An Introduction To Differential Flatness


Systems of ordinary differential equations (ODEs) can be divided into two classes, determined and underdetermined. The latter class can be defined as a system of $m$ ODEs with $n$ dependent variables where $n>m$. Systems of this sort are typically encountered by control engineers. They attempt to define $n-m$ of the dependent variables as functions of time in order force the system to act in a specified way. When the system of ODEs is linear, a number of methods exist to easily design control for the system. However, when the system of ODEs is nonlinear, control design is much more difficult and fewer design methods exist.

One method that has been developed to design control somewhat easily for nonlinear systems exploits a property of such systems known as differential flatness. A differentially flat system is a system of nonlinear ODEs whose solution curves are in a smooth one-one correspondence with arbitrary curves in a space whose dimension equals $n-m$. Hence, generating solution trajectories for the system is simplified by finding curves in a lower dimension that satisfy conditions at the endpoints.


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