Title:
Approximate Feedback Linearization: A Homotopy Operator Approach
Approximate Feedback Linearization: A Homotopy Operator Approach
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Author(s)
Banaszuk, Andrzej
Hauser, John
Hauser, John
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Abstract
In this paper, we present an approach for finding feedback linearizable systems that
approximate a given single-input nonlinear system on a given compact region of the
state space. First, we show that if the system is close to being involutive then it is
also close to being linearizable. Rather than working directly with the characteristic
distribution of the system, we work with characteristic one-forms, i.e., with the
one-forms annihilating the characteristic distribution. We show that homotopy operators
can be used to decompose a given characteristic one-form into an exact and antiexact
part. The exact part is used to define a change of coordinates to a normal form that
looks like a linearizable part plus nonlinear perturbation terms. The nonlinear terms in
this normal form depend continuously on the antiexact part and they vanish whenever
the antiexact part does. Thus, the antiexact part of a given characteristic one-form
is a measure of nonlinearizability of the system. If the nonlinear terms are small, by
neglecting them we obtain a linearizable system approximating the original system.
One can design control for the original system by designing it for the approximating
linearizable system and applying it to the original one. We apply this approach for
design of locally stabilizing feedback laws for nonlinear systems that are close to being
linearizable.
Sponsor
NSF under grant PYI ECS-9157835 and DMS-9207703
Date Issued
1995-01-12
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Pre-print