The analysis and control of nonlinear systems using Lyapunov stability theory

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Virginia Tech

Techniques based on Lyapunov theory for the stability analysis and control of nonlinear systems are developed. In the first part of this work, procedures for determining conditions of guaranteed asymptotic stability are developed for nonlinear, uncontrolled systems. A second order model of an aircraft which is potentially unstable in pitch is used to demonstrate these methods. This approach is then expanded for use with systems of arbitrary order and applied to the investigation of a nonlinear model describing the lateral-directional motion of a departure prone aircraft. These investigations show that concepts based on Lyapunov stability can be used to effectively analyze nonlinear systems.

A systematic method of finding an efficient controller is then developed for systems having controls which behave nonlinearly. These control techniques are demonstrated on a generic aircraft which exhibits nonlinear elevator behavior at high angles of attack. Although a conventional controller based on linear theory results in an unstable divergence, a Lyapunov based controller intelligently uses the available elevator power to augment the stability of the aircraft. A Lyapunov controller is then developed for use with the orbital Clohessy-Wiltshire equations of relative motion. When an engine unpredictably fails, this controller automatically accounts for the new conditions and the desired rendezvous is thus, still obtained. These Lyapunov based controllers are shown not only to perform well under highly nonlinear circumstances but to also maintain a high level of efficiency under more linear conditions.