Stabilization for feedforward systems with delay in the input

Abstract

We solve a family of uniform global asymptotic stabilization problems for time varying feedforward systems with a delayed input. Our solution relies on a time-varying change of coordinates and Lyapunov-Krasovskii functionals. It applies under any given constant delay, and provides controllers of arbitrarily small amplitude. We also prove input-to-state stability with respect to additive disturbances on the controllers under certain bounds on the disturbances. We demonstrate our work using a tracking problem for a model that is suitable for high level formation flight of unmanned air vehicles.

Theory, Methods, and Applications of Nonlinear Control

“…The significance of the PVTOL results [GMM11,GMM12b] is in (a) the global boundedness of the controllers in the decoupled coordinates, (b) their applicability to cases where the velocity measurements may be unavailable, when the original system is augmented by an observer for the unknown velocities, (c) the uniform global asymptotic stability of the tracking dynamics, (d) the applicability of the theory to a very general class of reference trajectories, and (e) the use of input-to-state stability to quantify performance under actuator errors of large amplitude. The work for UAVs in [GMM12a,GMM13,MM12a,MM12b] gave globally bounded tracking controllers under mild nondegeneracy conditions on the reference trajectory and the reference control, including (i) input-to-state stability with respect to small additive uncertainty on the controllers and (ii) compensation of arbitrarily long time delays in the state observations using Lyapunov-Krasovskii functionals. The UAV work can be applied when there are rate constraints on the controls (which are constant bounds on the rate of change of the controller values along all of the system trajectories) and can ensure that the UAV velocity remains above a desired minimal value.…”