Gramian assignment based on the Lyapunov equation

A theorem on the assignability of the controllability Gramian using state feedback is presented. The approach is based on the Lyapunov equation. Specifically, it shows that the set of Gramian matrices which are assignable using state feedback lies in a vector space and is precisely the intersection of that vector space with the set of positive definite matrices. The results show that it may be possible to reshape the structure of the system indirectly by specifying a desired closed-loop Lyapunov function, rather than by directly assigning eigenvalues or eigenvectors.<
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