# 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: http://www.cds.caltech.edu/~murray/preprints/mvn96-phd.pdf
- Project(s): Template:HTDB funding::AFOSR