Summary
The geometry of the space of real, proper, rational functions of a fixed degree and without common factors has been of interest in system theory for some time because of the central role transfer functions play in modeling linear time invariant systems. The 2n-dimensional manifold of real proper rational functions of degree n can also be identified with the product of the set of (2n − 1)-dimensional manifold of n-by-n real nonsingular Hankel matrices and the real line. The distinct possibilities for the signature of a nonsingular n-by-n Hankel matrix serves to characterize the distinct connected components of the correspond set of rational functions and, at the same time, serve to decompose the space into connected components. In this paper we consider the construction of the de Rham cohomology of the n-by-n real nonsingular Hankel matrices of signature n − 2 as a further step in the quest for more useful parameterizations of various families of rational functions.
This work was supported in part by the US Army Research Office under grant DAAG 55 97 1 0114.
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Brockett, R.W. (2010). Rational Functions and Flows with Periodic Solutions. In: Hu, X., Jonsson, U., Wahlberg, B., Ghosh, B. (eds) Three Decades of Progress in Control Sciences. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11278-2_4
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