## 4 Modified NLDI  
The flight controller based on NDI provides the commanded control surface deflections as a function of the error of control variables. The complete deflection of the control surfaces is computed per each sampling time of the flight controller. In contrast, the flight controller based on the modified NLDI only computes the increments of control surface deflections by taking the influence of the acting control surface deflections into account. Consequently, the modified NLDI (MNLDI) is more robust against model mismatch than conventional NLDI. As a result, the attitude Equation (5) must be expanded to include the variation of the flight control command using a Taylor expansion. The approximated first-order attitude equation is given in Equation (28), in which only the first three terms are considered and the higher-order terms (H.O.T) are neglected. \[\mathbf{I}_{B}\ddot{\omega}_{B}\approx\dot{\bar{\omega}}_{B0}+ \left(\frac{\partial(\vec{\omega}_{B}\times\mathbf{I}_{B}\vec{\omega}_{B}+ \vec{M}_{a}+\vec{M}_{c}\delta)}{\partial\omega}\left(\omega-\omega_{0}\right) \right)_{\delta_{0}}+ \tag{28}\] \[\left(\frac{\partial(\vec{\omega}_{B}\times\mathbf{I}_{B}\vec{ \omega}_{B}+\vec{M}_{a}+\vec{M}_{c}\delta)}{\partial\omega}\left(\delta- \delta_{0}\right)\right)_{\omega_{0}}+H.O.T\]  
where \(\dot{\vec{\omega}}_{B0}\) is the initial angular rates vector, and \(\omega\) and \(\delta\) are varying with time. The current control surface deflection command is the variation upon the previous control command. The moments  
Figure 4: Altitude control loop. Figure 5: Velocity control loop. generated by the control surface deflections \(M_{c}\) are independent of \(\omega\) and the aerodynamics moment \(M_{a}\) is not a function of \(\delta\). The (H.O.T) are small enough to be neglected, thus the approximated moment equation is given in Equation (29). \[\mathbf{I}_{B}\dot{\bar{\omega}}_{B}\approx\dot{\bar{\omega}}_{B0}+ \left(\frac{\partial(\bar{\omega}_{B}\times\mathbf{I}_{B}\bar{\omega}_{B}+ \vec{M}_{a})}{\partial\omega}\Delta\omega\right)_{\delta_{0}}+\left(\frac{ \partial(\vec{M}_{c}\delta)}{\partial\delta}\Delta\delta\right)_{\omega_{0}} \tag{29}\]  
Furthermore, by assuming the deflection of control surfaces only generates a variation in moments, the variation in forces is small enough to be neglected. This assumption is reasonable for aircraft with traditional configuration, so that control surfaces are far from the aircraft center of gravity. Consequently, the second term of Equation (29) is negligible compared to the third term so the attitude equation is given in Equation (30). \[\mathbf{I}_{B}\ddot{\bar{\omega}}_{B}\approx\dot{\bar{\omega}}_{B0}+\left( \frac{\partial(\vec{M}_{c}\delta)}{\partial\delta}\Delta\delta\right)_{\omega_ {0}} \tag{30}\]  
Assuming \(M_{c}\) is a linear function of \((\delta_{a},\delta_{e},\delta_{r})\), then the approximated attitude equation is given in Equation (31)  
\[\mathbf{I}_{B}\dot{\bar{\omega}}_{B}\approx\dot{\bar{\omega}}_{B0}+\left( \vec{M}_{c}\Delta\delta\right)_{\omega_{0}} \tag{31}\]  
Applying NLDI to the moment in Equation (31) results in an expression for the incremental change of the input control surface deflections \(\Delta\delta\) given in Equation (32). \[\Delta\delta=\vec{M}_{c}^{-1}\left[\mathbf{I}_{B}\
u-\ddot{\bar{\omega}}_{B0}\right] \tag{32}\]  
where \(\Delta\delta=\delta-\delta_{0}\), \(\
u=[\
u_{p},\
u_{q},\
u_{r}]^{T}\) are the designed control laws for roll, pitch, and yaw angular rates, respectively, which are given in Equations (7)-(9). The obtained control law is model independent, which means the uncertainties in the model are avoided by the modification of NLDI, therefore the robustness of using MNLDI is better than using NLDI. Moreover, practically MNLDI is valid by feeding back angular accelerations using the inertial navigation system.