# Adaptive Optimal Control of UAV Brake Based on Parameter Estimation
## 5 Simulations  
In order to verify the effectiveness of the proposed control law, simulation tests are carried out on the MATLAB platform with the model parameters in [35]. The reference slip rate is set as 0.1, the initial landing velocity \(v(x)=87\) m/s, and the initial main wheel angular velocity \(w(0)=210\) rad/s, and the initial slip rate is set as \(\lambda(0)=0\).  
The parameters in the controller are designed to \(k_{1}=3\), \(k_{2}=25\), \(k_{s1}=0.7\), \(k_{s2}=10\), \(\Theta=1\), \(\beta_{1}=4.5\), \(\beta_{2}=8\), and the critic NN weights converge to \([-1.982\), \(-1.553\), \(-4.006\), \(-3.86\), \(-1.210\), \(-0.567\), \(0.72\), \(-0.9234\), \(-0.34\), \(0.765\), \(1.076\), \(0.365\), \(3.023\)].  
The changes in the speed and distance of the UAV during braking are shown in Figures 2 and 3. It takes 21.6 s from the start of braking to the end of the anti-skid control, and the braking distance is 702.5 m. No significant overshoot occurs during the braking process, and the velocity and wheel speed decrease smoothly without large fluctuations or deep skid of the wheel.  
It can be seen from Figure 4 that the controller can track the optimal slip rate of real-time changes quickly and accurately and is always in the stable region. Figure 5 shows that the tracking error of the optimized controller based on RISE correction converges asymptotically to zero, meeting the high-precision requirements compared to the generic control with the gradient algorithm.  
To show the improved estimation and control performance, the method proposed in this paper is shown in Figure 6. Compared with the conventional control algorithm shown in Figure 7, the parameter estimation error can converge to zero and the convergence speed is fast.  
Figure 3: Stopping distance.  
Figure 2: Aircraft speed and wheel speed.  
Figure 4: Wheel slip ratio.  
Figure 5: The tracking error.  
The quadratic performance index function of the algorithm is designed in this paper and the conventional algorithm is estimated by MATLAB. The performance index function of the optimized controller is larger than that of the conventional controller, which indicates that the RISE-modified adaptive optimal controller ensures the comprehensive quasi-optimal of the system dynamic error and control energy consumption during the whole braking process. The control input is shown in Figure 8.  
Figure 6: Textual method.  
Figure 7: Gradient descent.  
Figure 9 shows the estimated weight of the NN. It can be seen that the weights converge to constant values. This result shows the convergence of the adaptive law and verifies the effectiveness of the proposed learning algorithm.  
Figure 8: Gradient descent.  
Figure 9: The control input.  
To show the improved performance of the proposed single critic NN-based ADP, compared with a critic-actor NN-based ADP method mentioned in [31], it is clear that the proposed single critic NN-based can achieve faster transient error convergence and smaller steady-state tracking errors.  
The simulation results show that the designed controller has a good control effect, realizes the tracking of the unknown optimal slip rate, and has a robust ability. By comparing the size of the secondary performance index, the modified optimized controller can meet the performance index, and the system's control input and control error are optimized comprehensively.