Bispectral analysis for measuring energy-orientation tradeoffs in the control of linear systems

Many characterizations of linear system controllability revolve around the eigenvalue spectrum of the controllability gramian, which is a function of the network dynamics. The gramian spectrum describes the minimum energies associated with inducing movement along orthogonal directions in Rⁿ. Here, we derive an enhanced interpretation of the spectral properties of the gramian in non-minimum energy regimes. Indeed, in a non-minimum energetic regime, an ‘excess’ of energy is available to the system for at least (n−1) orthogonal state transfers. We show that the utility of this excess energy can be quantified in terms of input orientation, or, simply, the angle between two competing inputs. Based on this notion, we derive the gramian bispectrum, which describes the relationship between energy and orientation among pairs of orthogonal state transfers. The bispectrum reflects a fundamental tradeoff between the energetic and orientation costs in the control of a linear system. We show how this bispectral analysis can provide control characterizations that are not apparent from inspection of the gramian spectrum alone.