Geometric Jacobian Linearization
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 geometri...
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| Format: | Others |
| Language: | en en |
| Published: |
2007
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| Online Access: | http://hdl.handle.net/1974/953 |
| Summary: | 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. === Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-12-19 16:59:47.76 |
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