This paper presents a formal method to design
a digital inertial control system for quad-rotor aircraft. In
particular, it formalizes how to use approximate passive models
in order to justify the initial design of passive controllers. Fundamental
limits are discussed with this approach – in particular,
how it relates to the control of systems consisting of cascades
of three or more integrators in which input actuator saturation
is present. Ultimately, two linear proportional derivative (PD)
passive controllers are proposed to be combined with a nonlinear
saturation element. It is also shown that yaw control can
be performed independently of the inertial controller, providing
a great deal of maneuverability for quad-rotor aircraft. A
corollary, based on the sector stability theorem provided by
Zames and later generalized for the multiple-input-output case
by Willems, provides the allowable range of k for the linear
negative feedback controller kI in which the dynamic system
H1 : x1 -> y1 is inside the sector [a1, b1], in which −1 < a1,
0 < b1 <= 1, and b1 > a1. This corollary provides a formal
method to verify stability, both in simulation and in operation
for a given family of inertial set-points given to the quadrotor
inertial controller. The controller is shown to perform
exceptionally well when simulated with a detailed model of the
STARMAC, which includes blade flapping dynamics.