## 2 Aircraft Energy Equations and Dynamic Model
### Aircraft Energy Equations  
The derivation of energy equations is based on Lambregts [4]. The total energy of an aircraft can be expressed as the sum of the potential energy and kinetic energy as follows:  
\[E=E_{\text{KE}}+E_{\text{PE}} \tag{1}\]  
\[E=\frac{1}{2}mV^{2}+mgh \tag{2}\]  
where g is the acceleration of gravity, \(h\) is altitude, \(m\) is aircraft weight, \(V\) is the longitudinal velocity, \(E_{\text{KE}}\) is kinetic energy, and \(E_{\text{PE}}\) is potential energy.  
Assuming constant weight, Equation (2) can be rewritten as:  
\[E=mg\left(\frac{1}{2}\frac{V^{2}}{g}+h\right) \tag{3}\]  
Similar to the derivation of aircraft equations of motion, by using the small-disturbance theory, the energy of aircraft is linearized at steady flight conditions, as shown below:  
\[E_{0}+\Delta E=mg\left(\frac{1}{2}\frac{\left(V_{0}+\Delta V\right)^{2}}{g}+ \left(h_{0}+\Delta h\right)\right) \tag{4}\]  
where \(V_{0}\) is trimmed airspeed. Assume that \(\left(\Delta V\right)^{2}=0\), and differentiate Equation (4) with respect to time, yielding:  
\[\frac{\text{d}}{\text{d}t}\left(E_{0}+\Delta E\right)=mg\frac{\text{d}}{\text {d}t}\left[\frac{1}{2}\left(\frac{{V_{0}}^{2}}{g}+\frac{2V_{0}\Delta V}{g} \right)+\left(h_{0}+\Delta h\right)\right] \tag{5}\]  
\[\Delta\dot{E}=mg\left(\frac{V_{0}\Delta\dot{V}}{g}+\Delta\dot{h}\right) \tag{6}\]Dividing by the trimmed airspeed, \(V_{0}\), yielding:  
\[\frac{\Delta\dot{E}}{V_{0}}=mg\left(\frac{\Delta\dot{V}}{g}+\Delta\gamma\right) \tag{7}\]  
where \(\gamma\) is flight path angle.  
Thus, at a given airspeed, the rate of change of aircraft energy is dependent only upon the change of longitudinal acceleration, \(\Delta\dot{V}\), and the change of flight path angle, \(\Delta\gamma\). Since the total energy is determined by the thrust of engine, the thrust can be derived from aircraft longitudinal equation of motion,  
\[m\Delta\dot{V}=\Delta T-\Delta D-m\text{g sin}\Delta\gamma \tag{8}\]  
Assuming that \(\Delta\gamma\) is small, therefore \(\text{sin}\Delta\gamma=\Delta\gamma\), then  
\[mg\left(\Delta\gamma+\frac{\Delta\dot{V}}{g}\right)=\Delta T-\Delta D \tag{9}\]  
Thus, the aircraft rate of change of energy is proportional to the difference between thrust, \(T\), and drag, \(D\). By rearranging the equation above, the required thrust change as:  
\[\Delta T=mg\left(\Delta\gamma+\frac{\Delta\dot{V}}{g}\right)+\Delta D=\frac{ \Delta\dot{E}}{V_{0}}+\Delta D=\frac{\Delta\dot{E_{\text{s}}}}{V_{0}}m\text{g }+\Delta D \tag{10}\]  
where \(\Delta\dot{E_{\text{s}}}\) is total specific energy rate.  
Assume that during the steady flight condition, the change of drag is small and negligible. The change of thrust is proportional to the sum of change of longitudinal acceleration and change of flight path angle:  
\[\Delta T\propto\left(\Delta\gamma+\frac{\Delta\dot{V}}{g}\right) \tag{11}\]  
Conversely, at a specific thrust level, it is possible to trade the change of flight path angle for change of longitudinal acceleration and vice versa using the elevator control only. Finally, the change of total specific energy rate error, driven by throttle, is derived as  
\[\Delta\dot{E_{\text{s}}}=\dot{E_{\text{s}}}-\dot{E_{\text{s0}}} \tag{12}\]  
where \(\Delta\dot{E_{\text{s}}}=\dot{E_{\text{s}}}-\dot{E_{\text{s0}}}\) and \(\dot{E_{\text{s0}}}=0\) is the total specific energy rate of an aircraft at steady flight condition.  
In addition, the change of specific energy rate distribution error, driven by elevator, is derived as  
\[\Delta\dot{L_{\text{s}}}=-\Delta\gamma+\frac{\Delta a_{\text{long}}}{g} \tag{14}\]  
where \(\Delta\dot{L_{\text{s}}}\) is the change of specific energy rate distribution error and \(a_{\text{long}}\) is longitudinal acceleration along flight path. The detailed derivation of \(\Delta\dot{L_{\text{s}}}\) refers to the study in [14].