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The focus of this paper is the connection between two foundational areas ofLTI systems theory: geometric control and eigenstructure assignment. Inparticular, we study the properties of the null-spaces of the reachabilitymatrix pencil and of the Rosenbrock system matrix, which have been extensivelyused as two computational building blocks for the calculation of pole placingstate feedback matrices and pole placing friends of output-nulling subspaces.Our objective is to show that the subspaces in the chains of kernels obtainedin the construction of these feedback matrices interact with each other in waysthat are entirely independent from the choice of eigenvalues. So far, thesechains of subspaces have only been studied in the case of stationarity. In thiscase, it is known that these chains converge to the classic Kalman reachablesubspace for the reachability matrix pencil and to the largest reachabilitysubspace in the case of the Rosenbrock matrix, respectively. Here we areinterested in showing that even before stationarity has been reached, thepartial chains are linked to structural properties of the system, and aretherefore independent of the closed-loop eigenvalues that we wish to assign. Wefurther characterize these subspaces by investigating the notion of largestsubspace on which it is possible to assign the closed-loop spectrum (possiblymaintaining the output at zero) without resorting to non-trivial Jordan forms.

New geometric results in eigenstructure assignment