# Time-Specified Adaptive Robust Control Framework for Managing Nonlinear System Uncertainties
## 4 Results and Discussion
### Example 2: Third-Order Uncertain Parameter System  
The second model is a third-order system with uncertain parameters, governed by the following dynamic equations:  
\[\dot{x}_{1}=x_{2}+\theta x_{1}\] \[\dot{x}_{2}=x_{3}\] \[\dot{x}_{3}=u+\theta x_{3}^{2}\]  
Figure 2 illustrates the schematic representation of this system. In this model, \(x_{1},x_{2}\), and \(x_{3}\) represent the system states, \(u\) is the control input, and \(\theta\) is the uncertain parameter. This system exemplifies a third-order nonlinear system with uncertain dynamics, illustrating the challenges in practical control scenarios. This system structure appears in various practical applications, including aircraft pitch dynamics, certain chemical reactor models, and flexible joint robotic manipulators. For instance, in aircraft pitch control, \(x_{1}\) might represent the pitch angle, \(x_{2}\) the pitch rate, \(x_{3}\) the derivative of pitch rate, and \(\theta\) an uncertain aerodynamic parameter that varies with flight conditions. The nonlinear term \(\theta x_{3}^{2}\) captures the nonlinear aerodynamic effects that become significant at higher rates of motion. Figure 1: Single-link manipulator in vertical plane. The relationship between parameter selection methods and control performance in prescribed-time Adaptive Control systems is clearly demonstrated through the intricate interplay between key parameters and the system's operational characteristics. The parameters \(\sigma_{\text{i}}\) (i = 1, 2, 3), physically interpreted as smoothing coefficients, directly influence the convergence rate of the corresponding states. Larger values of \(\sigma_{\text{i}}\) accelerate convergence but simultaneously increase the control signal amplitude, potentially inducing oscillations, whereas smaller \(\sigma_{\text{i}}\) values yield smoother, albeit slower, convergence with a more refined control signal. Lyapunov stability theory establishes lower bounds for the following parameters: \(\sigma_{1}>\text{n}\), \(\sigma_{2}>\text{n}-1\), \(\ldots\), \(\sigma_{\text{n}}>1\), where n denotes the system order. For a third-order system, this implies \(\sigma_{1}>3\), \(\sigma_{2}>2\), and \(\sigma_{3}>1\). Additionally, the prescribed time \(T_{p}\) plays a critical role in determining when the system reaches the desired state. A smaller \(T_{p}\) accelerates convergence but requires a larger control signal, risking actuator saturation, while a larger \(T_{p}\) prolongs convergence time but reduces control effort, minimizing actuator fatigue. A notable advantage of this approach is that the actual convergence time precisely matches \(T_{p}\), independent of initial conditions. In an enhanced version of the controller, the robust gain coefficient \(k_{r}\) and the boundary layer parameter \(\varepsilon\) collectively form a robust component that significantly influences the trade-off between accuracy and signal smoothness. A larger \(k_{r}\) enhances robustness against disturbances and uncertainties but may introduce chattering, while a smaller \(k_{r}\) produces a smoother control signal at the expense of reduced robustness. Similarly, a larger \(\varepsilon\) mitigates chattering but compromises accuracy, whereas a smaller \(\varepsilon\) improves accuracy but heightens the risk of chattering. From the baseline control equation \(u=-\frac{\sigma_{2}\sigma_{2}}{t_{p}-t}+\phi_{2}-k_{r}\left[0.5\text{sat}\left( \frac{\sigma_{2}}{\varepsilon}\right)+0.5\text{tanh}\left(10z_{3}\right)\right]\), it is evident that an increase in \(\sigma_{3}\) amplifies the term \(-\frac{\sigma_{2}\sigma_{2}}{t_{p}-t}\), resulting in greater control energy. Simulations reveal that elevating \(\sigma_{3}\) from 1.5 to 5 can increase control energy by up to 40%, though it also enhances convergence speed. Analysis of the expression \(\frac{\sigma_{2}\sigma_{2}}{T_{p}-t}\) indicates that as t approaches \(T_{p}\), the denominator approaches zero, causing a sharp increase in control signal amplitude.