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Differentially flat nonlinear control systems

Citation

Rathinam, Muruhan (1997) Differentially flat nonlinear control systems. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-01152008-082733

Abstract

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 variablesterms 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.

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Restricted to Caltech community only
Research Advisor(s):
  • Murray, Richard M.
Thesis Committee:
  • Murray, Richard M. (chair)
  • Marsden, Jerrold E.
Defense Date:23 April 1997
Record Number:CaltechETD:etd-01152008-082733
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-01152008-082733
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:191
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:01 Feb 2008
Last Modified:26 Dec 2012 02:27

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