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Differentially flat nonlinear control systems

Rathinam, Muruhan (1997) Differentially flat nonlinear control systems. California Institute of Technology , Pasadena, CA. (Unpublished) http://resolver.caltech.edu/CaltechCDSTR:1997.008

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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 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.


Item Type:Report or Paper (Technical Report)
Group:Control and Dynamical Systems Technical Reports
Record Number:CaltechCDSTR:1997.008
Persistent URL:http://resolver.caltech.edu/CaltechCDSTR:1997.008
Usage Policy:You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format.
ID Code:28129
Collection:CaltechCDSTR
Deposited By: Imported from CaltechCDSTR
Deposited On:15 Dec 2007
Last Modified:26 Dec 2012 14:31

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