# Robust Adaptive Control for an Aircraft Landing
## 4 Algorithm Control Designation
### Sliding Mode Controller-Adaptive Hybrid Control  
Sliding Mode Control (SMC) is the most common algorithm used to manage uncertainties. The main task of SMC is to drive the plant according to the reference model. Figure 6a shows the reference module to calculate the history of the reference signal (\(z^{d}{}_{1}\)). Before the landing gear touches down, the mathematical model with an adaptive hybrid control algorithm is executed. This progress depends on the estimated mass and sink speed explained in the previous chapter. Then, the tracking error is calculated as the difference between the signal sensor (\(z_{1}\)) and the reference signal (\(z^{d}{}_{1}\)), which is  
Figure 4: Force curve and target damping force of the force control: (**a**) force-stroke curve; (**b**) the relationship between target damping force and landing condition parameters.  
shown in Figure 6b. Next, the SMC system is developed based on the tracking error and the operating principle of the MR damper.  
Following [28], the sliding surface can be expressed by:  
\[S_{f}=\left(\lambda+\frac{d}{dt}\right)\overline{z}_{1} \tag{24}\]  
where \(\overline{z}_{1}=z_{1}-z_{1}^{d}\) is the tracking error, \(\lambda\) is the slope of the sliding surface line. Figure 7 shows the state phase of a typical aircraft. The motion starts from (0, v) passes (z\({}_{1}\).max, 0) and ends at (z\({}_{1}\).stable, 0).  
Figure 5: Skyhook gain surface.  
Figure 6: Sliding mode control-adaptive hybrid control algorithm. (**a**) Reference part. (**b**) Main part.  
Because the start point is on the vertical line, the slope value can be chosen \(\lambda=0\) to reduce the time to reach the sliding surface. The control law can be given as:  
\[u=\hat{u}-k\ sat(S_{f}/\xi) \tag{25}\]  
where sat is the saturation function [28], and \(\xi=0.1\) is the boundary thickness. The estimated control law, \(\hat{u}\) is designed as:  
\[\hat{u}=\hat{m}_{1}\left(g-\tilde{z}_{1}^{d}\right)-\left(\hat{F}_{d}\right) \tag{26}\]  
and \(\xi=\)0.1 is boundary thickness, discontinuous control gain, \(k\) is selected satisfying the inequality below:  
\[k\geq F_{d}-\hat{F}_{d} \tag{27}\]  
Because MR damper is a semi-active system, the control law must use the following rule:  
\[u=\left\{\begin{matrix}\hat{u}-k\ sat(S_{f}/\xi)&if\ \imath\hat{s}\geq 0\\ 0&if\ \imath\hat{s}<0\end{matrix}\right. \tag{28}\]  
To check the stability of the controller, the candidate of the Lyapunov function is given by:  
\[V=\frac{1}{2}S_{f}^{2}\geq 0 \tag{29}\]  
The derivative of Equation (29) is:  
\[\begin{array}{l}\dot{V}=S_{f}\dot{S}_{f}=S_{f}(\tilde{z}_{1}^{d}-\ddot{z}_{ 1})=S_{f}\left(\tilde{z}_{1}^{d}-\left(g-\frac{F_{d}+\imath\hat{m}_{1}}{m_{1} }\right)\right)\\ \dot{V}=S_{f}\left(\tilde{z}_{1}^{d}-\left(g-\frac{F_{d}+\imath\hat{m}_{1}(g- \tilde{z}_{1}^{d})-\hat{F}_{d}-k\ sat(S_{f}/\xi)}{m_{1}}\right)\right)\\ \dot{V}=S_{f}\left(\left(1-\frac{\imath\hat{m}_{1}}{m_{1}}\right)\left(\tilde {z}_{1}^{d}-g\right)+\frac{1}{m_{1}}\left(F_{d}-\hat{F}_{d}-k\ sat(S_{f}/\xi) \right)\right)\\ \dot{V}=\frac{1}{m_{1}}S_{f}\left(F_{d}-\hat{F}_{d}-k\ sat(S_{f}/\xi)\right) \leq 0\end{array} \tag{30}\]  
The Lyapunov condition is satisfied; thus, the controller is stable to deal with uncertainties. However, the control input can be infinite to make the system reside in the sliding surface. Because the physical force output from an MR damper is bounded as in Table 1, the stability proof of the proposed control algorithm might be valid with only a limited level of uncertainties and disturbances.  
Figure 7: State phase of damper.