The zeroth and first laws of thermodynamics define the concepts of thermal equilibrium and thermal energy. The second law of thermodynamics determines whether a particular transfer of thermal energy can occur. Collectively, these fundamental laws of nature imply that a closed collection of thermodynamic subsystems will tend to thermal equilibrium. This paper generalizes the concepts of energy, entropy, and temperature to undirected and directed networks of single integrators, and demonstrates how thermodynamic principles can be applied to the design of distributed consensus control algorithms for networked dynamical systems.
Issue Section:
Research Papers
References
1.
Mesbahi
, M.
, and Egerstedt
, M.
, 2010
, Graph Theoretic Methods in Multiagent Systems
, Princeton University
, Princeton, NJ
.2.
Fax
, J. A.
, and Murray
, R. M.
, 2004
, “Information Flow and Cooperative Control of Vehicle Formations
,” IEEE Trans. Autom. Control.
, 49
(9
), pp. 1465
–1476
.10.1109/TAC.2004.8344333.
Olfati-Saber
, R.
, and Murray
, R. M.
, 2004
, “Consensus Problems in Networks of Agents With Switching Topology and Time-Delays
,” IEEE Trans. Autom. Control
, 49
(9
), pp. 1520
–1533
.10.1109/TAC.2004.8341134.
Olfati-Saber
, R.
, Fax
, J. A.
, and Murray
, R. M.
, 2007
, “Consensus and Cooperation in Networked Multi-Agent Systems
,” Proc. IEEE
, 95
(1
), pp. 215
–233
.10.1109/JPROC.2006.8872935.
Cortés
, J.
, 2008
, “Distributed Algorithms for Reaching Consensus on General Functions
,” Automatica
, 44
, pp. 726
–737
.10.1016/j.automatica.2007.07.0226.
Bullo
, F.
, Cortés
, J.
, and Martnez
, S.
, 2009
, Distributed Control of Robotic Networks
, Princeton University
, Princeton NJ
.7.
Haddad
, W. M.
, Chellaboina
, V. S.
, and Nersesov
, S. G.
, 2005
, Thermodynamics: A Dynamical Systems Approach
, Princeton University
, Princeton, NJ
.8.
Haddad
, W. M.
, and Hui
, Q.
, 2009
, “Complexity, Robustness, Self-Organization, Swarms, and System Thermodynamics
,” Nonlinear Anal.: Real World Appl.
, 10
(1
), pp. 531
–543
.10.1016/j.nonrwa.2008.02.0369.
Haddad
, W. M.
, Chellaboina
, V.
, Hui
, Q.
, and Nersesov
, S. G.
, 2007
, “Energy- and Entropy-Based Stabilization for Lossless Dynamical Systems via Hybrid Controllers
,” IEEE Trans. Autom. Control.
, 52
(9
), pp. 1604
–1614
.10.1109/TAC.2007.90445210.
Haddad
, W. M.
, Hui
, Q.
, Chellaboina
, V.
, and Nersesov
, S. G.
, 2007
, “Hybrid Decentralized Maximum Entropy Control for Large-Scale Dynamical Systems
,” Nonlinear Anal.
, 1
(2
), pp. 244
–263
.10.1016/j.nahs.2006.10.00311.
Haddad
, W. M.
, Hui
, Q.
, Nersesov
, S. G.
, and Chellaboina
, V.
, 2005
, “Thermodynamic Modeling, Energy Equipartition, and Nonconservation of Entropy for Discrete-Time Dynamical Systems
,” Adv. Differ. Equ.
, 2005
(3
), pp. 275
–318
.10.1155/ADE.2005.27512.
Haddad
, W. M.
, and Chellaboina
, V.
, 2008
, Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach
, Princeton University
, Princeton, NJ
.13.
Hui
, Q.
, and Haddad
, W. M.
, 2008
, “Distributed Nonlinear Control Algorithms for Network Consensus
,” Automatica
, 44
(9
), pp. 2375
–2381
.10.1016/j.automatica.2008.01.01114.
Boyling
, J. B.
, 1972
, “An Axiomatic Approach to Classical Thermodynamics
,” Proc. R. Soc. London, Ser. A
, 329
(1576
), pp. 35
–70
.10.1098/rspa.1972.010015.
Kondepudi
, D.
, and Prigogine
, I.
, 1998
, Modern Thermodynamics: From Heat Engines to Dissipative Structures
, Wiley
, New York
.16.
Godsil
, C.
, and Royle
, G. F.
, 2001
, Algebraic Graph Theory
, Springer-Verlag
, Berlin
.17.
Bang-Jensen
, J.
, and Gutin
, G.
, 2002
, Digraphs: Theory, Algorithms and Applications
, Springer
, New York
.18.
Hardy
, G. H.
, Littlewood
, J. E.
, and Pólya
, G.
, 1952
, Inequalities
, Cambridge University
, Cambridge, England
.19.
Vidyasagar
, M.
, 2002
, Nonlinear Systems Analysis: Second Edition
, SIAM
, Philadelphia
.20.
Andelic
, M.
, and da Fonseca
, C. M.
, 2001
, “Sufficient Conditions for Positive Definiteness of Tridiagonal Matrices Revisited
,” Positivity
, 15
(1
), pp. 155
–159
.10.1007/s11117-010-0047-yCopyright © 2013 by ASME
You do not currently have access to this content.
