Chapter 6, Problem 64P

It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by

$s^3=\left(\frac{mgh}{1}\left(q_2+k{k}_d\right)+q_{1}g\right)s^2+k\frac{mgh}{1}s+\frac{mgh}{1}\left(k{k}_i+q_1\right)=0$

where m is the mass of the helicopter, g is the gravitational constant, I is the rotational inertia of the helicopter, h is the height of the rotor plane above the center of gravity, q₁and q₂are stabilizer flapping parameters, k, k_i, and k_d, are controller parameters: all constants > 0. The UAV is supposed to pick up a payload: when this occurs, the mass, height. and inertia change to m’, h’, and I’, respectively, all still > 0. Show that the helicopter will remain stable as long as

$\frac{m^{\prime }g{h}^{\prime }}{I^{\prime }}>\frac{q_1+k{k}_i-q_{1}gk}{k\left(q_2+k{k}_d\right)}$