We consider the equations of motion of an airplane in a vertical plane. If the
units are scaled appropriately (forward speed equal to one for instance), then
these equations are approximately
$$
\begin{aligned}
&\dot{\gamma}=\sin \alpha, \\
&\ddot{\theta}=-(\alpha-u), \\
&\dot{h}=\sin \gamma,
\end{aligned}
$$
where
\- \(h\) is the height of the airplane with respect to a certain reference
height;
\- \(\gamma=\theta-\alpha\) is the flight angle;
\- \(\theta\) is the angle between the reference axis of the airplane and the
horizontal;
\- \(u\) is the rudder control.
One must design an automatic pilot to keep \(h\) constant (and equal to zero) in
the presence of all kinds of perturbations such as vertical gusts.
\- Linearize the equations of motion and write them as a set of first order
differential equations.
\- Show that the designer who proposes a feedback of the form \(u=k h\), where
\(k\) is a suitably chosen constant, cannot be successful.
\- Prove that a feedback of the form \(u=k_{1} h+k_{2} \theta\), with suitably
chosen constants \(k_{1}\) and \(k_{2}\), 'does the job', i.e. the resulting
closed-loop system is asymptotically stable.
