The computational burden of parameter exploration of nonlinear dynamical systems can become a costly exercise. A computationally efficient lower dimensional representation of a higher dimensional dynamical system is achieved by developing a reduced order model (ROM). Proper orthogonal decomposition (POD) is usually the preferred method in projection-based nonlinear model reduction. POD seeks to find a set of projection modes that maximize the variance between the full-scale state variables and its reduced representation through a constrained optimization problem. Here, we investigate the benefits of an ROM, both qualitatively and quantitatively, by the inclusion of time derivatives of the state variables. In one formulation, time derivatives are introduced as a constraint in the optimization formulation—smooth orthogonal decomposition (SOD). In another formulation, time derivatives are concatenated with the state variables to increase the size of the state space in the optimization formulation—extended state proper orthogonal decomposition (ESPOD). The three methods (POD, SOD, and ESPOD) are compared using a periodically, periodically forced with measurement noise, and a randomly forced beam on a nonlinear foundation. For both the periodically and randomly forced cases, SOD yields a robust subspace for model reduction that is insensitive to changes in forcing amplitudes and input energy. In addition, SOD offers continual improvement as the size of the dimension of the subspace increases. In the periodically forced case where the ROM is developed with noisy data, ESPOD outperforms both SOD and POD and captures the dynamics of the desired system using a lower dimensional model.

References

References
1.
Kerschen
,
G.
,
Golinval
,
J.-C.
,
Vakakis
,
A. F.
, and
Bergman
,
L. A.
,
2005
, “
The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview
,”
Nonlinear Dyn.
,
41
(
1–3
), pp.
147
169
.
2.
Holmes
,
P.
,
Lumley
,
J. L.
, and
Berkooz
,
G.
,
1998
,
Turbulence, Coherent Structures, Dynamical Systems and Symmetry
,
Cambridge University Press
,
New York
.
3.
Liang
,
Y. C.
,
Lee
,
H. P.
,
Lim
,
S. P.
,
Lin
,
W. Z.
,
Lee
,
K. H.
, and
Wu
,
C. G.
,
2002
, “
Proper Orthogonal Decomposition and Its Applications—Part I: Theory
,”
J. Sound Vib.
,
252
(
3
), pp.
527
544
.
4.
Loeve
,
M.
,
1955
,
Probability Theory
,
D. Van Nostrand Company
,
Princeton, NJ
.
5.
Kosambi
,
D. D.
,
1943
, “
Statistics in Function Space
,”
J. Indian Math. Soc.
,
7
, pp.
76
88
.
6.
Hotelling
,
H.
,
1933
, “
Analysis of a Complex of Statistical Variables Into Principal Components
,”
J. Educ. Psychol.
,
24
(
6
), pp.
417
441
.
7.
North
,
G. R.
,
Bell
,
T. L.
,
Cahalan
,
R. F.
, and
Moeng
,
F. J.
,
1982
, “
Sampling Errors in the Estimation of Empirical Orthogonal Functions
,”
Mon. Weather Rev.
,
110
(
7
), pp.
699
706
.
8.
Behzad
,
F.
,
Helenbrook
,
B. T.
, and
Ahmadi
,
G.
,
2015
, “
On the Sensitivity and Accuracy of Proper-Orthogonal-Decomposition-Based Reduced Order Models for Burgers Equation
,”
Comput. Fluids
,
106
, pp.
19
32
.
9.
Feeny
,
B.
,
2002
, “
On the Proper Orthogonal Modes and Normal Modes of a Continuous Vibration System
,”
ASME J. Sound Vib.
,
124
(
1
), pp.
157
160
.
10.
Feeny
,
B. F.
, and
Kappagantu
,
R.
,
1998
, “
On the Physical Interpretation of Proper Orthogonal Modes in Vibrations
,”
J. Sound Vib.
,
211
(
4
), pp.
607
616
.
11.
Chatterjee
,
A.
,
2000
, “
An Introduction to the Proper Orthogonal Decomposition
,”
Current Sci.
,
78
(
7
), pp.
808
817
.
12.
Kerfriden
,
P.
,
Gosselet
,
P.
,
Adhikari
,
S.
, and
Bordas
,
S. P.-A.
,
2011
, “
Bridging Proper Orthogonal Decomposition Methods and Augmented Newton–Krylov Algorithms: An Adaptive Model Order Reduction for Highly Nonlinear Mechanical Problems
,”
Comput. Methods Appl. Mech. Eng.
,
200
(
5
), pp.
850
866
.
13.
Hung
,
E. S.
, and
Senturia
,
S. D.
,
1999
, “
Generating Efficient Dynamical Models for Microelectromechanical Systems From a Few Finite-Element Simulation Runs
,”
J. Microelectromech. Syst.
,
8
(
3
), pp.
280
289
.
14.
Xie
,
D.
,
Xu
,
M.
, and
Dowell
,
E. H.
,
2014
, “
Proper Orthogonal Decomposition Reduced-Order Model for Nonlinear Aeroelastic Oscillations
,”
AIAA J.
,
52
(
2
), pp.
229
241
.
15.
Xie
,
D.
,
Xu
,
M.
, and
Dowell
,
E. H.
,
2014
, “
Projection-Free Proper Orthogonal Decomposition Method for a Cantilever Plate in Supersonic Flow
,”
J. Sound Vib.
,
333
(
23
), pp.
6190
6208
.
16.
Xie
,
D.
,
Xu
,
M.
,
Dai
,
H.
, and
Dowell
,
E. H.
,
2015
, “
Proper Orthogonal Decomposition Method for Analysis of Nonlinear Panel Flutter With Thermal Effects in Supersonic Flow
,”
J. Sound Vib.
,
337
, pp.
263
283
.
17.
Ihrle
,
S.
,
Lauxmann
,
M.
,
Eiber
,
A.
, and
Eberhard
,
P.
,
2013
, “
Nonlinear Modelling of the Middle Ear as an Elastic Multibody System-Applying Model Order Reduction to Acousto-Structural Coupled Systems
,”
J. Comput. Appl. Math.
,
246
, pp.
18
26
.
18.
Li
,
X.
,
Chen
,
X.
,
Hu
,
B. X.
, and
Navon
,
I. M.
,
2013
, “
Model Reduction of a Coupled Numerical Model Using Proper Orthogonal Decomposition
,”
J. Hydrol.
,
507
, pp.
227
240
.
19.
Rewienski
,
M. J.
,
2003
, “
A Trajectory Piecewise-Linear Approach to Model Order Reduction of Nonlinear Dynamical Systems
,” Ph.D. thesis,
Massachusetts Institute of Technology
,
Boston, MA
.
20.
Rewieński
,
M.
, and
White
,
J.
,
2006
, “
Model Order Reduction for Nonlinear Dynamical Systems Based on Trajectory Piecewise-Linear Approximations
,”
Linear Algebra Appl.
,
415
(
2
), pp.
426
454
.
21.
Chaturantabut
,
S.
, and
Sorensen
,
D. C.
,
2010
, “
Nonlinear Model Reduction Via Discrete Empirical Interpolation
,”
SIAM J. Sci. Comput.
,
32
(
5
), pp.
2737
2764
.
22.
Henneron
,
T.
, and
Clenet
,
S.
,
2014
, “
Model Order Reduction of Non-Linear Magnetostatic Problems Based on POD and DEI Methods
,”
IEEE Trans. Mag.
,
50
(
2
), pp.
33
36
.
23.
Chelidze
,
D.
, and
Liu
,
M.
,
2005
, “
Dynamical Systems Approach to Fatigue Damage Identification
,”
J. Sound Vib.
,
281
(
3–5
), pp.
887
904
.
24.
Chelidze
,
D.
, and
Zhou
,
W.
,
2006
, “
Smooth Orthogonal Decomposition Based Modal Analysis
,”
J. Sound Vib.
,
292
(
3–5
), pp.
461
473
.
25.
Rezaee
,
M.
,
Shaterian-Alghalandis
,
V.
, and
Banan-Nojavani
,
A.
,
2013
, “
Development of the Smooth Orthogonal Decomposition Method to Derive the Modal Parameters of Vehicle Suspension System
,”
J. Sound Vib.
,
332
(
7
), pp.
1829
1842
.
26.
Segala
,
D. B.
,
Gates
,
D. H.
,
Dingwell
,
J. B.
, and
Chelidze
,
D.
,
2011
, “
Nonlinear Smooth Orthogonal Decomposition of Kinematic Features of Sawing Reconstructs Muscle Fatigue Evolution as Indicated by Electromyography
,”
ASME J. Biomech. Eng.
,
133
(
3
), p.
031009
.
27.
Kuehl
,
J.
,
DiMarco
,
S.
,
Spencer
,
L.
, and
Guinasso
,
N.
,
2014
, “
Application of the Smooth Orthogonal Decomposition to Oceanographic Data Sets
,”
Geophys. Res. Lett.
,
41
(
11
), pp.
3966
3971
.
28.
Feeny
,
B.
, and
Farooq
,
U.
,
2008
, “
A Nonsymmetric State-Variable Decomposition for Modal Analysis
,”
J. Sound Vib.
,
310
(
4–5
), pp.
792
800
.
29.
Farooq
,
U.
, and
Feeny
,
B. F.
,
2012
, “
An Experimental Investigation of State-Variable Modal Decomposition for Modal Analysis
,”
ASME J. Vib. Acoust.
,
134
(
2
), p.
021017
.
30.
Segala
,
D. B.
, and
Chelidze
,
D.
,
2014
, “
Robust and Dynamically Consistent Model Order Reduction for Nonlinear Dynamic Systems
,”
ASME J. Dyn. Syst. Meas. Control
,
137
(
2
), p.
021011
.
31.
Ilbeigi
,
S.
, and
Chelidze
,
D.
,
2016
,
Model Order Reduction of Nonlinear Euler-Bernoulli Beam
,
Springer International Publishing
,
Cham, Switzerland
, pp.
377
385
.
32.
Ilbeigi
,
S.
, and
Chelidze
,
D.
,
2017
,
Reduced Order Models for Systems With Disparate Spatial and Temporal Scales
,
Springer International Publishing
,
Cham, Switzerland
, pp.
447
455
.
33.
Golub
,
G. H.
, and
Van Loan
,
C. F.
,
1996
,
Matrix Computations
,
Johns Hopkins University Press
,
Baltimore, MD
.
34.
Rathinam
,
M.
, and
Petzold
,
L. R.
,
2003
, “
A New Look at Proper Orthogonal Decomposition
,”
J. Numer. Anal.
,
41
(
5
), pp.
1893
1925
.
You do not currently have access to this content.