# Time-Specified Adaptive Robust Control Framework for Managing Nonlinear System Uncertainties
## 3 Method  
This research paper upgrades the controller to bring about higher efficiency based on the controller proposed in reference [44]. Consider a nonlinear system with unknown parameters represented as  
\[\dot{x}(t)=f(t,x(t),u(t),\theta) \tag{4}\]  
In this process, we aim to design an adaptive prescribed-time controller for a specified time \(T_{p}\), breaking the design into two cases: \(0\leq t<T_{p}\) and \(t\geq T_{p}\). We start by introducing a state transformation:  
\[z_{1}=x_{1},z_{i}=x_{i}-\alpha_{i-1},i=2,\ldots,n \tag{5}\]  
where \(\alpha_{i-1}\) is the virtual controller to be designed later. Before proceeding, we select the Lyapunov function as follows:  
\[V=V_{1}+V_{2} \tag{6}\]  
where \(V_{1}\) and \(V_{2}\) are defined as  
\[V_{1}=\sum_{j=1}^{n}z_{j}^{2},V_{2}=\widetilde{\theta}^{T}\Gamma^{-1} \widetilde{\theta} \tag{7}\]  
with \(\widetilde{\theta}=\theta-\widehat{\theta},\widehat{\theta}\) being the estimated value of \(\theta\), and \(\Gamma\) a positive definite matrix. Using the properties of the Lyapunov function, it can be shown that \(V,V_{1}\), and \(V_{2}\) satisfy the conditions outlined in above. The controller is developed using the backstepping approach, consisting of \(n\) steps. For the derivative of \(z_{1}\), we have the following:  
\[\dot{z}_{1}=z_{2}+\alpha_{1}+\theta^{T}f_{1}\]  
Thus, the virtual controller \(\alpha_{1}\) is designed as follows:  
\[\alpha_{1}=-\sigma_{1}\frac{z_{1}}{T_{p}-t}-\widehat{\theta}^{T}f_{1} \tag{8}\]  
where \(\sigma_{1}>n\) is a design parameter. Then, the derivatives of \(z_{1}\) and \(W_{1}=z_{1}^{2}+V_{2}\) are given by the following:  
\[\dot{z}_{1}=-\sigma_{1}\frac{z_{1}}{T_{p}-t}+\Omega_{1} \tag{9}\]  
\[\dot{W}_{1}=-2\sigma_{1}\frac{z_{1}^{2}}{T_{p}-t}+2z_{1}z_{2}+2\widetilde{ \theta}^{T}\bigg{(}\tau_{1}-\Gamma^{-1}\dot{\widetilde{\theta}}\bigg{)} \tag{10}\]  
\[\dot{\widehat{\theta}}=\Gamma\tau_{n} \tag{21}\]  
where \(\tau_{n}=\tau_{n-1}+z_{n}\Big{(}f_{n}-\sum_{j=1}^{n-1}\frac{\partial\alpha_{n-1} }{\partial x_{j}}f_{j}\Big{)},\sigma_{n}>1\) is a design parameter, and  
\[\Phi=-\sigma_{n}\frac{z_{n}}{T_{p}-t}-z_{n-1}+\frac{\partial\alpha_{n-1}}{ \partial t}+\frac{\partial\alpha_{n-1}}{\partial\widehat{\theta}}\Gamma\tau_{ n}+\sum_{j=1}^{n-1}\frac{\partial\alpha_{n-1}}{\partial x_{j}}\Big{(}x_{j+1}+ \widehat{\theta}^{T}f_{j}\Big{)}+\left(f_{n}-\sum_{j=1}^{n-1}\frac{\partial \alpha_{n-1}}{\partial x_{j}}f_{j}\right)\Gamma \tag{22}\]  
Finally, we have the below:  
\[z_{1}=x_{1},z_{i}=x_{i}-\alpha_{i-1},i=2,\ldots,n \tag{5}\]0  
\[z_{1}=x_{1},z_{i}=x_{i}-\alpha_{i-1},i=2,\ldots,n \tag{5}\]1  
where \(\Omega_{n}\) is a complex expression, and \(\mu(t)=\sigma_{1}\frac{1}{T_{p}-t}\) is a prescribed-time function. For \(t\geq T_{p}\), the control signal and adaptive law are set to  
\[z_{1}=x_{1},z_{i}=x_{i}-\alpha_{i-1},i=2,\ldots,n \tag{5}\]2  
In cases with large or rapidly varying uncertainties, a robust term is added to the original controller. The standard procedure for defining \(z_{1},\alpha_{1},z_{2}\), and \(\Phi_{1}\) remains unchanged, but the final control law includes a supplementary term:  
\[z_{1}=x_{1},z_{i}=x_{i}-\alpha_{i-1},i=2,\ldots,n \tag{5}\]3  
where \(k_{r}\) is the robust gain, \(\epsilon\) is the boundary layer thickness, \(\text{sat}(s)\) is the saturation function, and \(\text{tanh}(s)\) smooths transitions near the boundary manifold. This results in the updated control law:  
\[z_{1}=x_{1},z_{i}=x_{i}-\alpha_{i-1},i=2,\ldots,n \tag{5}\]4  
This improves the system's robustness to external disturbances and modeling inaccuracies. The adaptive law for \(\widehat{\theta}\) continues to estimate uncertain parameters, preserving the backstepping structure while reducing chattering and extreme control signals. A similar approach is applied in the following section, where a robust block is added after finalizing the nominal design. The backstepping errors and virtual controls \(\alpha_{i}\) remain, and the final control action incorporates a robust component using saturation and \(\text{tanh}\).