TY - JOUR

T1 - Bispectral analysis for measuring energy-orientation tradeoffs in the control of linear systems

AU - Menolascino, Delsin

AU - Ching, Shi Nung

N1 - Funding Information:
ShiNung Ching holds a Career Award at the Scientific Interface from the Burroughs-Wellcome Fund. This work was partially supported by AFOSR15RT0189, NSF ECCS1509342 and and NSF CMMI1537015, from the US Air Force Office of Scientific Research (AFOSR) and the US National Science Foundation (NSF), respectively.
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - 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.

AB - 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.

KW - Bispectrum

KW - Linear systems

KW - Network analysis

KW - Network control

UR - http://www.scopus.com/inward/record.url?scp=85013335910&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2017.01.001

DO - 10.1016/j.sysconle.2017.01.001

M3 - Article

AN - SCOPUS:85013335910

SN - 0167-6911

VL - 102

SP - 68

EP - 73

JO - Systems and Control Letters

JF - Systems and Control Letters

ER -