Feedback canonical forms of singular systems

@article{Loiseau1991FeedbackCF,
  title={Feedback canonical forms of singular systems},
  author={Jean Jacques Loiseau and Kadri {\"O}zçaldiran and Michel Malabre and Nicos Karcanias},
  journal={Kybernetika},
  year={1991},
  volume={27},
  pages={289-305}
}
Ex = Ax + Bu (1.1) where x e 3C ~ R" e % ~ R, (E, A, B)e R x R x R. Note that E and A are non square. We can always assume without loss of generality that B is monic. In the sequel, (1.1) will be referred to by the triple (E, A, B). Now, let I denote the set of all such (E, A, B)'s. Define T P D = {(W, V, G, FP, FD)eR qXq x R x x R x R x R with W, F a n d G non singular}. The action of this group on I is defined by: 

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