Abstract

This article proposes a nonparametric system identification technique to discover the governing equation of nonlinear dynamic systems with the focus on practical aspects. The algorithm builds on Brunton’s work in 2016 and combines the sparse regression with an algebraic calculus to estimate the required derivatives of the measurements. This reduces the required derivative data for the system identification. Furthermore, we make use of the concepts of K-fold cross validation from machine learning and information criteria for model selection. This allows the system identification with less measurements than the typically required data for the sparse regression. The result is an optimal model for the underlining system of the data with a minimum number of terms. The proposed nonparametric system identification method is applicable for multiple-input–multiple-output systems. Two examples are presented to demonstrate the proposed method. The first one makes use of the simulated data of a nonlinear oscillator to show the effectiveness and accuracy of the proposed method. The second example is a nonlinear rotary flexible beam. Experimental responses of the beam are used to identify the underlining model. The Coulomb friction in the servo motor together with other nonlinear terms of the system variables are found to be important components of the model. These are, otherwise, not available in the theoretical linear model of the system.

References

References
1.
Gautrais
,
J.
,
Ginelli
,
F.
,
Fournier
,
R.
,
Blanco
,
S.
,
Soria
,
M.
,
Hugues
,
C.
, and
Guy
,
T.
,
2012
, “
Deciphering Interactions in Moving Animal Groups
,”
PLOS Comput. Biol
,
8
(
9
), p.
e1002678
. 10.1371/journal.pcbi.1002678
2.
Zienkiewicz
,
A.
,
Barton
,
D. A. W.
,
Porfiri
,
M.
, and
di Bernardo
,
M.
,
2015
, “
Data-Driven Stochastic Modelling of Zebrafish Locomotion
,”
J. Math. Biol.
,
71
(
5
), pp.
1081
1105
. 10.1007/s00285-014-0843-2
3.
Burbano
,
L.
,
Daniel
,
A.
, and
Porfiri
,
M.
,
2020
, “
Data-Driven Modeling of Zebrafish Behavioral Response to Acute Caffeine Administration
,”
J. Theor. Biol.
,
485
, p.
110054
. 10.1016/j.jtbi.2019.110054
4.
Hernandez
,
A. M.
,
Casado Magana
,
E. J.
, and
Berna
,
A. G.
,
2018
, “
Data-Driven Aircraft Trajectory Predictions Using Ensemble Meta-Estimators
,”
Proceedings of the 37th Digital Avionics Systems Conference
,
London, UK
,
Sept. 23–27
, pp.
1
10
.
5.
Altchë
,
F.
, and
de La Fortelle
,
A.
,
2017
, “
An LSTM Network for Highway Trajectory Prediction
,”
Proceedings of the 20th International Conference on Intelligent Transportation Systems
, pp.
353
359
.
6.
Wang
,
W.-X.
,
Lai
,
Y.-C.
, and
Grebogi
,
C.
,
2016
, “
Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems
,”
Phys. Rep.
,
644
, pp.
1
76
. 10.1016/j.physrep.2016.06.004
7.
Ljung
,
L.
,
2010
, “
Perspectives on System Identification
,”
Ann. Rev. Control
,
34
(
1
), pp.
1
12
. 10.1016/j.arcontrol.2009.12.001
8.
Noël
,
J. P.
, and
Kerschen
,
G.
,
2017
, “
Nonlinear System Identification in Structural Dynamics: 10 More Years of Progress
,”
Mech. Syst. Signal Proc.
,
83
, pp.
2
35
. 10.1016/j.ymssp.2016.07.020
9.
Schoukens
,
J.
,
Vaes
,
M.
, and
Pintelon
,
R.
,
2016
, “
Linear System Identification in a Nonlinear Setting: Nonparametric Analysis of the Nonlinear Distortions and Their Impact on the Best Linear Approximation
,”
IEEE Control Syst. Magaz.
,
36
(
3
), pp.
38
69
. 10.1109/MCS.2016.2535918
10.
Sracic
,
M. W.
, and
Allen
,
M. S.
,
2014
, “
Identifying Parameters of Multi-Degree-of-Freedom Nonlinear Structural Dynamic Systems Using Linear Time Periodic Approximations
,”
Mechan. Syst. Signal Process.
,
46
(
2
), pp.
325
343
. 10.1016/j.ymssp.2014.01.014
11.
Moaveni
,
B.
, and
Asgarieh
,
E.
,
2012
, “
Deterministic-Stochastic Subspace Identification Method for Identification of Nonlinear Structures As Time-Varying Linear Systems
,”
Mech. Syst. Signal Process.
,
31
, pp.
40
55
. 10.1016/j.ymssp.2012.03.004
12.
Lai
,
Z.
, and
Nagarajaiah
,
S.
,
2019
, “
Sparse Structural System Identification Method for Nonlinear Dynamic Systems With Hysteresis/Inelastic Behavior
,”
Mech. Syst. Signal Process.
,
117
, pp.
813
842
. 10.1016/j.ymssp.2018.08.033
13.
Haroon
,
M.
, and
Adams
,
D. E.
,
2009
, “
A Modified H2 Algorithm for Improved Frequency Response Function and Nonlinear Parameter Estimation
,”
J. Sound. Vib.
,
320
(
4
), pp.
822
837
. 10.1016/j.jsv.2008.09.015
14.
Magnevall
,
M.
,
Josefsson
,
A.
,
Ahlin
,
K.
, and
Broman
,
G.
,
2012
, “
Nonlinear Structural Identification by the Reverse Path Spectral Method
,”
J. Sound. Vib.
,
331
(
4
), pp.
938
946
. 10.1016/j.jsv.2011.10.029
15.
Vazquez Feijoo
,
J. A.
,
Worden
,
K.
,
Montes Garcia
,
P.
,
Lagunez Rivera
,
L.
,
Juarez Rodriguez
,
N.
, and
Pech Pérez
,
A.
,
2010
, “
Analysis of MDOF Nonlinear Systems Using Associated Linear Equations
,”
Mech. Syst. Signal Proc.
,
24
(
8
), pp.
2824
2843
. 10.1016/j.ymssp.2010.04.008
16.
Noël
,
J. P.
, and
Kerschen
,
G.
,
2013
, “
Frequency-Domain Subspace Identification for Nonlinear Mechanical Systems
,”
Mech. Syst. Signal Process.
,
40
(
2
), pp.
701
717
. 10.1016/j.ymssp.2013.06.034
17.
Noël
,
J.-P.
,
Renson
,
L.
,
Grappasonni
,
C.
, and
Kerschen
,
G.
,
2016
, “
Identification of Nonlinear Normal Modes of Engineering Structures Under Broadband Forcing
,”
Mech. Syst. Signal Process.
,
74
, pp.
95
110
. 10.1016/j.ymssp.2015.04.016
18.
Clement
,
S.
,
Bellizzi
,
S.
,
Cochelin
,
B.
, and
Ricciardi
,
G.
,
2014
, “
Sliding Window Proper Orthogonal Decomposition: Application to Linear and Nonlinear Modal Identification
,”
J. Sound. Vib.
,
333
(
21
), pp.
5312
5323
. 10.1016/j.jsv.2014.05.035
19.
Worden
,
K.
, and
Green
,
P. L.
,
2017
, “
A Machine Learning Approach to Nonlinear Modal Analysis
,”
Mech. Syst. Signal Process.
,
84
, pp.
34
53
. 10.1016/j.ymssp.2016.04.029
20.
Noël
,
J. P.
,
Marchesiello
,
S.
, and
Kerschen
,
G.
,
2014
, “
Subspace-Based Identification of a Nonlinear Spacecraft in the Time and Frequency Domains
,”
Mech. Syst. Signal Process.
,
43
(
1
), pp.
217
236
. 10.1016/j.ymssp.2013.10.016
21.
Peng
,
Z. K.
,
Lang
,
Z. Q.
,
Wolters
,
C.
,
Billings
,
S. A.
, and
Worden
,
K.
,
2011
, “
Feasibility Study of Structural Damage Detection Using NARMAX Modelling and Nonlinear Output Frequency Response Function Based Analysis
,”
Mech. Syst. Signal Process.
,
25
(
3
), pp.
1045
1061
. 10.1016/j.ymssp.2010.09.014
22.
Pai
,
P. F.
, and
Palazotto
,
A. N.
,
2008
, “
Detection and Identification of Nonlinearities by Amplitude and Frequency Modulation Analysis
,”
Mech. Syst. Signal Process.
,
22
(
5
), pp.
1107
1132
. 10.1016/j.ymssp.2007.11.006
23.
Brewick
,
P. T.
,
Masri
,
S. F.
,
Carboni
,
B.
, and
Lacarbonara
,
W.
,
2016
, “
Data-Based Nonlinear Identification and Constitutive Modeling of Hysteresis in NiTiNOL and Steel Strands
,”
J. Engin. Mech.
,
142
(
12
), p.
04016107
. 10.1061/(ASCE)EM.1943-7889.0001170
24.
Lusch
,
B.
,
Kutz
,
J. N.
, and
Brunton
,
S. L.
,
2018
, “
Deep Learning for Universal Linear Embeddings of Nonlinear Dynamics
,”
Nat. Commun.
,
9
(
1
), p.
4950
. 10.1038/s41467-018-07210-0
25.
Champion
,
K.
,
Lusch
,
B.
,
Kutz
,
J. N.
, and
Brunton
,
S. L.
,
2019
, “
Data-Driven Discovery of Coordinates and Governing Equations
,”
Proc. Natl. Acad. Sci. USA
,
116
(
45
), p.
22445
. 10.1073/pnas.1906995116
26.
Raissi
,
M.
,
Perdikaris
,
P.
, and
Karniadakis
,
G.
,
2018
, “
Multistep Neural Networks for Data-Driven Discovery of Nonlinear Dynamical Systems
,” arXiv.org .
27.
Billings
,
S. A.
,
2013
,
Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains
,
John Wiley & Sons
,
New York
.
28.
Fu
,
L.
, and
Li
,
P.
,
2013
, “
The Research Survey of System Identification Method
,”
Proceedings of the 5th International Conference on Intelligent Human-Machine Systems and Cybernetics
, Vol.
2
, pp.
397
401
.
29.
Vlachas
,
P. R.
,
Byeon
,
W.
,
Wan
,
Z. Y.
,
Sapsis
,
T. P.
, and
Koumoutsakos
,
P.
,
2018
, “
Data-Driven Forecasting of High-Dimensional Chaotic Systems With Long Short-Term Memory Networks
,”
Proc. R. Soc. Math. Phys. Engin. Sci.
,
474
(
2213
). 10.1098/rspa.2017.0844
30.
Pathak
,
J.
,
Hunt
,
B.
,
Girvan
,
M.
,
Lu
,
Z.
, and
Ott
,
E.
,
2018
, “
Model-Free Prediction of Large Spatiotemporally Chaotic Systems From Data: A Reservoir Computing Approach
,”
Phys. Rev. Lett.
,
120
(
2
), p.
024102
. 10.1103/PhysRevLett.120.024102
31.
Muto
,
M.
, and
Beck
,
J. L.
,
2008
, “
Bayesian Updating and Model Class Selection for Hysteretic Structural Models Using Stochastic Simulation
,”
J. Vib. Control
,
14
(
1–2
), pp.
7
34
. 10.1177/1077546307079400
32.
Beck
,
J. L.
, and
Katafygiotis
,
L. S.
,
1998
, “
Updating Models and Their Uncertainties. I: Bayesian Statistical Framework
,”
J. Engin. Mech.
,
124
(
4
), pp.
455
461
. 10.1061/(ASCE)0733-9399(1998)124:4(455)
33.
Green
,
P. L.
, and
Worden
,
K.
,
2015
, “
Bayesian and Markov Chain Monte Carlo Methods for Identifying Nonlinear Systems in the Presence of Uncertainty
,”
Proc. R. Soc. A. Math. Phys. Engin. Sci.
,
373
, p.
20140405
. 10.1098/rsta.2014.0405
34.
Kerschen
,
G.
,
Golinval
,
J.-C.
, and
Hemez
,
F. M.
,
2003
, “
Bayesian Model Screening for the Identification of Nonlinear Mechanical Structures
,”
ASME J. Vib. Acoust.
,
125
(
3
), pp.
389
397
. 10.1115/1.1569947
35.
Yosida
,
K.
,
1984
,
Operational Calculus—A Theory of Hyperfunctions
,
Springer Science+Business Media
,
New York
.
36.
Sira-Ramìrez
,
H.
,
Garcìa-Rodrìguez
,
C.
,
Cortés-Romero
,
J.
, and
Luviano-Juárez
,
A.
,
2014
,
Algebraic Identification and Estimation Methods in Feedback Control System
,
John Wiley & Sons
,
New York
.
37.
Fliess
,
M.
, and
Sira-Ramìrez
,
H.
,
2003
, “
An Algebraic Framework for Linear Identification
,”
ESAIM Control Optim. Calc. Variat.
,
9
, pp.
151
168
. 10.1051/cocv:2003008
38.
Garrido
,
R.
, and
Concha
,
A.
,
2013
, “
An Algebraic Recursive Method for Parameter Identification of a Servo Model
,”
IEEE/ASME Trans. Mechatron.
,
18
(
5
), pp.
1572
1580
. 10.1109/TMECH.2012.2208197
39.
Trapero
,
J. R.
,
Sira-Ramìrez
,
H.
, and
Batlle
,
V. F.
,
2007
, “
A Fast On-line Frequency Estimator of Lightly Damped Vibrations in Flexible Structures
,”
J. Sound. Vib.
,
307
(
1
), pp.
365
378
. 10.1016/j.jsv.2007.07.005
40.
Becedas
,
J.
,
Mamani
,
G.
,
Feliu-Batlle
,
V.
, and
Sira-Ramìrez
,
H.
,
2007
, “
Algebraic Identification Method for Mass-Spring-Damper System
,”
Proceedings of World Congress on Engineering and Computer Science
,
San Francisco, CA
,
Oct. 24–26
.
41.
Brunton
,
S. L.
,
Proctor
,
J. L.
, and
Kutz
,
J. N.
,
2016
, “
Discovering Governing Equations From Data by Sparse Identification of Nonlinear Dynamical Systems
,”
Proc. Natl. Acad. Sci. USA
,
113
(
15
), p.
3932
. 10.1073/pnas.1517384113
42.
Mangan
,
N.
,
Kutz
,
J.
,
Brunton
,
S.
, and
Proctor
,
J.
,
2017
, “
Model Selection for Dynamical Systems Via Sparse Regression and Information Criteria
,”
Proc. R. Soc. A. Math. Phys. Engin. Sci.
,
473
, p.
20170009
. 10.1098/rspa.2017.0009
43.
Kaiser
,
E.
,
Kutz
,
J.
, and
Brunton
,
S.
,
2017
, “
Sparse Identification of Nonlinear Dynamics for Model Predictive Control in the Low-Data Limit
,”
Proc. R. Soc. A. Math. Phys. Engin. Sci.
,
474
, p.
20180335
. 10.1098/rspa.2018.0335
44.
Mangan
,
N.
,
Askham
,
T.
,
Brunton
,
S.
,
Kutz
,
J.
, and
Proctor
,
J.
,
2019
, “
Model Selection for Hybrid Dynamical Systems Via Sparse Regression
,”
Proc. R. Soc. A. Math. Phys. Engin. Sci.
,
475
, p.
20180534
. 10.1098/rspa.2018.0534
45.
Chartrand
,
R.
,
2011
, “
Numerical Differentiation of Noisy, Nonsmooth Data
,”
ISRN Appl. Math. J.
,
2011
, p.
164564
.
46.
Rudin
,
L. I.
,
Osher
,
S.
, and
Fatemi
,
E.
,
1992
, “
Nonlinear Total Variation Based Noise Removal Algorithms
,”
Physica D
,
60
, pp.
259
268
. 10.1016/0167-2789(92)90242-F
47.
Schaeffer
,
H.
, and
McCalla
,
S. G.
,
2017
, “
Sparse Model Selection Via Integral Terms
,”
Phys. Rev. E
,
96
(
2
), p.
023302
. 10.1103/PhysRevE.96.023302
48.
Masri
,
S. F.
, and
Caughey
,
T. K.
,
1979
, “
A Nonparametric Identification Technique for Nonlinear Dynamic Problems
,”
ASME J. Appl. Mech.
,
46
(
2
), pp.
433
447
. 10.1115/1.3424568
49.
Hastie
,
T.
,
Tibshirani
,
R.
, and
Friedman
,
J.
,
2009
,
The Elements of Statistical Learning: Data Mining, Inference, and Prediction
,
Springer Series in Statistics
,
New York
.
50.
Burnham
,
K. P.
, and
Anderson
,
D. R.
,
2002
,
Model Selection and Multi-model Inference
,
Springer
,
Berlin, Germany
.
51.
Apkarian
,
J.
,
Karam
,
P.
, and
Lévis
,
M.
,
2011
,
Flexible Link Experiment for MatLab/Simulink Users
,
Quanser Inc.
,
Markham, Ontario, Canada
.
52.
Han
,
J.
,
1995
, “
A Class of Extended State Observers for Uncertain Systems
,”
Control Decision
,
10
, pp.
85
88
.
53.
Guo
,
B.
, and
Zhao
,
Z. L.
,
2016
,
Active Disturbance Rejection Control for Nonlinear Systems: An Introduction
,
John Wiley & Sons
,
New York
.
54.
Apkarian
,
J.
,
Lévis
,
M.
, and
Gurocak
,
H.
,
2011
,
User Manual SRV02 Rotary Servo Base Unit Set Up and Configuration
,
Quanser Inc.
,
Markham, Ontario
.
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