Relay feedback systems are strongly nonlinear due to their switching properties. Some nonlinear properties of relay feedback systems have been verified to be preferable to modern control engineering, whereas others might drive the system to be more complex or even unpredictable. An alternative criterion is proposed to investigate the pitchfork bifurcations of the limit cycle of relay feedback systems in this paper. The proposed critical criterion is explicitly formulated by the coefficients of the characteristic polynomial equation instead of the eigenvalues of the Jacobian matrix. It is more convenient and efficient for detecting the existence of this type of bifurcation than the classical critical criterion. Numerical simulations show the pitchfork bifurcation behaviors in relay feedback systems and demonstrate that the proposed criterion is a general and exact analytic method for determining pitchfork bifurcations in maps.

References

References
1.
Tsypkin
,
Y. Z.
,
1984
,
Relay Control Systems
,
Cambridge University Press
,
Cambridge, UK
.
2.
Schuck
,
O. H.
,
1959
, “
Honeywell's History and Philosophy in the Adaptive Control Field
,”
Proceedings of the Self Adaptive Flight Control Symposium
,
P. C.
Gregory
, ed.,
Wright- Patterson AFB
,
Ohio
.
3.
Åström
,
K. J.
, and
Hägglund
,
T.
,
1995
,
PID Controllers: Theory, Design and Tuning
,
second ed.
,
Instrument Society of America, Research Triangle Park
,
NC
.
4.
Yu
,
C. C.
,
1999
,
Autotuning of PID Controllers: A Relay Feedback Approach
,
Springer-Verlag
,
Berlin
.
5.
Wang
,
Q. G.
,
Lee
,
T. H.
, and
Lin
,
C.
,
2003
,
Relay Feedback: Analysis, Identification and Control
,
Springer-Verlag
,
London
.
6.
Palmor
,
Z. J.
,
Halevi
,
Y.
, and
Efrati
,
T.
,
1995
, “
A General and Exact Method for Determining Limit Cycles in Decentralized Relay Systems
,”
Automatica
,
31
, pp.
1333
1339
.10.1016/0005-1098(95)00031-Q
7.
Varigonda
,
S.
, and
Georgiou
,
T. T.
,
2001
, “
Dynamics of Relay Relaxation Oscillators
,”
IEEE Trans. Autom. Control
,
46
, pp.
65
77
.10.1109/9.898696
8.
Lin
,
C.
,
Wang
,
Q. G.
,
Lee
,
T. H.
, and
Lam
,
J.
,
2002
, “
Local Stability of Limit Cycles for Time-Delay Relay-Feedback Systems
,”
IEEE Trans. Circuits Syst., I: Fundam. Theory Appl.
,
49
, pp.
1870
1875
.10.1109/TCSI.2002.805732
9.
Goncalves
,
J. M.
,
Megretski
,
A.
, and
Dahleh
,
M. A.
,
2001
, “
Global Stability of Relay Feedback Systems
,”
IEEE Trans. Autom. Control
,
45
, pp.
550
562
.10.1109/9.917657
10.
Kowalczyk
,
P.
, and
di Bernardo
,
M.
,
2001
, “
On a Novel Class of Bifurcations in Hybrid Dynamical Systems—The Case of Relay Feedback Systems
,”
Proceedings of the Hybrid Systems: Computation and Control
,
Springer-Verlag, Berlin
.
11.
Wen
,
G. L.
,
Wang
,
Q. G.
, and
Lee
,
T. H.
,
2007
, “
Quasi-Period Oscillations of Relay Feedback Systems
,”
Chaos, Solitons Fractals
,
34
, pp.
405
411
.10.1016/j.chaos.2006.03.059
12.
Amrani
,
D.
, and
Atherthon
,
D. P.
,
1989
, “
Designing Autonomous Relay Systems With Chaotic Motion
,”
Proceedings 28th IEEE Conference on Decision and Control
,
Tampa, FL
, pp.
932
936
.
13.
Cook
,
P. A.
,
1985
, “
Simple Feedback Systems With Chaotic Behaviour
,”
Syst. Control Lett.
,
6
, pp.
223
227
.10.1016/0167-6911(85)90071-4
14.
Genesio
,
R.
, and
Tesi
,
A.
,
1990
, “
Chaos Prediction in a Third-Order Relay System
,” Dipartimento di Sistemi ed Informatica, University of Florence, Italy, Internal Report No. RT 29/90.
15.
Kuznetsov
,
Y. A.
,
1998
,
Elements of Applied Bifurcation Theory
,
second ed.
,
Springer-Verlag
,
New York
.
16.
Perko
,
L.
,
2001
,
Differential Equations and Dynamical Systems (Texts in Applied Mathematics)
,
third ed.
,
Springer-Verlag
,
New York
.
17.
Guckenheimer
,
J.
, and
Holmes
,
P.
,
1986
,
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
,
Springer-Verlag
,
New York
.
18.
Champanerkar
,
J.
, and
Blackmore
,
D.
,
2007
, “
Pitchfork Bifurcations of Invariant Manifolds
,”
Topol. Appl
,
154
, pp.
1650
1663
.10.1016/j.topol.2006.12.014
19.
Xu
,
K. D.
,
1995
, “
Stochastic Pitchfork Bifurcation: Numerical Simulations and Symbolic Calculations Using MAPLE
,”
Math. Comput. Simul.
,
38
, pp.
199
209
.10.1016/0378-4754(93)E0083-H
20.
Varela
,
S.
,
Masoller
,
C.
, and
Sicardi
,
A. C.
,
2000
, “
Numerical Simulations of the Effect of Noise on a Delayed Pitchfork Bifurcation
,”
Physica A
,
283
, pp.
228
232
.10.1016/S0378-4371(00)00158-8
21.
Lasalle
,
J. P.
,
1986
,
The Stability and Control of Discrete Processes
,
Springer
,
Berlin
.
22.
Brown
,
B. M.
,
1965
, “
On the Distribution of the Zeros of a Polynomial
,”
Q. J. Math.
,
16
, pp.
241
256
.10.1093/qmath/16.3.241
You do not currently have access to this content.