System Dynamics

Although the simulation models the entire airplane dynamics, the adaptive controller is only concerned with the pitching moment equation. The pitching moment equation for a symmetric airplane in longitudinal motion only is given, quite simply, by Equation 1.

$$\dot q = M/I_{yy}$$ (1)

The pitching moment $M$ is more easily modeled when nondimensionalized, so we introduce the pitching moment coefficient, $C_M$:

$$M = \frac12 \rho V^2 S \bar c C_M$$ (2)

Here, $\rho$ is the outside air density, $V$ is the true airspeed, $S$ is the wing area, and $\bar c$ is the wing chord. These are all known values.

The pitching moment coefficient is a function of the states and controls, and is modeled by the higher-order polynomial given by Equation 3.

$$C_M = m_0 + m_1\alpha + m_2\delta_e + m_3\alpha\delta_e + m_4\delta_e^2$$ (3)

$$+ m_5\alpha^2\delta_e + m_6\delta_e^3 + m_7\alpha\delta_e^2 + C_{M_q}(\alpha) \tilde q$$

The polynomial coefficients $m_0$, $m_1$, etc., are, of course, the parameters to be identified. $\alpha$ is the angle of attack, and is a function of the states and is considered known. The term $C_{M_q}(\alpha) \tilde q$ is of no relevance to this report.