Given a general local differentiable family of pairs of matrices, we obtain
a local differentiable family of feedbacks solving the pole assignment
problem, that is to say, shifting the spectrum into a prefixed one. We point
out that no additional hypothesis is needed. In fact, simple approaches
work in particular cases (controllable pairs, constancy of the dimension of
the controllable subspace, and so on). Here the general case is proved by
means of Arnold’s techniques: the key point is to re...
Given a general local differentiable family of pairs of matrices, we obtain
a local differentiable family of feedbacks solving the pole assignment
problem, that is to say, shifting the spectrum into a prefixed one. We point
out that no additional hypothesis is needed. In fact, simple approaches
work in particular cases (controllable pairs, constancy of the dimension of
the controllable subspace, and so on). Here the general case is proved by
means of Arnold’s techniques: the key point is to reduce the construction to
a versal deformation of the central pair; in fact to a quite singular
miniversal one for which the family of feedbacks can be explicitly
constructed. As a direct application, a differentiable family of stabilizing
feedbacks is obtained, provided that the central pair is stabilizable.
Citation
Compta, A.; Ferrer, J.; Peña, M. Local differentiable pole assignment. "Linear and multilinear algebra", 2010, vol. 58, núm. 5, p. 563-569.