A direct integration method (DIM) for time-delayed control (TDC) is proposed in this research. For a second-order dynamic system with time-delayed controllers, a Volterra integral equation of the second kind is used instead of a state derivative equation. With the proposed DIM where matrix exponentials are avoided, semi-analytical representation of the Floquet transition matrix for stability analysis can be derived, the stability region on the parametric space comprising control variables can also be plotted. Within this stability region, optimal control variables are subsequently obtained using a multilevel conjugate gradient optimization method. Further simulation examples demonstrated the superiority of the proposed DIM in terms of computational efficiency and accuracy, as well as the effectiveness of the optimization-based controller design approach.

References

References
1.
Ulsoy
,
A. G.
,
2015
, “
Time-Delayed Control of Siso Systems for Improved Stability Margins
,”
ASME J. Dyn. Syst., Meas., Control
,
137
(
4
), p.
041014
.
2.
Stépán
,
G.
,
1989
, “
Retarded Dynamical Systems: Stability and Characteristic Functions
,”
Pitman Research Notes
(Mathematics Series),
Longman Scientific & Technical
,
New York
.
3.
Gu
,
K.
,
Chen
,
J.
, and
Kharitonov
,
V.
,
2003
, “
Stability of Time-Delay Systems
,”
Control Engineering
,
Birkhäuser Boston
,
Cambridge, MA
.
4.
Marshall
,
J.
,
1992
, “
Time-Delay Systems: Stability and Performance Criteria With Applications
,”
Mathematics and Its Applications
(Ellis Horwood Series),
Ellis Horwood
,
Hemel Hempstead, UK
.
5.
Pyragas
,
K.
,
1992
, “
Continuous Control of Chaos by Self-Controlling Feedback
,”
Phys. Lett. A
,
170
(
6
), pp.
421
428
.
6.
Gu
,
K.
, and
Niculescu
,
S.-I.
,
2003
, “
Survey on Recent Results in the Stability and Control of Time-Delay Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
125
(
2
), pp.
158
165
.
7.
Udwadia
,
F. E.
,
von Bremen
,
H. F.
,
Kumar
,
R.
, and
Hosseini
,
M.
,
2003
, “
Time Delayed Control of Structural Systems
,”
Earthquake Eng. Struct. Dyn.
,
32
(
4
), pp.
495
535
.
8.
Udwadia
,
F. E.
,
Von Bremen
,
H.
, and
Phohomsiri
,
P.
,
2007
, “
Time-Delayed Control Design for Active Control of Structures: Principles and Applications
,”
Struct. Control Health Monit.
,
14
(
1
), pp.
27
61
.
9.
Chung
,
L.
,
Lin
,
C.
, and
Lu
,
K.
,
1995
, “
Time-Delay Control of Structures
,”
Earthquake Eng. Struct. Dyn.
,
24
(
5
), pp.
687
701
.
10.
Yang
,
B.
, and
Mote
,
C.
, Jr.
,
1990
, “
Vibration Control of Band Saws: Theory and Experiment
,”
Wood Sci. Technol.
,
24
(
4
), pp.
355
373
.
11.
Yang
,
B.
, and
Mote
,
C.
,
1992
, “
On Time Delay in Noncolocated Control of Flexible Mechanical Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
114
(
3
), pp.
409
415
.
12.
San-Millan
,
A.
,
Russell
,
D.
,
Feliu
,
V.
, and
Aphale
,
S. S.
,
2015
, “
A Modified Positive Velocity and Position Feedback Scheme With Delay Compensation for Improved Nanopositioning Performance
,”
Smart Mater. Struct.
,
24
(
7
), p.
075021
.
13.
Suh
,
I.
, and
Bien
,
Z.
,
1979
, “
Proportional Minus Delay Controller
,”
IEEE Trans. Autom. Control
,
24
(
2
), pp.
370
372
.
14.
Villafuerte
,
R.
,
Mondie
,
S.
, and
Garrido
,
R.
,
2013
, “
Tuning of Proportional Retarded Controllers: Theory and Experiments
,”
IEEE Trans. Control Syst. Technol.
,
21
(
3
), pp.
983
990
.
15.
Hövel
,
P.
,
2010
,
Control of Complex Nonlinear Systems With Delay
,
Springer-Verlag, Berlin/Heidelberg
.
16.
Ramírez
,
A.
,
Garrido
,
R.
, and
Mondié
,
S.
,
2015
, “
Velocity Control of Servo Systems Using an Integral Retarded Algorithm
,”
ISA Trans.
,
58
, pp.
357
366
.
17.
Goyal
,
V.
,
Deolia
,
V. K.
, and
Sharma
,
T. N.
,
2015
, “
Robust Sliding Mode Control for Nonlinear Discrete-Time Delayed Systems Based on Neural Network
,”
Intell. Control Autom.
,
6
(
1
), p.
75
.
18.
Niculescu
,
S.-I.
, and
Gu
,
K.
,
2012
,
Advances in Time-Delay Systems
, Vol.
38
,
Springer-Verlag, Berlin/Heidelberg
.
19.
Niculescu
,
S.-I.
,
Dion
,
J.-M.
,
Dugard
,
L.
, and
LI
,
H.
,
1997
, “
Stability of Linear Systems With Delayed State: An LMI Approach
,”
J. Eur. Syst. Autom.
,
31
(
6
), pp.
955
969
.
20.
Fridman
,
E.
, and
Shaked
,
U.
,
2003
, “
Delay-Dependent Stability and h∞ Control: Constant and Time-Varying Delays
,”
Int. J. Control
,
76
(
1
), pp.
48
60
.
21.
Fridman
,
E.
, and
Shaked
,
U.
,
2002
, “
An Improved Stabilization Method for Linear Time-Delay Systems
,”
IEEE Trans. Autom. Control
,
47
(
11
), pp.
1931
1937
.
22.
Wu
,
M.
,
He
,
Y.
, and
She
,
J.-H.
,
2004
, “
New Delay-Dependent Stability Criteria and Stabilizing Method for Neutral Systems
,”
IEEE Trans. Autom. Control
,
49
(
12
), pp.
2266
2271
.
23.
Feng
,
Z.
,
Lam
,
J.
, and
Gao
,
H.
,
2011
, “
α-Dissipativity Analysis of Singular Time-Delay Systems
,”
Automatica
,
47
(
11
), pp.
2548
2552
.
24.
Dorf
,
R. C.
, and
Bishop
,
R. H.
,
2001
,
Modern Control Systems
,
9th ed.
,
Prentice-Hall
,
Upper Saddle River, NJ
.
25.
Wang
,
D.
,
2013
,
Design of Lower-Order Controllers for Time-Delay Systems: A Parametric Space Approach
,
Science Press
,
Beijing
.
26.
Olgac
,
N.
,
Ergenc
,
A. F.
, and
Sipahi
,
R.
,
2005
, “
Delay Scheduling: A New Concept for Stabilization in Multiple Delay Systems
,”
J. Vib. Control
,
11
(
9
), pp.
1159
1172
.
27.
Yi
,
S.
,
2009
, “
Time-Delay Systems: Analysis and Control Using the Lambert w Function
,” Ph.D. thesis,
The University of Michigan
,
Ann Arbor, MI
.
28.
Duan
,
S.
,
Ni
,
J.
, and
Ulsoy
,
A. G.
,
2011
, “
Decay Function Estimation for Linear Time Delay Systems Via the Lambert W Function
,”
J. Vib. Control
,
18
(
10
), pp.
1462
1473
.
29.
Farkas
,
M.
,
2013
,
Periodic Motions
, Vol.
104
,
Springer Science & Business Media
,
New York
.
30.
Insperger
,
T.
, and
Stépán
,
G.
,
2002
, “
Semi-Discretization Method for Delayed Systems
,”
Int. J. Numer. Methods Eng.
,
55
(
5
), pp.
503
518
.
31.
Insperger
,
T.
, and
Stépán
,
G.
,
2004
, “
Updated Semi-Discretization Method for Periodic Delay-Differential Equations With Discrete Delay
,”
Int. J. Numer. Methods Eng.
,
61
(
1
), pp.
117
141
.
32.
Insperger
,
T.
, and
Stépán
,
G.
,
2011
,
Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications
, Vol.
178
,
Springer Science & Business Media
,
New York
.
33.
Ding
,
Y.
,
Zhu
,
L.
,
Zhang
,
X.
, and
Ding
,
H.
,
2011
, “
Numerical Integration Method for Prediction of Milling Stability
,”
ASME J. Manuf. Sci. Eng.
,
133
(
3
), p.
031005
.
34.
Dong
,
W.
,
Ding
,
Y.
,
Zhu
,
X.
, and
Ding
,
H.
,
2015
, “
Optimal Proportional–Integral–Derivative Control of Time-Delay Systems Using the Differential Quadrature Method
,”
ASME J. Dyn. Syst., Meas., Control
,
137
(
10
), p.
101005
.
35.
Delves
,
L.
, and
Mohamed
,
J.
,
1988
,
Computational Methods for Integral Equations
,
Cambridge University Press
,
Cambridge, UK
.
36.
Li
,
X.
,
2008
,
Integral Equations
,
Science Press
,
Beijing, China
.
37.
Yang
,
W. Y.
,
Cao
,
W.
,
Chung
,
T.-S.
, and
Morris
,
J.
,
2005
,
Applied Numerical Methods Using MATLAB
,
Wiley
,
New York
.
38.
Bellman
,
R. E.
, and
Cooke
,
K. L.
,
1963
,
Differential-Difference Equations
,
Rand Corporation
,
Academic Press, Cambridge, MA
.
39.
Graham
,
D.
, and
Lathrop
,
R. C.
,
1953
, “
The Synthesis of Optimum Transient Response: Criteria and Standard Forms
,”
Trans. Am. Inst. Electr. Eng., Part II: Appl. Ind.
,
72
(
5
), pp.
273
288
.
40.
Lax
,
P.
,
2007
, “
Linear Algebra and Its Applications
,”
No. 10 in Linear Algebra and Its Applications
,
Wiley
,
New York
.
41.
Rao
,
S. S.
, and
Rao
,
S.
,
2009
,
Engineering Optimization: Theory and Practice
,
Wiley
,
New York
.
42.
Li
,
X.
, and
De Souza
,
C. E.
,
1997
, “
Criteria for Robust Stability and Stabilization of Uncertain Linear Systems With State Delay
,”
Automatica
,
33
(
9
), pp.
1657
1662
.
43.
Park
,
P.
,
1999
, “
A Delay-Dependent Stability Criterion for Systems With Uncertain Time-Invariant Delays
,”
IEEE Trans. Autom. Control
,
44
(
4
), pp.
876
877
.
44.
Sheng
,
J.
, and
Sun
,
J.
,
2005
, “
Feedback Controls and Optimal Gain Design of Delayed Periodic Linear Systems
,”
J. Vib. Control
,
11
(
2
), pp.
277
294
.
You do not currently have access to this content.