The trouble with imaginary eigenvalues. Optimal control, controllability, frequency response

Abstract

We investigate a few problems in the theory of linear time-invariant systems that arise when the open-loop system possesses imaginary eigenvalues. The problems are: (1) the infinite horizon, minimum energy optimal control problem; (2) the matter of defining the controllability Gramian for systems that are not stable; (3) the regularity as a distribution of the Fourier transform of the state impulse response. Apart from separately investigating these problems, we also indicate how they are interconnected.

There is an error in the manner in which the stabilising feedback is implemented in the case of multiple canonical blocks