Abstract

Large-scale dynamical systems, no matter whether possessing interconnected appearances, are frequently modeled as networks. For instance, graphs, multi-agent systems, and materials' intricate behaviors are often treated as networked dynamical systems. However, only a few studies have approached the problem in the frequency domain, mostly due to the complexity of evaluating their frequency response. That gap is filled by this paper, which proposes algorithms computing a general class of self-similar networks' frequency response and transfer functions, no matter they are finite or infinite, damaged or undamaged. In addition, this paper shows that for infinite self-similar networks, even when they are damaged, fractional-order and irrational dynamics naturally come into sight. Most importantly, this paper illustrates that for a network under different operating conditions, its frequency response would form a set of neighboring plants, which sets the basis of applying robust control methods to dynamic networks.

References

1.
Biggs
,
N. L.
,
Lloyd
,
E. K.
, and
Wilson
,
R. J.
,
1976
,
Graph Theory 1736-1936
,
Clarendon Press
,
Oxford
.
2.
Newman
,
M. E. J.
,
2003
, “
The Structure and Function of Complex Networks
,”
SIAM Rev.
,
45
(
2
), pp.
167
256
.10.1137/S003614450342480
3.
Bejan
,
A.
, and
Merkx
,
G. W.
,
2007
,
Constructal Theory of Social Dynamics
,
Springer
, Berlin.
4.
Huberman
,
B. A.
,
2001
,
The Laws of the Web: Patterns in the Ecology of Information
,
MIT Press
,
Cambridge, MA
.
5.
Wang
,
S.
, and
Ran
,
C.
,
2016
, “
Rethinking Cellular Network Planning and Optimization
,”
IEEE Wireless Commun.
,
23
(
2
), pp.
118
125
.10.1109/MWC.2016.7462493
6.
Stelling
,
J.
,
Klamt
,
S.
,
Bettenbrock
,
K.
,
Schuster
,
S.
, and
Gilles
,
E. D.
,
2002
, “
Metabolic Network Structure Determines Key Aspects of Functionality and Regulation
,”
Nature
,
420
(
6912
), pp.
190
193
.10.1038/nature01166
7.
Erdős
,
P.
, and
Rényi
,
A.
,
1960
, “
On the Evolution of Random Graphs
,”
Publ. Math. Inst. Hung. Acad. Sci.
,
5
(
1
), pp.
17
60
.http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.153.5943
8.
Frank
,
O.
, and
Strauss
,
D.
,
1986
, “
Markov Graphs
,”
J. Am. Stat. Assoc.
,
81
(
395
), pp.
832
842
.10.1080/01621459.1986.10478342
9.
Watts
,
D. J.
, and
Strogatz
,
S. H.
,
1998
, “
Collective Dynamics of ‘Small-World’ Networks
,”
Nature
,
393
(
6684
), pp.
440
442
.10.1038/30918
10.
Barabási
,
A.-L.
, and
Albert
,
R.
,
1999
, “
Emergence of Scaling in Random Networks
,”
Science
,
286
(
5439
), pp.
509
512
.10.1126/science.286.5439.509
11.
Pastor-Satorras
,
R.
,
Castellano
,
C.
,
Van Mieghem
,
P.
, and
Vespignani
,
A.
,
2015
, “
Epidemic Processes in Complex Networks
,”
Rev. Mod. Phys.
,
87
(
3
), pp.
925
979
.10.1103/RevModPhys.87.925
12.
Chen
,
F.
, and
Ren
,
W.
,
2019
, “
On the Control of Multi-Agent Systems: A Survey
,”
Found. Trends Syst. Control
,
6
(
4
), pp.
339
499
.10.1561/2600000019
13.
Ren
,
W.
,
Beard
,
R. W.
, and
Atkins
,
E. M.
,
2005
, “
A Survey of Consensus Problems in Multi-Agent Coordination
,”
Proceedings of American Control Conference
, Vol. 3, Portland, OR, June 8–10, pp.
1859
1864
.10.1109/ACC.2005.1470239
14.
Choi
,
H.-L.
,
Brunet
,
L.
, and
How
,
J. P.
,
2009
, “
Consensus-Based Decentralized Auctions for Robust Task Allocation
,”
IEEE Trans. Rob.
,
25
(
4
), pp.
912
926
.10.1109/TRO.2009.2022423
15.
Aragues
,
R.
,
Cortes
,
J.
, and
Sagues
,
C.
,
2012
, “
Distributed Consensus on Robot Networks for Dynamically Merging Feature-Based Maps
,”
IEEE Trans. Rob.
,
28
(
4
), pp.
840
854
.10.1109/TRO.2012.2192012
16.
Rubenstein
,
M.
,
Cabrera
,
A.
,
Werfel
,
J.
,
Habibi
,
G.
,
McLurkin
,
J.
, and
Nagpal
,
R.
,
2013
, “
Collective Transport of Complex Objects by Simple Robots: Theory and Experiments
,”
Proceedings of the International Conference on Autonomous Agents and Multi-Agent Systems
, Saint Paul, MN, May 6–10, pp.
47
54
.https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.304.4575&rep=rep1&type=pdf
17.
Schenato
,
L.
, and
Fiorentin
,
F.
,
2011
, “
Average Timesynch: A Consensus-Based Protocol for Clock Synchronization in Wireless Sensor Networks
,”
Automatica
,
47
(
9
), pp.
1878
1886
.10.1016/j.automatica.2011.06.012
18.
Spanos
,
D. P.
,
Olfati-Saber
,
R.
, and
Murray
,
R. M.
,
2005
, “
Distributed Sensor Fusion Using Dynamic Consensus
,”
IFAC World Congress
,
Citeseer
, Prague, Czech Republic, July 3–8, pp.
1
6
.
19.
Simonetto
,
A.
, and
Leus
,
G.
,
2014
, “
Distributed Maximum Likelihood Sensor Network Localization
,”
IEEE Trans. Signal Process.
,
62
(
6
), pp.
1424
1437
.10.1109/TSP.2014.2302746
20.
Forero
,
P. A.
,
Cano
,
A.
, and
Giannakis
,
G. B.
,
2010
, “
Consensus-Based Distributed Support Vector Machines
,”
J. Mach. Learn. Res.
,
11
(
5
), pp.
1663
1707
.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.407.6366&rep=rep1&type=pdf
21.
Hatanaka
,
T.
,
Zhang
,
X.
,
Shi
,
W.
,
Zhu
,
M.
, and
Li
,
N.
,
2017
, “
Physics-Integrated Hierarchical/Distributed HVAC Optimization for Multiple Buildings With Robustness Against Time Delays
,” IEEE 56th Annual Conference on Decision and Control (
CDC
), Melbourne, Australia, Dec. 12–15, pp.
6573
6579
.10.1109/CDC.2017.8264650
22.
Mlynek
,
P.
,
Misurec
,
J.
,
Koutny
,
M.
, and
Silhavy
,
P.
,
2012
, “
Two-Port Network Transfer Function for Power Line Topology Modeling
,”
Radioengineering
,
21
(
1
), pp.
356
363
.https://www.radioeng.cz/fulltexts/2012/12_01_0356_0363.pdf
23.
Zhao
,
L.
,
Guo
,
J.
,
Li
,
H.
, and
Liu
,
W.
,
2009
, “
The Simulation Analysis of Influence on Jointless Track Circuit Signal Transmission From Compensation Capacitor Based on Transmission-Line Theory
,”
Third IEEE International Symposium on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications
, Beijing, China, Oct. 27–29, pp.
1113
1118
.
24.
Sandidzadeh
,
M. A.
, and
Dehghani
,
M.
,
2013
, “
Intelligent Condition Monitoring of Railway Signaling in Train Detection Subsystems
,”
J. Intell. Fuzzy Syst.
,
24
(
4
), pp.
859
869
.10.3233/IFS-2012-0604
25.
Chen
,
J.
,
Roberts
,
C.
, and
Weston
,
P.
,
2008
, “
Fault Detection and Diagnosis for Railway Track Circuits Using Neuro-Fuzzy Systems
,”
Control Eng. Pract.
,
16
(
5
), pp.
585
596
.10.1016/j.conengprac.2007.06.007
26.
Karimi
,
H. R.
,
Palacios-Quiñonero
,
F.
,
Rossell
,
J. M.
, and
Rubió-Massegú
,
J.
,
2013
, “
Sequential Design of Multioverlapping Controllers for Structural Vibration Control of Tall Buildings Under Seismic Excitation
,”
Proc. Inst. Mech. Eng., Part I J. Syst. Control Eng.
,
227
(
2
), pp.
176
183
.10.1177/0959651812464026
27.
Fu
,
T. S.
, and
Johnson
,
E. A.
,
2011
, “
Distributed Mass Damper System for Integrating Structural and Environmental Controls in Buildings
,”
J. Eng. Mech.
,
137
(
3
), pp.
205
213
.10.1061/(ASCE)EM.1943-7889.0000211
28.
Gudarzi
,
M.
,
2015
, “
μ-Synthesis Controller Design for Seismic Alleviation of Structures With Parametric Uncertainties
,”
J. Low Freq. Noise, Vib. Active Control
,
34
(
4
), pp.
491
511
.10.1260/0263-0923.34.4.491
29.
Hoang
,
N.
,
Fujino
,
Y.
, and
Warnitchai
,
P.
,
2008
, “
Optimal Tuned Mass Damper for Seismic Applications and Practical Design Formulas
,”
Eng. Struct.
,
30
(
3
), pp.
707
715
.10.1016/j.engstruct.2007.05.007
30.
Jonscher
,
A. K.
,
1977
, “
The ‘Universal’ Dielectric Response
,”
Nature
,
267
(
5613
), pp.
673
679
.10.1038/267673a0
31.
Jonscher
,
A. K.
,
1999
, “
Dielectric Relaxation in Solids
,”
J. Phys. D Appl. Phys.
,
32
(
14
), pp.
R57
R70
.10.1088/0022-3727/32/14/201
32.
McCullen
,
N. J.
,
Almond
,
D. P.
,
Budd
,
C. J.
, and
Hunt
,
G. W.
,
2009
, “
The Robustness of the Emergent Scaling Property of Random rc Network Models of Complex Materials
,”
J. Phys. D Appl. Phys.
,
42
(
6
), p.
064001
.10.1088/0022-3727/42/6/064001
33.
Almond
,
D. P.
,
Budd
,
C. J.
,
Freitag
,
M. A.
,
Hunt
,
G. W.
,
McCullen
,
N. J.
, and
Smith
,
N. D.
,
2013
, “
The Origin of Power-Law Emergent Scaling in Large Binary Networks
,”
Phys. A Stat. Mech. Appl.
,
392
(
4
), pp.
1004
1027
.10.1016/j.physa.2012.10.035
34.
Zhai
,
C.
,
Hanaor
,
D.
, and
Gan
,
Y.
,
2017
, “
Universality of the Emergent Scaling in Finite Random Binary Percolation Networks
,”
PLoS One
,
12
(
2
), p.
e0172298
.10.1371/journal.pone.0172298
35.
Murphy
,
K. D.
,
Hunt
,
G. W.
, and
Almond
,
D. P.
,
2006
, “
Evidence of Emergent Scaling in Mechanical Systems
,”
Philos. Mag.
,
86
(
21–22
), pp.
3325
3338
.10.1080/14786430500197934
36.
Buła
,
D.
,
Grabowski
,
D.
,
Lewandowski
,
M.
,
Maciążek
,
M.
, and
Piwowar
,
A.
,
2020
, “
Software Solution for Modeling, Sizing, and Allocation of Active Power Filters in Distribution Networks
,”
Energies
,
14
(
1
), p.
133
.10.3390/en14010133
37.
Nasir
,
M.
,
Hayat
,
M. F.
,
Jamal
,
A.
, and
Ahmed
,
Z.
,
2021
, “
Frequency Domain Consensus Control Analysis of the Networked Multi-Agent System With Controller Area Network Bus-Induced Delay
,”
J. Vib. Control
, epub.10.1177/10775463211022476
38.
Ahmed
,
Z.
,
Saeed
,
M. A.
,
Jenabzadeh
,
A.
, and
Weidong
,
Z.
,
2021
, “
Frequency Domain Analysis of Resilient Consensus in Multi-Agent Systems Subject to an Integrity Attack
,”
ISA Trans.
,
111
, pp.
156
170
.10.1016/j.isatra.2020.11.014
39.
Leyden
,
K.
,
Sen
,
M.
, and
Goodwine
,
B.
,
2019
, “
Large and Infinite Mass–Spring–Damper Networks
,”
ASME J. Dyn. Syst., Meas., Control
,
141
(
6
), p.
061005
.10.1115/1.4042466
40.
Guel-Cortez
,
A.-J.
,
Méndez-Barrios
,
C.-F.
,
Kim
,
E-J.
, and
Sen
,
M.
,
2021
, “
Fractional-Order Controllers for Irrational Systems
,”
IET Control Theory Appl.
,
15
(
7
), pp.
965
977
.10.1049/cth2.12095
41.
Sabatier
,
J.
,
2020
, “
Beyond the Particular Case of Circuits With Geometrically Distributed Components for Approximation of Fractional Order Models: Application to a New Class of Model for Power Law Type Long Memory Behaviour Modelling
,”
J. Adv. Res.
,
25
, pp.
243
255
.10.1016/j.jare.2020.04.004
42.
Nigmatullin
,
R. R.
, and
Baleanu
,
D.
,
2010
, “
Is It Possible to Derive Newtonian Equations of Motion With Memory?
,”
Int. J. Theor. Phys.
,
49
(
4
), pp.
701
708
.10.1007/s10773-010-0249-x
43.
Doehring
,
T. C.
,
Freed
,
A. D.
,
Carew
,
E. O.
, and
Vesely
,
I.
,
2005
, “
Fractional Order Viscoelasticity of the Aortic Valve Cusp: An Alternative to Quasilinear Viscoelasticity
,”
ASME J. Biomech. Eng.
,
127
(
4
), pp.
700
708
.10.1115/1.1933900
44.
Heymans
,
N.
, and
Bauwens
,
J.-C.
,
1994
, “
Fractal Rheological Models and Fractional Differential Equations for Viscoelastic Behavior
,”
Rheol. Acta
,
33
(
3
), pp.
210
219
.10.1007/BF00437306
45.
Ni
,
X.
,
2021
, “
Frequency Response of Self-Similar Dynamic Networks With Applications to Health Monitoring and Control
,” Ph.D. thesis,
University of Notre Dame
, Notre Dame, IN.
46.
Goodwine
,
B.
,
2014
, “
Modeling a Multi-Robot System With Fractional-Order Differential Equations
,” IEEE International Conference on Robotics and Automation (
ICRA
), Hong Kong, China, May 31–June 7, pp.
1763
1768
.10.1109/ICRA.2014.6907089
47.
Leyden
,
K.
, and
Goodwine
,
B.
,
2016
, “
Using Fractional-Order Differential Equations for Health Monitoring of a System of Cooperating Robots
,” IEEE International Conference on Robotics and Automation (
ICRA
), Stockholm, Sweden, May 16–21, pp.
366
371
.10.1109/ICRA.2016.7487154
48.
Mayes
,
J.
,
2012
, “
Reduction and Approximation in Large and Infinite Potential-Driven Flow Networks
,” Ph.D. thesis,
University of Notre Dame
, Notre Dame, IN.
49.
Chen
,
Y.
,
Petráš
,
I.
, and
Xue
,
D.
,
2009
, “
Fractional Order Control - A Tutorial
,”
American Control Conference
, St. Louis, MO, June 10–12, pp.
1397
1411
.10.1109/ACC.2009.5160719
50.
McClure
,
T.
,
2013
, “
Numerical Inverse Laplace Transform
,” accessed 27 June, 2021, https://www.mathworks.com/matlabcentral/fileexchange/39035-numerical-inverse-laplace-transform
51.
Joseph Abate
,
W. W. A.
,
2006
, “
Unified Framework for Numerically Inverting Laplace Transforms
,”
INFORMS J. Comput.
, 18(4), pp.
408
421
.10.1287/ijoc.1050.0137
You do not currently have access to this content.