Abstract

Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems (TDS). In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

References

1.
Yi
,
S.
,
Nelson
,
P. W.
, and
Ulsoy
,
A. G.
,
2010
,
Time-Delay Systems: Analysis and Control Using the Lambert W Function
,
World Scientific
,
Hackensack, NJ
.
2.
Sadath
,
A.
, and
Vyasarayani
,
C. P.
,
2015
, “
Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
6
), p.
061024
.10.1115/1.4030153
3.
Asl
,
F. M.
, and
Ulsoy
,
A. G.
,
2003
, “
Analysis of a System of Linear Delay Differential Equations
,”
ASME J. Dyn. Syst., Meas., Control
,
125
(
2
), pp.
215
223
.10.1115/1.1568121
4.
Jarlebring
,
E.
, and
Damm
,
T.
,
2007
, “
The Lambert W Function and the Spectrum of Some Multidimensional Time-Delay Systems
,”
Automatica
,
43
(
12
), pp.
2124
2128
.10.1016/j.automatica.2007.04.001
5.
Yi
,
S.
,
Nelson
,
P. W.
, and
Ulsoy
,
A. G.
,
2007
, “
Survey on Analysis of Time Delayed Systems Via the Lambert W Function
,”
Dyn. Contin. Discrete Impulsive Syst.
,
14
(
S2
), pp.
296
301
.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.422.1716&rep=rep1&type=pdf
6.
Wahi
,
P.
, and
Chatterjee
,
A.
,
2005
, “
Asymptotics for the Characteristic Roots of Delayed Dynamic Systems
,”
ASME J. Appl. Mech.
,
72
(
4
), pp.
475
483
.10.1115/1.1875492
7.
Vyasarayani
,
C. P.
,
2012
, “
Galerkin Approximations for Higher Order Delay Differential Equations
,”
ASME J. Comput. Nonlinear Dyn.
,
7
(
3
), p.
031004
.10.1115/1.4005931
8.
Kalmár-Nagy
,
T.
,
2009
, “
Stability Analysis of Delay-Differential Equations by the Method of Steps and Inverse Laplace Transform
,”
Differ. Equations Dyn. Syst.
,
17
(
1–2
), pp.
185
200
.10.1007/s12591-009-0014-x
9.
Insperger
,
T.
, and
Stépán
,
G.
,
2011
,
Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications
,
Springer
,
New York
.
10.
Wahi
,
P.
, and
Chatterjee
,
A.
,
2005
, “
Galerkin Projections for Delay Differential Equations
,”
ASME J. Dyn. Syst., Meas., Control
,
127
(
1
), pp.
80
87
.10.1115/1.1870042
11.
Butcher
,
E. A.
,
Ma
,
H.
,
Bueler
,
E.
,
Averina
,
V.
, and
Szabo
,
Z.
,
2004
, “
Stability of Linear Time-Periodic Delay-Differential Equations Via Chebyshev Polynomials
,”
Int. J. Numer. Methods Eng.
,
59
(
7
), pp.
895
922
.10.1002/nme.894
12.
Breda
,
D.
,
Maset
,
S.
, and
Vermiglio
,
R.
,
2005
, “
Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations
,”
SIAM J. Sci. Comput.
,
27
(
2
), pp.
482
495
.10.1137/030601600
13.
Wu
,
Z.
, and
Michiels
,
W.
,
2012
, “
Reliably Computing All Characteristic Roots of Delay Differential Equations in a Given Right Half Plane Using a Spectral Method
,”
J. Comput. Appl. Math.
,
236
(
9
), pp.
2499
2514
.10.1016/j.cam.2011.12.009
14.
Pekar
,
L.
, and
Gao
,
Q.
,
2018
, “
Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays: A Literature Overview of Some Recent Results
,”
IEEE Access
,
6
, pp.
35457
35491
.10.1109/ACCESS.2018.2851453
15.
Sadath
,
A.
, and
Vyasarayani
,
C. P.
,
2015
, “
Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Delays
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
6
), p.
061008
.10.1115/1.4028631
16.
Ahsan
,
Z.
,
Sadath
,
A.
,
Uchida
,
T. K.
, and
Vyasarayani
,
C. P.
,
2015
, “
Galerkin–Arnoldi Algorithm for Stability Analysis of Time-Periodic Delay Differential Equations
,”
Nonlinear Dyn.
,
82
(
4
), pp.
1893
1904
.10.1007/s11071-015-2285-9
17.
Altintas
,
Y.
, and
Chan
,
P. K.
,
1992
, “
In-Process Detection and Suppression of Chatter in Milling
,”
Int. J. Mach. Tools Manuf.
,
32
(
3
), pp.
329
347
.10.1016/0890-6955(92)90006-3
18.
Radulescu
,
R.
,
Kapoor
,
S. G.
, and
DeVor
,
R. E.
,
1997
, “
An Investigation of Variable Spindle Speed Face Milling for Tool-Work Structures With Complex Dynamics, Part 1: Simulation Results
,”
ASME J. Manuf. Sci. Eng.
,
119
(
3
), pp.
266
272
.10.1115/1.2831103
19.
Long
,
X.
,
Insperger
,
T.
, and
Balachandran
,
B.
,
2009
, “
Systems With Periodic Coefficients and Periodically Varying Delays: Semidiscretization-Based Stability Analysis
,”
Delay Differential Equations: Recent Advances and New Directions
,
D. E.
Gilsinn
,
T.
Kalmár-Nagy
, and
B.
Balachandran
, eds.,
Springer
,
Boston
, pp.
131
153
.
20.
Long
,
X.
, and
Balachandran
,
B.
,
2010
, “
Stability of Up-Milling and Down-Milling Operations With Variable Spindle Speed
,”
J. Vib. Control
,
16
(
7–8
), pp.
1151
1168
.10.1177/1077546309341131
21.
Sastry
,
S.
,
Kapoor
,
S. G.
,
DeVor
,
R. E.
, and
Dullerud
,
G. E.
,
2001
, “
Chatter Stability Analysis of the Variable Speed Face-Milling Process
,”
ASME J. Manuf. Sci. Eng.
,
123
(
4
), pp.
753
756
.10.1115/1.1373649
22.
Yilmaz
,
A.
,
Emad
,
A.-R.
, and
Ni
,
J.
,
2002
, “
Machine Tool Chatter Suppression by Multi-Level Random Spindle Speed Variation
,”
ASME J. Manuf. Sci. Eng.
,
124
(
2
), pp.
208
216
.10.1115/1.1378794
23.
Liu
,
Z.
, and
Liao
,
L.
,
2004
, “
Existence and Global Exponential Stability of Periodic Solution of Cellular Neural Networks With Time-Varying Delays
,”
J. Math. Anal. Appl.
,
290
(
1
), pp.
247
262
.10.1016/j.jmaa.2003.09.052
24.
Jiang
,
M.
,
Shen
,
Y.
, and
Liao
,
X.
,
2006
, “
Global Stability of Periodic Solution for Bidirectional Associative Memory Neural Networks With Varying-Time Delay
,”
Appl. Math. Comput.
,
182
(
1
), pp.
509
520
.10.1016/j.amc.2006.04.012
25.
Engelborghs
,
K.
,
Luzyanina
,
T.
, and
Roose
,
D.
,
2002
, “
Numerical Bifurcation Analysis of Delay Differential Equations Using DDE-BIFTOOL
,”
ACM Trans. Math. Software
,
28
(
1
), pp.
1
21
.10.1145/513001.513002
26.
Butcher
,
E.
, and
Mann
,
B.
,
2009
, “
Stability Analysis and Control of Linear Periodic Delayed Systems Using Chebyshev and Temporal Finite Element Methods
,”
Delay Differential Equations: Recent Advances and New Directions
,
D. E.
Gilsinn
,
T.
Kalmár-Nagy
, and
B.
Balachandran
, eds.,
Springer
,
Boston
, pp.
93
129
.
27.
Zhang
,
J.
, and
Sun
,
J.-Q.
,
2009
, “
Robustness Analysis of Optimally Designed Feedback Control of Linear Periodic Systems With Time-Delay
,”
ASME
Paper No. DETC2007-34438.10.1115/DETC2007-34438
28.
Butcher
,
E. A.
,
Bobrenkov
,
O.
,
Nazari
,
M.
, and
Torkamani
,
S.
,
2013
, “
Estimation and Control in Time-Delayed Dynamical Systems Using the Chebyshev Spectral Continuous Time Approximation and Reduced Liapunov-Floquet Transformation
,”
Advances in Analysis and Control of Time-Delayed Dynamical Systems
,
J.-Q.
Sun
and
Q.
Ding
, eds.,
World Scientific
,
Singapore
, pp.
219
264
.
29.
Nazari
,
M.
,
Butcher
,
E. A.
, and
Bobrenkov
,
O. A.
,
2014
, “
Optimal Feedback Control Strategies for Periodic Delayed Systems
,”
Int. J. Dyn. Control
,
2
(
1
), pp.
102
118
.10.1007/s40435-013-0053-6
30.
Ma
,
H.
,
Deshmukh
,
V.
,
Butcher
,
E.
, and
Averina
,
V.
,
2003
, “
Controller Design for Linear Time-Periodic Delay Systems Via a Symbolic Approach
,”
Proceedings of the 2003 American Control Conference
, Denver, CO, June 4–6, pp.
2126
2131
.10.1109/ACC.2003.1243388
31.
Ma
,
H.
,
Deshmukh
,
V.
,
Butcher
,
E.
, and
Averina
,
V.
,
2005
, “
Delayed State Feedback and Chaos Control for Time-Periodic Systems Via a Symbolic Approach
,”
Commun. Nonlinear Sci. Numer. Simul.
,
10
(
5
), pp.
479
497
.10.1016/j.cnsns.2003.12.007
32.
Borgioli
,
F.
,
Hajdu
,
D.
,
Insperger
,
T.
,
Stepan
,
G.
, and
Michiels
,
W.
,
2020
, “
Pseudospectral Method for Assessing Stability Robustness for Linear Time-Periodic Delayed Dynamical Systems
,”
Int. J. Numer. Methods Eng.
,
121
(
16
), pp.
3505
3528
.10.1002/nme.6368
33.
Sadath
,
A.
, and
Vyasarayani
,
C. P.
,
2015
, “
Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), p.
021011
.10.1115/1.4026989
34.
Kandala
,
S. S.
,
2020
, “
Pole Placement and Reduced-Order Modelling of Time-Delayed Systems Using Galerkin Approximations
,” Ph.D. thesis,
Indian Institute of Technology Hyderabad
, Telangana, India.
35.
Vyasarayani
,
C. P.
,
Subhash
,
S.
, and
Kalmár-Nagy
,
T.
,
2014
, “
Spectral Approximations for Characteristic Roots of Delay Differential Equations
,”
Int. J. Dyn. Control
,
2
(
2
), pp.
126
132
.10.1007/s40435-014-0060-2
36.
Gottlieb
,
D.
, and
Orszag
,
S. A.
,
1977
,
Numerical Analysis of Spectral Methods: Theory and Applications
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.
37.
Kovacic
,
I.
,
Rand
,
R.
, and
Sah
,
S. M.
,
2018
, “
Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features
,”
ASME Appl. Mech. Rev.
,
70
(
2
), p.
020802
.10.1115/1.4039144
38.
Apkarian
,
J.
,
Lévis
,
M.
, and
Martin
,
P.
,
2016
,
Instructor Workbook: QUBE-Servo 2 Experiment for MATLAB/Simulink Users
,
Quanser
, Markham, ON, Canada.
39.
Apkarian
,
J.
,
Karam
,
P.
, and
Lévis
,
M.
,
2012
,
Student Workbook: Inverted Pendulum Experiment for LabVIEW Users
,
Quanser
, Markham, ON, Canada.
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