= \frac{1}{m}(Thr(h) - D(V,h) - m \cdot g \cdot \sin(\gamma))$$ where $\sin ( \gamma )=\frac{\dot h}{V}$. So, from eq. \[eq:acceleration\], we can see that the acceleration evolves independently of the mode given the rate of climb. Also, when $\dot h$ is high, $\sin( \gamma )$ is high and the acceleration $\dot V$ is low. This goes along with the hypothesis the total-energy model. Finally, the next section will present the evolution of the variable $q$.

Mode Definition
---------------

In this section, we define the modes and their transitions based on BADA 3.10 User Manual [@Bada2012]. First, let $Q = \{CAS,MACH\} \times \{LOW,HIGH\} \times \{DEC,CST,ACC\}$ be the mode space. The first two modes, transition altitude and tropopause, depend solely on the altitude: $$q_1(h) = \left\{ 
  \begin{array}{l l}
    \text{CAS} & \quad \text{if $h \le H_{trans}$}\\
    \text{MACH} & \quad \text{otherwise}\\
  \end{array} \right.$$

$$q_2(h) = \left\{ 
  \begin{array}{l l}
    \text{LOW} & \quad \text{if $h \le H_{trop}$}\\
    \text{HIGH} & \quad \text{otherwise}\\
  \end{array} \right.$$ where $H_{trans}$ is the transition altitude and $H_{trop}$ is the tropopause geopotential pressure altitude. Finally, the last feature $q_3$ is the mode of acceleration. The simplest controller of this mode is given by: $$q_3(V,h) = \left\{ 
  \begin{array}{l l}
    \text{ACC} & \quad \text{if $V 