# Robust Adaptive Control for an Aircraft Landing
## 3 Problem Definition
### Control Targets  
The aircraft must be able to touch down in different landing scenarios. The landing scenarios can be characterized by two basic parameters: aircraft mass and sink speed. The ranges of varying aircraft mass and sink speed considered in this paper are shown in Table 1. To build the control system easily, accurately measuring those parameters is crucial. Whereas it is possible to estimate the aircraft sink speed with a ground proximity sensor or an accelerometer, it is unlikely to obtain highly accurate aircraft mass estimates because it depends on the remaining fuel mass and payload. In this paper, two methods for mass estimation are given. * First method: Before take-off, only the pneumatic force from the landing gear damper sustains the aircraft mass. From Equations (3) and (6), the mass of aircraft \(\hat{m}_{1}^{\mathit{take-off}}\) can be estimated by: \[\hat{m}_{1}^{\mathit{take-off}}=\frac{F_{\mathit{air}}}{g}=\left(p_{0}\left( \frac{V_{0}}{V_{0}-A_{p}s}\right)^{n}-P_{\mathit{ATM}}\right)\frac{A_{p}}{g}\] (9)  
Before landing, the mass of aircraft \(\hat{m}_{1}^{\mathit{landing}}\) reduces by the fuel consumption \(\Delta m_{\mathit{gas}}\). Thus, the aircraft mass can be calculated by:  
\[\hat{m}_{1}^{\mathit{landing}}=\hat{m}_{1}^{\mathit{take-off}}-\Delta m_{ \mathit{gas}} \tag{10}\]  
The advantage of this method is that it is fast, and the estimated value can be predicted before landing. However, the accuracy of this method depends on the measurement of the remaining fuel. * Second method: The mass can be estimated by rearranging Equation (6), \[\hat{m}_{1}=\frac{F_{d}}{g-\tilde{z}_{1}}\] (11) where \(\hat{m}_{1}\) is the estimate of the unknown parameter \(m_{1}\). Because of sensor noise, this estimation is hardly stable, and the denominator can be zero. Appling the least squares method, the mass estimation is given by: \[\hat{m}_{1}=\frac{\int F_{d}(g-\tilde{z}_{1})dt}{\int\left(g-\tilde{z}_{1} \right)^{2}dt}\] (12)  
The estimate is \(\hat{m}_{1}\) updated over time. In some sense, it is averaging the effects of measurement noise. Note that the denominator in Equation (12) is always increasing and non-zero. The convergence of the estimation can be proved as follows. Equation (12) is rewritten as:  
\[\hat{m}_{1}\int\left(g-\tilde{z}_{1}\right)^{2}dt=\int F_{d}(g-\tilde{z}_{1} )dt \tag{13}\]  
The derivative of Equation (13) is given by:  
\[\dot{\hat{m}}_{1}\int\left(g-\tilde{z}_{1}\right)^{2}dt+\hat{m}_{1}(g-\tilde{ z}_{1})^{2}=F_{d}(g-\tilde{z}_{1}) \tag{14}\]  
Combining Equations (6) and (14), we further obtain the time derivative of the estimate \(\hat{m}_{1}\):  
\[\dot{\hat{m}}_{1}=\frac{\left(m_{1}-\hat{m}_{1}\right)(g-\tilde{z}_{1})^{2}}{ \int\left(g-\tilde{z}_{1}\right)^{2}dt} \tag{15}\]Then, a Lyapunov function candidate can be proposed as below:  
\[V=\frac{1}{2}(m_{1}-\hat{m}_{1})^{2} \tag{16}\]  
From Equations (15) and (16), the derivative of the Lyapunov function is found to be:  
\[\dot{V}=-\dot{\hat{m}}_{1}(m_{1}-\hat{m}_{1})^{2}=-(m_{1}-\hat{m}_{1})^{2}\frac {\left(g-\ddot{z}_{1}\right)^{2}}{\int\left(g-\ddot{z}_{1}\right)^{2}dt}\leq 0 \tag{17}\]  
The Lyapunov condition is satisfied, and, thus, the estimation converges (\(\hat{m}_{1}\to m_{1}\)). Although this method can estimate aircraft mass online, however, the accurate estimation is limited by the model validity and accuracy. Some parametric uncertainties are impossible to estimate without incurring excessive cost. Such factors as undesirable wind, ground effect, and runway unevenness can impact the landing performance of an aircraft landing gear. In addition, some parameters are impossible to measure or estimate accurately, such as viscous damping coefficient and friction force. In this research, friction force is assumed to be much less than the other forces, and its uncertainty can be ignored. However, viscous force is a critical factor in absorbing kinetic energy at landing, and thus, its parametric uncertainty is considered.