On the existence of constant-ratio trajectories in nominally time- optimal control systems subject to parameter variation

In a previous paper it was found that certain control systems with nominally time-optimal feedback controllers are highly sensitive to changes of plant parameter. When the plant consists of several pure integrators, changes of plant parameter can cause what are termed constant-ratio trajectories. Such trajectories yield slow settling, or even instability. The previous paper used, in part, computational methods to show the existence of constant-ratio trajectories. In the present paper some simple conditions are shown to be sufficient for existence of the constant-ratio trajectories.