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Differentially Flat Nonlinear Control Systems

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Muruhan Rathinam

PhD Dissertation, Caltech, May 1997

Differentially flat systems are underdetermined systems of (nonlinear) ordinary differential equations
(ODEs) whose solution curves are in smooth one-one correspondence with arbitrary curves in a space
whose dimension equals the number of equations by which the system is underdetermined. For control
systems this is the same as the number of inputs. The components of the map from the system space to
the smaller dimensional space are referred to as the flat outputs. Flatness allows one to systematically
generate feasible trajectories in a relatively simple way. Typically the flat outputs may depend on the
original independent and dependent variables in terms of which the ODEs are written as well as finitely
many derivatives of the dependent variables. Flatness of systems underdetermined by one equation is
completely characterised by Elie Cartan's work. But for general underdetermined systems no complete
characterisation of flatness exists.

In this dissertation we describe two different geometric frameworks for studying flatness and provide
constructive methods for deciding the flatness of certain classes of nonlinear systems and for finding
these flat outputs if they exist. We first introduce the concept of ``absolute equivalence'' due to Cartan
and define flatness in this frame work. We provide a method of testing for the flatness of systems, which
involves making a guess for all but one of the flat outputs after which the problem is reduced to the case
solved by Cartan. Secondly we present an alternative geometric approach to flatness which uses ``jet
bundles'' and present a theorem which partially characterises flat outputs that depend only on the
original variables but not on their derivatives, for the case of systems described by two independent
one-forms in arbitrary number of variables. Finally, for the class of Lagrangian mechanical systems whose
number of control inputs is one less than the number of degrees of freedom, we provide a
characterisation of flat outputs that depend only on the configuration variables, but not on their
derivatives. This characterisation makes use of the Riemannian metric provided by the kinetic energy of
the system.

CDS Technical Report
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Richard Murray (murray@cds.caltech.edu)
Last modified: Tue Aug 30 07:42:18 2005