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.

References

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.2004.834433
3.
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.2004.834113
4.
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.2006.887293
5.
Cortés
,
J.
,
2008
, “
Distributed Algorithms for Reaching Consensus on General Functions
,”
Automatica
,
44
, pp.
726
737
.10.1016/j.automatica.2007.07.022
6.
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.2008.02.036
9.
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.2007.904452
10.
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.2006.10.003
11.
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.2005.275
12.
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.2008.01.011
14.
Boyling
,
J. B.
,
1972
, “
An Axiomatic Approach to Classical Thermodynamics
,”
Proc. R. Soc. London, Ser. A
,
329
(
1576
), pp.
35
70
.10.1098/rspa.1972.0100
15.
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-y
You do not currently have access to this content.