## 5 Simulation Results
### Example 1  
For the first example, we consider the FO Lorenz system to demonstrate the advantage of the proposed desired controller by utilizing the Lyapunov-like function. The system can be described by the following equations, which capture the FO dynamics:  
\[ \begin{array}{l}D^{v}x=\sigma(y-x)+\text{sat}(k_{1}x)\\ D^{v}y=x(\rho-z)-y+\text{sat}(k_{2}y)\\ D^{v}z=xy-\beta z+\text{sat}(k_{3}z)\end{array}  \tag{49}\]  
where \(x\), \(y\), and \(z\) represent the state variables, and \(\sigma\), \(\rho\), and \(\beta\) are system parameters. By designing a suitable controller based on the Lyapunov-like function and applying it to the FO Lorenz system, we aim to demonstrate the ability of the proposed control approach to stabilize and control the chaotic behavior of the system. The analysis and simulation results will provide insights into the effectiveness and performance of the controller in achieving stabilization and desired system behavior. The model parameters are given by \(\sigma=10\), \(\rho=28\), \(\beta=\frac{8}{3}\), \(v=0.993\), and \(K= \begin{bmatrix}k_{1}&k_{2}&k_{3}\end{bmatrix} = \begin{bmatrix}0&0&0\end{bmatrix} \). The open-loop system is depicted in Figure 2.  
Next, we consider the condition of Theorem 2 as follows:  
\[A_{cl}= \begin{bmatrix}k_{1}-\sigma&0&0\\ 0&k_{2}-1&0\\ 0&0&k_{3}-\beta\end{bmatrix} ,\quad F(x)= \begin{bmatrix}0\\ -xz\\ xy\end{bmatrix} ,\quad G(x)= \begin{bmatrix}\sigma\\ \rho\\ 0\end{bmatrix} .\]  
We can now design the controller using the proposed method outlined in Section 4 and solve the optimization problem given by (48). We then consider \(\|u\|\):  
\[ \begin{array}{ll}\text{Minimize}&\|u\|\\ \text{subject to:}\\ \left\{\begin{bmatrix}\Theta&P&P&P\\ P&-k_{f}I&0&0\\ P&0&-k_{g}I&0\\ P&0&0&-k_{s}I\end{bmatrix}<0\\ \end{array} \right.\end{array} \tag{50}\]  
We achieve the following results:  
\[ \begin{array}{ll}P=\begin{bmatrix}1&0.001&0.001\\ 0.001&1&0.001\\ 0.001&0.001&1\end{bmatrix}\\ \\ K=\text{diag}\Big{[}-68.8939&-77.8701&-57.2020\Big{]}=\text{diag}\Big{[}k_{1} &k_{2}&k_{3}\Big{]}\\ \\ k_{f}=0.8132,\quad k_{g}=0.0295,\quad k_{s}=0.0825\end{array}  \tag{51}\]  
Figures 3 and 4 exhibit the phase plots of the system and demonstrate the behavior of the closed-loop system. These plots provide valuable insights into the dynamics and convergence characteristics of the system. In Figure 3, the phase plot illustrates the trajectory of the system's state variables over time, allowing us to visualize the system's attractor and observe its overall behavior. A well-defined and structured attractor indicates the stability and regular behavior in the system. The plot reveals the desirable convergence speed of the system, with the trajectory approaching a steady state or periodic orbit. Similarly, in Figure 4, the behavior of the closed-loop system with initial states \(x(0)=[10,14,7]\) is depicted. The closed-loop response reflects the impact of the control input on the system's dynamics. The figure demonstrates the effectiveness of the control strategy in guiding the system toward a desired state or trajectory. The convergence speed, as evident from  
Figure 2: Phase plot of the open-loop FO Lorenz system for \(v=0.993\). the plot, indicates that the closed-loop system can reach the desired equilibrium or follow a desired trajectory in a relatively short time. The rapid convergence speed observed in Figures 3 and 4 is a positive outcome, signifying the effectiveness and efficiency of the control approach. A fast convergence speed implies that the system can quickly adjust and stabilize itself in response to perturbations or changes in initial conditions. In Figure 5, we present a comparison of the states between FO and integer-order system modes. The figure clearly demonstrates that the states of the FO system converge faster than those of the integer-order system in [46]. This observation can be attributed to two possible reasons. First, the stability region of the FO system, when \(v\in(0,1)\), is inherently broader than that of the integer-order system.