}(x)u(t)+D(t) \tag{10}\]  
To simplify the control, the lumped uncertainty is defined by  
\[w_{l}=\widetilde{\gamma}_{f}\xi_{f}+\widetilde{\gamma}_{g}\xi_{g}u+D \tag{11}\]  
where, \(\widetilde{\gamma}_{f}=f_{0}(x)-f_{00}^{*}(x)\), \(\widetilde{\gamma}_{g}=g_{0}(x)-g_{00}^{*}(x)\).