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On nonlinear controllability of homogeneous systems linear in control

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3 Author(s)
J. Melody ; Coordinated Sci. Lab., Univ. of Illinois, Urbana, IL, USA ; T. Basar ; F. Bullo

This paper considers small-time local controllability (STLC) of single- and multiple-input systems, x˙=f0(x)+Σi=1mfiui where f0(x) contains homogeneous polynomials and f1,...,fm are constant vector fields. For single-input systems, it is shown that even-degree homogeneity precludes STLC if the state dimension is larger than one. This, along with the obvious result that for odd-degree homogeneous systems STLC is equivalent to accessibility, provides a complete characterization of STLC for this class of systems. In the multiple-input case, transformations on the input space are applied to homogeneous systems of degree two, an example of this type of system being motion of a rigid-body in a plane. Such input transformations are related via consideration of a tensor on the tangent space to congruence transformation of a matrix to one with zeros on the diagonal. Conditions are given for successful neutralization of bad type (1,2) brackets via congruence transformations.

Published in:

IEEE Transactions on Automatic Control  (Volume:48 ,  Issue: 1 )