|
David
R. Tyner:
|
Title: Geometric Jacobian linearization (64 pages)
Author: David R. Tyner
Detail: Ph.D. Thesis, Queen's University
Original manuscript: 21/12/2007
For control systems that evolve on Euclidean spaces, Jacobian linearization is a common technique in many control applications, analysis, and controller design methodologies. However, the standard linearization method along a non-trivial reference trajectory does not directly apply in a geometric theory where the state space is a differentiable manifold. Indeed, the standard constructions involving the Jacobian are dependent on a choice of coordinates.
The procedure of linearizing a control affine system along a
non-trivial
reference trajectory is studied from a differential geometric
perspective. A
coordinate-invariant setting for linearization is presented. With the
linearization in hand, the controllability of the geometric
linearization is
characterized using an alternative version of the usual controllability
test
for time-varying linear systems. The various types of stability are
defined
using a metric on the fibers along the reference trajectory and
Lyapunov's
second method is recast for linear vector fields on tangent bundles.
With the
necessary background stated in a geometric framework, Kalman's theory
of
quadratic optimal control is understood from the perspective of the
Maximum
Principle. Finally, following Kalman, the resulting feedback from
solving the
infinite time optimal control problem is shown to uniformly
asymptotically
stabilize the linearization using Lyapunov's second method.
Download: PhD Thesis