A controlled invariarice problem for the VTOL aircraft with bounded internal dynamics

This paper discusses a problem of controlled invariance for the VTOL aircraft, namely we require the output of the system to stay on a given curve on the output space, with the requirement that the curve must be followed completely. The problem is solved essentially by means of a Lyapunov technique which recalls some results from the theory for slowly varying nonautonomous linear systems. The main obstacle encountered lies in the fact that the internal dynamics of the constrained system lose controllability in some "critical points" along the reference curve (close to the point where the reference curve is vertical). The overall closed loop system is characterized by an alternations of periods of stability and instability and its state is proved to be globally stable under some conditions. Simulations results are provided for the cases where the curve is a straight line and a circle.