A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented in this paper. The independence of delays makes the problem much more challenging compared to systems with commensurate time delays (where the delays have rational relations). We uncover some wonderful features for such systems. For instance, all the imaginary characteristic roots of these systems can be found exhaustively along a set of surfaces in the domain of the delays. They are called the “kernelsurfaces (curves for two-delay cases), and it is proven that the number of the kernel surfaces is manageably small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie either on this kernel or its infinitely many “offspring” surfaces. Another hidden feature is that the root tendencies along these surfaces exhibit an invariance property. From these outstanding characteristics an efficient, exact, and exhaustive methodology results for the stability assessment. As an added uniqueness of this method, the systems under consideration do not have to be stable for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology known to the authors.

1.
Hsu
,
C. S.
, and
Bhatt
,
K. L.
, 1996, “
Stability Charts for Second-Order Dynamical Systems With Time Lag
,”
J. Appl. Mech.
0021-8936,
33
, pp.
119
124
.
2.
Rekasius
,
Z. V.
, 1980, “
A Stability Test for Systems With Delays
,” presented at
Proceedings of the Joint Automatic Control Conference
, Paper No. TP9-A.
3.
Hsu
,
C. S.
, 1970, “
Application of the Tau-Decomposition Method to Dynamical Systems Subjected to Retarded Follower Forces
,”
ASME J. Appl. Mech.
0021-8936,
37
, pp.
258
266
.
4.
Cooke
,
K. L.
, and
van den Driessche
,
P.
, 1986, “
On Zeros of Some Transcendental Equations
,”
Funkc. Ekvac.
0532-8721,
29
, pp.
77
90
.
5.
Chen
,
J.
,
Gu
,
G.
, and
Nett
,
C. N.
, 1995, “
A New Method for Computing Delay Margins for Stability of Linear Delay Systems
,”
Syst. Control Lett.
0167-6911,
26
, pp.
107
117
.
6.
Bellman
,
R. E.
, and
Cooke
,
K. L.
, 1963,
Differential-Difference Equations
,
Academic
, New York.
7.
Olgac
,
N.
, and
Sipahi
,
R.
, 2002, “
An Exact Method for the Stability Analysis of Time Delayed Lti Systems
,”
IEEE Trans. Autom. Control
0018-9286,
47
, pp.
793
797
.
8.
Sipahi
,
R.
, and
Olgac
,
N.
, 2006, “
A Novel Stability Study on Multiple Time-Delay Systems Using the Root Clustering Paradigm
,”
Syst. Control Lett.
0167-6911,
55
, pp.
819
825
.
9.
Sipahi
,
R.
, and
Olgac
,
N.
, 2003, “
Degenerate Cases in Using Direct Method
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
125
, pp.
194
201
.
10.
Sipahi
,
R.
, and
Olgac
,
N.
, 2003, “
Active Vibration Suppression With Time Delayed Feedback
,”
ASME J. Vibr. Acoust.
0739-3717,
125
, pp.
384
388
.
11.
Niculescu
,
S.-I.
, 2002, “
On Delay Robustness Analysis of a Simple Control Algorithm in High-Speed Networks
,”
Automatica
0005-1098,
38
, pp.
885
889
.
12.
Stepan
,
G.
, 1989,
Retarded Dynamical Systems: Stability and Characteristic Function
,
Longman Scientific & Technical and Wiley
, New York.
13.
MacDonald
,
N.
, 1987, “
An Interference Effect of Independent Delays
,”
Proc. IEEE
0018-9219,
134
, pp.
38
42
.
14.
Hale
,
J. K.
, and
Huang
,
W.
, 1993, “
Global Geometry of the Stable Regions for Two Delay Differential Equations
,”
J. Math. Anal. Appl.
0022-247X,
178
, pp.
344
362
.
15.
Breda
,
D.
,
Maset
,
S.
, and
Vermiglio
,
R.
, 2004, “
Computing the Characteristic Roots for Delay Differential Equations
,”
IMA J. Numer. Anal.
0272-4979,
24
, pp.
1
19
.
16.
Rekasius
,
Z. V.
, 1980, “
A Stability Test for Systems With Delays
,” presented at
Proceedings of the Joint Automatic Control Conference
, TP9-A, San Francisco, CA.
17.
Thowsen
,
A.
, 1981, “
The Routh-Hurwitz Method for Stability Determination of Linear Differential-Difference Systems
,”
Int. J. Control
0020-7179,
33
, pp.
991
995
.
18.
Hertz
,
D.
,
Jury
,
E. I.
, and
Zeheb
,
E.
, 1984, “
Simplified Analytic Stability Test for Systems With Commensurate Time Delays
,”
IEE Proc.-D: Control Theory Appl.
0143-7054,
131
, pp.
52
56
.
19.
Ogata
,
K.
, 2004,
System Dynamics
,
4th ed.
,
Pearson-Prentice Hall
, New York.
20.
Hale
,
J. K.
, and
Verduyn Lunel
,
S. M.
, 2001, “
Strong Stabilization of Neutral Functional Differential Equations
,”
IMA J. Math. Control Inf.
0265-0754,
19
, pp.
1
19
.
21.
Hale
,
J. K.
, and
Verduyn Lunel
,
S. M.
, 2001, “
Effects of Small Delays on Stability and Control
,”
Operator Theory and Analysis; Advances and Applications
,
Birkhauser
,
122
, pp.
275
301
.
22.
Kolmanovskii
,
V. B.
, and
Nosov
,
V. R.
, 1989
Stability of Functional Differential Equations
,
Academic
, London.
23.
Sipahi
,
R.
, and
Olgac
,
N.
, 2005, “
Complete Stability Robustness of Third Order, LTI, Multiple Time-Delay Systems
,”
Automatica
0005-1098,
41
, pp.
1413
1422
.
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