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Trajectory Generation for Nonlinear Control Systems

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Michiel J. van Nieuwstadt

PhD Dissertation, Caltech, Jul 1996

This thesis explores the paradigm of two degree of freedom design for
nonlinear control systems. In two degree of freedom design one
generates an explicit trajectory for state and input around which the
system is linearized. Linear techniques are then used to stabilize the
system around the nominal trajectory and to deal with
uncertainty. This approach allows the use of the wealth of tools in
linear control theory to stabilize a system in the face of
uncertainty, while exploiting the nonlinearities to increase
performance. Indeed, this thesis shows through simulations and
experiments that the generation of a nominal trajectory allows more
aggressive tracking in mechanical systems.

The generation of trajectories for general systems involves the
solution of two point boundary value problems which are hard to solve
numerically. For the special class of differentially flat systems
there exists a unique correspondence between trajectories in the
output space and the full state and input space. This allows us to
generate trajectories in the lower dimensional output space where we
don't have differential constraints, and subsequently map these to the
full state and input space through an algebraic procedure. No
differential equations have to be solved in this process. This thesis
gives a definition of differential flatness in terms of differential
geometry, and proves some properties of flat systems. In particular,
it is shown that differential flatness is equivalent to dynamic
feedback linearizability in an open and dense set.

This dissertation focuses on differentially flat systems. We describe
some interesting trajectory generation problems for these systems, and
present software to solve them. We also present algorithms and
software for real time trajectory generation, that allow a
computational tradeoff between stability and performance. We prove
convergence for a rather general class of desired trajectories. If a
system is not differentially flat we can approximate it with a
differentially flat system, and extend the techniques for flat
systems. The various extensions for approximately flat systems are
validated in simulation and experiments on a thrust vectored
aircraft. A system may exhibit a two layer structure where the outer
layer is a flat system, and the inner system is not. We call this
structure \emph{outer flatness}. We investigate trajectory generation
for these systems and present theorems on the type of tracking we can
achieve. We validate the outer flatness approach on a model helicopter
in simulations and experiment.

CDS Technical Report
(PDF, 3139K, 160 pages)

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Richard Murray (murray@cds.caltech.edu)
Last modified: Tue Aug 30 07:42:19 2005