# Time-Specified Adaptive Robust Control Framework for Managing Nonlinear System Uncertainties
## 3 Method
For unmodeled uncertainties, a robust component is added:  
\[z_{1}=x_{1},z_{i}=x_{i}-\alpha_{i-1},i=2,\ldots,n \tag{5}\]5  
where  
\[\text{sat}(s)= \begin{cases}\text{sign}(s),&|s|>1\\ s,&|s|\leq 1\end{cases} \]  
\(\epsilon\) is a parameter to avoid excessive chattering, \(k_{r}\) is the robust gain, and \(\text{tanh}(s)\) smooths the control signal near the sliding manifold boundary. Thus, the enhanced control law becomes  
\[u=-\frac{\sigma_{i}z_{i}}{T_{p}-t}+\phi_{i-1}-k_{r}\Big{[}0.5\text{sat}\Big{(} \frac{z_{i}}{\epsilon}\Big{)}+0.5\text{tanh}(10z_{i})\Big{]} \tag{29}\]  
The relationship between parameter selection and control performance is crucial in prescribed-time control. The design parameters \(\sigma_{i}\ (i=1,\ 2,\ \ldots,n)\) directly influence the convergence rate and robustness. Higher values of \(\sigma_{i}\) accelerate convergence but may lead to increased control effort and potential chattering. The prescribed time \(T_{\text{P}}\) must be selected according to physical system constraints and desired performance metrics. Meanwhile, the robust gain \(k_{\text{r}}\) and boundary layer thickness \(\varepsilon\) in Equation (28) control the trade-off between robustness to uncertainties and the smoothness of the control signal. Smaller values of \(\varepsilon\) provide tighter tracking but may increase chattering, while larger values smoothen the control signal at the cost of tracking precision. Our simulations demonstrate that setting \(\sigma_{\text{i}}>n-i+1\), with moderate robust gain \(k_{\text{r}}\), provides an optimal balance between convergence speed and control effort. This ensures backstepping-style stability while handling unmodeled uncertainties and disturbances. The novelty in this upgrade is the addition of a robust term into the control law, which helps the system maintain high stability and more accurate control when facing unmodeled uncertainties or large disturbances. This robust term is built from saturation and tanh functions, which smooth the control signal, minimize large oscillations, and reduce chattering. This helps limit unwanted oscillations, creates more stable control signals, and improves the system's ability to withstand sudden changes in the operating environment. This upgrade is based on Adaptive Control theory and Lyapunov theory to ensure that the control signals and adaptive laws continue to maintain the stability and convergence of the system, even in the presence of parameter variations or disturbances. The control structure still retains the backstepping method, but with important improvements that allow the system to adapt better and remain more stable under changing conditions. The adaptability of this upgrade is clearly demonstrated through the reduction in chattering and maintaining control accuracy, especially when dealing with rapid or uncertain parameter changes in the system. The addition of the robust term allows the system to operate more effectively without requiring significant changes to the basic control structure, making it easier to apply to real-world systems. The backstepping-based design employed in our approach offers several key advantages for uncertain nonlinear systems:  
* Improved global stability: It enables the handling of nonlinear systems in a structured manner, ensuring stability at each design step. * Reduction in chattering: Smooth Lyapunov functions can be designed, using approximations such as tanh(s) instead of sign(s), thereby mitigating undesired chattering effects. * More precise control: Compared to traditional SMC, the backstepping approach offers more refined control, avoiding abrupt discontinuities in the control signal. * Greater flexibility: It is useful in systems where standard SMC cannot be applied directly due to physical or structural constraints. * However, this approach also presents certain challenges:
* Computational complexity: The backstepping design introduces additional differential equations, increasing the computational burden.