The limitations of the Taylor series approximations of the delayed variables have been documented by means of examples in several works. This note unifies these previous comments through a higher-level analysis of the issue. It is shown that the use of the truncated Taylor series expansion of the (inverse) advance operator does not feature the same drawbacks.

References

References
1.
Richard
,
J.-P.
,
2003
, “
Time-Delay Systems: An Overview of Some Recent Advances and Open Problems
,”
Automatica
,
39
(
10
), pp.
1667
1694
.
2.
Paraskevopoulos
,
P.
, and
Samiotakis
,
S.
,
1994
, “
State Space Analysis and Optimal Control of Time Delay Systems Via Taylor Series
,”
International Conference on Control
,
Control’94
, Mar. 21–24, Vol.
1
, pp.
70
74
.
3.
Mutharasan
,
R.
, and
Luus
,
R.
,
1975
, “
Analysis of Time-Delay Systems by Series Approximation
,”
AIChE J.
,
21
(
3
), pp.
567
571
.
4.
Leipholz
,
H.
, and
Abdel-Rohman
,
M.
,
1986
,
Control of Structures
(Mechanics of Elastic Stability Series),
Springer
,
London
.
5.
Seborg
,
D.
,
Mellichamp
,
D.
,
Edgar
,
T.
, and
Doyle
,
F.
,
2010
,
Process Dynamics and Control
,
3rd ed.
,
Wiley
,
Hoboken, NJ
.
6.
Mazanov
,
A.
, and
Tognetti
,
K. P.
,
1974
, “
Taylor Series Expansion of Delay Differential Equations—A Warning
,”
J. Theor. Biol.
,
46
(
1
), pp.
271
282
.
7.
Driver
,
R.
,
1977
,
Ordinary and Delay Differential Equations
(Applied Mathematical Sciences, Vol.
20
),
Springer
,
New York
.
8.
Zhang
,
W.
,
2011
,
Quantitative Process Control Theory
(Automation and Control Engineering),
CRC Press
,
Boca Raton, FL
.
9.
Romagnoli
,
J.
, and
Palazoglu
,
A.
,
2005
,
Introduction to Process Control
(Chemical Industries),
CRC Press
,
Boca Raton, FL
.
10.
Dorf
,
R.
, and
Bishop
,
R.
,
2013
,
Modern Control Systems
,
12th ed.
,
Pearson Education
,
Upper Saddle River, NJ
.
11.
Insperger
,
T.
,
2015
, “
On the Approximation of Delayed Systems by Taylor Series Expansion
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), pp.
1
4
.
12.
Isern
,
N.
,
Fort
,
J.
, and
Vander Linden
,
M.
,
2012
, “
Space Competition and Time Delays in Human Range Expansions. Application to the Neolithic Transition
,”
PLoS One
,
7
(
12
), p.
e51106
.
13.
Udwadia
,
F. E.
,
von Bremen
,
H. F.
,
Kumar
,
R.
, and
Hosseini
,
M.
,
2003
, “
Time Delayed Control of Structural Systems
,”
Int. J. Earthquake Eng. Struct. Dyn.
,
32
(
4
), pp.
495
535
.
14.
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
.
15.
Mori
,
T.
,
1985
, “
Asymptotic Stability Conditions for Linear Time-Delay Systems
,”
Electr. Eng. Jpn.
,
105
(
2
), pp.
138
144
.
16.
Shampine
,
L.
, and
Thompson
,
S.
,
2001
, “
Solving DDEs in Matlab
,”
Appl. Numer. Math.
,
37
(
4
), pp.
441
458
.
17.
Cahlon
,
B.
, and
Schmidt
,
D.
,
2004
, “
Stability Criteria for Certain Second-Order Delay Differential Equations With Mixed Coefficients
,”
J. Comput. Appl. Math.
,
170
(
1
), pp.
79
102
.
You do not currently have access to this content.