Controlling chaos is fundamental in many applications, and for this reason, many techniques have been proposed to address this problem. Here, we propose a strategy based on an optimal placement of the sensor and actuator providing global observability of the state space and global controllability to any desired state. The first of these two conditions enables the derivation of a model of the system by using a global modeling technique. In turn, this permits the use of feedback linearization for designing the control law based on the equations of the obtained model and providing a zero-flat system. The procedure is applied to three case studies, including two piecewise linear circuits, namely, the Carroll circuit and the Chua circuit whose governing equations are approximated by a continuous global model. The sensitivity of the procedure to the time constant of the dynamics is also discussed.
REFERENCES
1
C.-T.
Lin
, “Structural controllability
,” IEEE Trans. Autom. Control
19
, 201
–208
(1974
). 2
R.
Kalman
, “On the general theory of control systems
,” in ,” IFAC Proc. Vol.
1
, 491
–502
(1960
). 3
M.
Fliess
, J. L.
Lévine
, P.
Martin
, and P.
Rouchon
, “Flatness and defect of non-linear systems: Introductory theory and examples
,” Int. J. Control
61
, 1327
–1361
(1995
). 4
G.
Franklin
, J.
Powell
, and A.
Emami-Naeini
, Feedback Control of Dynamic Systems
, 8th ed. (Pearson
, 2015
).5
6
7
E. D.
Sontag
, Mathematical Control Theory: Deterministic Finite Dimensional Systems
, 2nd ed. (Springer
, New York
, 1998
).8
R.
Ortega
, “Passivity properties for stabilization of cascaded nonlinear systems
,” Automatica
27
, 423
–424
(1991
). 9
Z.-P.
Jiang
and L.
Praly
, “Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties
,” Automatica
34
, 825
–840
(1998
). 10
V.
Utkin
, Sliding Modes in Control and Optimization, Communications and Control Engineering (Springer-Verlag, Berlin, 1992).11
L. A.
Aguirre
and C.
Letellier
, “Modeling nonlinear dynamics and chaos: A review
,” Math. Probl. Eng.
2009
, 238960
(2009
). 12
S.
Billings
and I. J.
Leontaritis
, “Parameter estimation techniques for nonlinear systems
,” IFAC Proc. Vol.
15
, 505
–510
(1982
). 13
J. P.
Crutchfield
and B. S.
McNamara
, “Equations of motion from a data series
,” Complex Syst.
1
, 417
–452
(1987
).14
J.
Cremers
and A.
Hübler
, “Construction of differential equations from experimental data
,” Z. Naturforsch. A
42
, 797
–802
(1987
). 15
G.
Gouesbet
and C.
Letellier
, “Global vector-field reconstruction by using a multivariate polynomial approximation on nets
,” Phys. Rev. E
49
, 4955
–4972
(1994
). 16
S.
Mangiarotti
, R.
Coudret
, L.
Drapeau
, and L.
Jarlan
, “Polynomial search and global modeling: Two algorithms for modeling chaos
,” Phys. Rev. E
86
, 046205
(2012
). 17
S.
Mangiarotti
and M.
Huc
, “Can the original equations of a dynamical system be retrieved from observational time series?
,” Chaos
29
, 023133
(2019
). 18
R.
Rico-Martínez
, K.
Krischer
, I. G.
Kevrekidis
, M. C.
Kube
, and J. L.
Hudson
, “Discrete-vs continuous time nonlinear signal processing of Cu electrodissolution data
,” Chem. Eng. Commun.
118
, 25
–48
(1992
). 19
C.
Letellier
, L. L.
Sceller
, P.
Dutertre
, G.
Gouesbet
, Z.
Fei
, and J. L.
Hudson
, “Topological characterization and global vector field reconstruction from an experimental electrochemical system
,” J. Phys. Chem.
99
, 7016
–7027
(1995
). 20
C.
Letellier
, J.
Maquet
, H.
Labro
, L. L.
Sceller
, G.
Gouesbet
, F.
Argoul
, and A.
Arnéodo
, “Analyzing chaotic behaviour in a Belousov-Zhabotinskii reaction by using a global vector field reconstruction
,” J. Phys. Chem. A
102
, 10265
–10273
(1998
). 21
C.
Letellier
, L. L.
Sceller
, G.
Gouesbet
, F.
Lusseyran
, A.
Kemoun
, and B.
Izrar
, “Recovering deterministic behavior from experimental time series in mixing reactor
,” AIChE J.
43
, 2194
–2202
(1997
). 22
R.
Brown
, N. F.
Rulkov
, and E. R.
Tracy
, “Modeling and synchronizing chaotic systems from experimental data
,” Phys. Lett. A
194
, 71
–76
(1994
). 23
J.
Maquet
, C.
Letellier
, and L. A.
Aguirre
, “Global models from the Canadian lynx cycles as a first evidence for chaos in real ecosystems
,” J. Math. Biol.
55
, 21
–39
(2007
). 24
L. A.
Aguirre
, C.
Letellier
, and J.
Maquet
, “Forecasting the time series of sunspot numbers
,” Sol. Phys.
249
, 103
–120
(2008
). 25
S.
Mangiarotti
, L.
Drapeau
, and C.
Letellier
, “Two chaotic global models for cereal crops cycles observed from satellite in northern Morocco
,” Chaos
24
, 023130
(2014
). 26
S.
Mangiarotti
, M.
Peyre
, and M.
Huc
, “A chaotic model for the epidemic of Ebola virus disease in West Africa (2013–2016)
,” Chaos
26
, 113112
(2016
). 27
C.
Letellier
, R.
Abraham
, D. L.
Shepelyansky
, O. E.
össler
, P.
Holmes
, R.
Lozi
, L.
Glass
, A.
Pikovsky
, L. F.
Olsen
, I.
Tsuda
, C.
Grebogi
, U.
Parlitz
, R.
Gilmore
, L. M.
Pecora
, and T. L.
Carroll
, “Some elements for a history of the dynamical systems theory
,” Chaos
31
, 053110
(2021
). 28
C.
Letellier
, P.
Dutertre
, and B.
Maheu
, “Unstable periodic orbits and templates of the Rössler system: Toward a systematic topological characterization
,” Chaos
5
, 271
–282
(1995
). 29
T. L.
Carroll
, “A simple circuit for demonstrating regular and synchronized chaos
,” Am. J. Phys.
63
, 377
–379
(1995
). 30
C.
Letellier
and J.-P.
Barbot
, “Flatness for an optimal control of chaotic systems using a minimal numbers of sensors and actuators
,” Chaos
31
, 103114
(2021
). 31
C.
Letellier
, L.
Minati
, and J.-P.
Barbot
, “Optimal placement of sensor and actuator for controlling the piecewise linear Chua circuit via a discretized controller
,” J. Differ. Discrete Equ.
(in press) (2022
).32
R.
Hermann
and A.
Krener
, “Nonlinear controllability and observability
,” IEEE Trans. Autom. Control
22
, 728
–740
(1977
). 33
C.
Lobry
, “Contrôlabilité des systèmes non linéaires
,” SIAM J. Control Optim.
8
, 573
–605
(1970
). 34
H. J.
Sussmann
and V.
Jurdjevic
, “Controllability of nonlinear systems
,” J. Differ. Equ.
12
, 95
–116
(1972
). 35
B.
Jakubczyk
and W.
Respondek
, “On linearization of control systems
,” Bull. Acad. Pol. Sér. Sci. Math.
28
, 517
–522
(1980
).36
L.
Hunt
, Renjeng
Su
, and G.
Meyer
, “Global transformations of nonlinear systems
,” IEEE Trans. Autom. Control
28
, 24
–31
(1983
). 37
A.
Isidori
, A.
Krener
, C.
Gori-Giorgi
, and S.
Monaco
, “Nonlinear decoupling via feedback: A differential geometric approach
,” IEEE Trans. Autom. Control
26
, 331
–345
(1981
). 38
J.
Pathak
, Z.
Lu
, B. R.
Hunt
, M.
Girvan
, and E.
Ott
, “Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data
,” Chaos
27
, 121102
(2017
). 39
P.
Rajendra
and V.
Brahmajirao
, “Modeling of dynamical systems through deep learning
,” Biophys. Rev.
12
, 1311
–1320
(2020
). 40
L. A.
Aguirre
, U. S.
Freitas
, C.
Letellier
, L. L.
Sceller
, and J.
Maquet
, “State space parsimonious reconstruction of attractor produced by an electronic oscillator
,” AIP Conf. Proc.
502
, 649
–654
(2000
). 41
L. A.
Aguirre
, “Some remarks on structure selection for nonlinear models
,” Int. J. Bifurcation Chaos
04
, 1707
–1714
(1994
). 42
C. S. M.
Lainscsek
, C.
Letellier
, and F.
Schürrer
, “Ansatz library for global modeling with a structure selection
,” Phys. Rev. E
64
, 016206
(2001
). 43
D. A.
Smirnov
, B. P.
Bezruchko
, and Y. P.
Seleznev
, “Choice of dynamical variables for global reconstruction of model equations from time series
,” Phys. Rev. E
65
, 026205
(2002
). 44
S.
Chen
, S. A.
Billings
, and W.
Luo
, “Orthogonal least squares methods and their application to non-linear system identification
,” Int. J. Control
50
, 1873
–1896
(1989
). 45
J. L.
Breeden
and A.
Hübler
, “Reconstructing equations of motion from experimental data with unobserved variables
,” Phys. Rev. A
42
, 5817
–5826
(1990
). 46
M.
Giona
, F.
Lentini
, and V.
Cimagalli
, “Functional reconstruction and local prediction of chaotic time series
,” Phys. Rev. A
44
, 3496
–3502
(1991
). 47
L. A.
Aguirre
and S. A.
Billings
, “Retrieving dynamical invariants from chaotic data using NARMAX models
,” Int. J. Bifurcation Chaos
5
, 449
–474
(1995
). 48
K.
Judd
and A.
Mees
, “Embedding as a modeling problem
,” Physica D
120
, 273
–286
(1998
). 49
C.
Letellier
and J.-M.
Ginoux
, “Development of the nonlinear dynamical systems theory from radio-engineering to electronics
,” Int. J. Bifurcation Chaos
19
, 2131
–2163
(2009
). 50
C.
Letellier
and L. A.
Aguirre
, “Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables
,” Chaos
12
, 549
–558
(2002
). 51
C.
Letellier
, L.
Aguirre
, and J.
Maquet
, “How the choice of the observable may influence the analysis of nonlinear dynamical systems
,” Commun. Nonlinear Sci. Numer. Simul.
11
, 555
–576
(2006
). 52
C.
Letellier
, L. A.
Aguirre
, and J.
Maquet
, “Relation between observability and differential embeddings for nonlinear dynamics
,” Phys. Rev. E
71
, 066213
(2005
). 53
H. D. I.
Abarbanel
and M. B.
Kennel
, “Local false nearest neighbors and dynamical dimensions from observed chaotic data
,” Phys. Rev. E
47
, 3057
–3068
(1993
). 54
L.
Cao
, “Practical method for determining the minimum embedding dimension of a scalar time series
,” Phys. D
110
, 43
–50
(1997
). 55
See https://cran.r-project.org/web/packages/GPoM/index.html where all packages can be found
.56
C.
Letellier
, “Caractérisation topologique et reconstruction des attracteurs étranges,” Ph.D. thesis (University of Paris VII, Paris, France, 1994).57
O. E.
Rössler
, “An equation for continuous chaos
,” Phys. Lett. A
57
, 397
–398
(1976
). 58
R. W.
Brockett
and C. I.
Byrnes
, “Multivariable Nyquist criteria, root loci, and pole placement: A geometric viewpoint
,” IEEE Trans. Autom. Control
26
, 271
–284
(1981
). 59
C.
Letellier
, S.
Mangiarotti
, I.
Sendiña-Nadal
, and O. E.
Rössler
, “Topological characterization versus synchronization for assessing (or not) dynamical equivalence
,” Chaos
28
, 045107
(2018
). 60
P.
Dorato
, “A historical review of robust control
,” IEEE Control Syst. Mag.
7
, 44
–47
(1987
). 61
T.
Matsumoto
, “A chaotic attractor from Chua’s circuit
,” IEEE Trans. Circuits Syst.
CAS-31
, 1055
–1058
(1984
). 62
T.
Matsumoto
, L. O.
Chua
, and M.
Komuro
, “The double scroll
,” IEEE Trans. Circuits Syst.
CAS-32
, 798
–818
(1985
). 63
M.
Rosalie
and C.
Letellier
, “Systematic template extraction from chaotic attractors: I. Genus-one attractors with an inversion symmetry
,” J. Phys. A
46
, 375101
(2013
). 64
R. W.
Ghrist
and P. J.
Holmes
, “An ODE whose solutions contain all knots and links
,” Int. J. Bifurcations Chaos
6
, 779
–800
(1996
). 65
R. W.
Ghrist
, “Branched two-manifolds supporting all links
,” Topology
36
, 423
–448
(1997
). 66
C.
Letellier
and R.
Gilmore
, “The universal template is a subtemplate of the double-scroll template
,” J. Phys. A
46
, 065102
(2013
). 67
C.
Letellier
, P.
Dutertre
, J.
Reizner
, and G.
Gouesbet
, “Evolution of multimodal map induced by an equivariant vector field
,” J. Phys. A
29
, 5359
–5373
(1996
). 68
A. I.
Khitnik
, D.
Rose
, and L. O.
Chua
, “On periodic orbits and homoclinic bifurcations in Chua’s circuit with a smooth non linearity
,” Int. J. Bifurcations Chaos
3
, 363
–384
(1993
). 69
E. N.
Lorenz
, “Deterministic nonperiodic flow
,” J. Atmos. Sci.
20
, 130
–141
(1963
). 70
C.
Letellier
and R.
Gilmore
, “Covering dynamical systems: Two-fold covers
,” Phys. Rev. E
63
, 016206
(2001
). 71
L.
Rosier
, “Homogeneous Lyapunov function for homogeneous continuous vector field
,” Syst. Control Lett.
19
, 467
–473
(1992
). 72
A.
Polyakov
, “On homogeneous controllability functions
,” Visnyk V. N. Karazin Kharkiv Nat. Univ. Ser. Math. Appl. Math. Mech.
94
, 24
–39
(2021
). 73
C.
Letellier
, E.
Ringuet
, B.
Maheu
, J.
Maquet
, and G.
Gouesbet
, “Global vector field reconstruction of chaotic attractors from one unstable periodic orbit
,” Entropie
202/203
, 147
–153
(1997
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
