47
Proceedings, 5th Chaotic Modeling and Simulation International
Conference, 12 – 15 June 2012, Athens Greece
PRNG shows that statistical properties of the randomness ofthe sequences
generated via the our PRNG and the RC4 PRNG do not have signicant
dierences.
Our results conrm onceagainthat suitabledesigned chaos-based PNGs
may generate sound random sequences, in particular for a replacement for
the one-timepad system[9]. Furtherresearch along this lineispromising.
Acknowledgements
L. Min would like to thank Professor Leon O. Chua at the UB Berkeley
for directing him to study the fascinating chaos eld. This work is jointly
supported by theNNSF of China (Nos. 61074192, 61170037).
References
1.K.Binder, andD.W.Heermann, Monte Carlo SimulationinStatisticalPhysics:
AnIntroduction(4thedition). 2002. Springer.
2.N.Ferguson,B.Schneier,andT.Kohno,CryptographyEngineering: DesignPrin-
ciplesand PracticalApplications.,2010.Wiley Publishing.
3.S.Wegenkittl,Gamblingtestsforpseudorandomnumbergenerator,Mathematics
and ComputersinSimulation, 55: 281-288,2001.
4.NIST,FIPSPUB 140-2, securityrequirements forcryptographic modules.2001.
5.G.Marsaglia,http://www.stat.fsu.edu/pub/diehard/,1996[2012-03-30].
6.R. Rukhin, J. Soto, J. Nechvatal et al., A statistical test suite for random and
pseudorandomnumbergeneratorforcryptographicapplications,page64,NIST
SpecialPublication,2001.
7.E. N. Lorenz, Deterministic nonperiodic
ow, J. of Atmospheric Sciences, Vol.
20(2): 2130{148, 1963.
8.T. Y. Li and J. A. York, Period three implies chaos,American Mathematical
Monthly,82(10): 481{485, 1975.
9.R. A. J. Matthews, On the derivation of achaotic encryption algorithm, Cryp-
tologia, XIII(1): 29{42,1989.
10.T.Stojanovskiand L.Kocarev01, Choas-basedrandomnumber generators-part
I:analysis,IEEETransactiononCircuittsandSystems-I: FundamentalTheory
and Applications,48(3): 281{288, 2001.
11.L. Gamez-Guzman, C. Cruz-Hernandez, R.M. Lerrez et al., Synchronization
of Chua’s circuits with multi-scroll attractors Application to communication,
CommumNonlinearSci Numer Simulat,14: 2765{2775,2009.
12.C. Li, S. Li, G. Alvarez et al., Cryptanalysis of a chaotic block cipher with
externalkeyanditsimprovedversion,ChaosSolitons& Fractals,37: 299{307,
2008.
13.X. Yu, L. Min, and T. Chen, Chaos criterion on some quadric polynomial
maps and design for chaoticpseudorandom number generator.InProc. of the
2011 Sewventh Int. Conf. on Nutural Computation(26-28July 2011, Shaghai,
China), Vol.3: 1399{1402,2011.
14.S. W. Golomb, Shift Register Sequences. Revised edition, CA: Aegean Park,
1982.LagunaHills.
15.J. C. Sprott, Chaos and Time-Sries Analysis, page 427, Oxford. 2003. Oxford
UniversityPress.
49
Proceedings, 5th Chaotic Modeling and Simulation International
Conference, 12 – 15 June 2012, Athens Greece
AMulti-Input Multi-Output Delayed
Feedback Controller for Stabilizing Periodic
Solutions of the Lorenz System
Soraia Moradi
1;2
,Ali Khaki Sedigh
1
and Nastaran Vasegh
1;3
1
Advanced Process Automation and Control(APAC) Research Group, K. N.
ToosiUniversityof Technology,Tehran,Iran
2
Automation and ControlInstitute (ACIN),ViennaUniversityof Technology,
Vienna, Austria(E-mail: e1126740@student.tuwien.ac.at)
3
Department of Electrical Engineeringand Computer, Shahid RajaeeTeacher
TrainingUniversity,Lavizan, Tehran, Iran
Abstract. In this paper the idea of harmonic balance method is used in a new
frameworktoanalyzeand predict the periodic solutions of the Lorenz system. An
analyticequationhasbeenderived for thesepredictedlimitcyclesforthersttime.
The proposed method is fairly straightforward avoiding complicated calculations.
Amulti-input multi-output Delayed Feedback Controller (DFC) is designed and
implemented for stabilizing unstable periodic solutions of the Lorenz system. All
previous works done on stabilization of periodic solutions of this system, using
asimple DFC (without adding a new dynamic to the system) were unsuccessful.
Choosingan appropriate signal tousein the delayed feedback loop and an appro-
priate point for introducing the control signal are very important tasks in DFC
implementation. Consideringthese facts, we overcome the mentioned problem by
choosing the third state variable of the Lorenz system that toour knowledge has
notbeen used before, in thedelayedfeedbackloop andintroducingthecontrolsig-
naltothesystemin adierentpointfromprevious works. Ourproposed controller
is also able to stabilize the equilibrium points (EPs) of the system. The stability
analysis is alsodone.
Keywords: Lorenz system, delay feedback control, harmonic balance.
1 Introduction
DFC is an ecient method of chaos control, which stabilizes Unstable Pe-
riodic Orbits (UPO) embedded in a chaotic attractor. In 1994 researchers
found outthatDFC isnotableto stabilizesystemswith oddnumberofFlo-
quetexponents. Inotherwords,theythoughtitisimpossibleto stabilizeany
UPOs with odd-number of real characteristic multipliers greater than unity
[1{3]. So they tried to overcome this limitation. In [4] authors used an ex-
pandedDFC. In[5] itwasshownthatthisstablecontrollercannotovercome
all the DFCs limitations. Since they thought these limitations were due to
the odd number of positive Floquet exponents, in later studies, researchers
tried to solve this problem by adding an unstable term to change the total
number of real-positive Floquet exponentto an even number[6,7]. Another
58
Proceedings, 5th Chaotic Modeling and Simulation International
Conference, 12 – 15 June 2012, Athens Greece
method was using some dierentvalues of delay in the feedback to increase
the controllersdegrees of freedom[8,9].
Variety of methods were suggested to eliminate this problem till 2007,
but in[10] it was shown that, this limitation that scientists were trying to
overcome for more than fteen years did not exist at all and theoretical
analysis and simulations conrmed this fact too [11]. One of the systems
thatwasthoughtitcannotbestabilizedwiththeuseoftheDFC,duetothe
\odd-number limitation" was theLorenzsystem. Several studies weredone
toavoidthislimitation(see[7,12,13]. Inallthesestudies, itwastriedtoavoid
thelimitationbyintroducinganunstabledegreeoffreedominafeedbackloop
to changethenumberofunstabletorsion-freemodesto anevennumberand
the control signal was just applied to one of the state variables (the second
one).
In this paper we use a simple DFC to stabilize an unstable periodic so-
lution of the Lorenz system. The key idea of our work is using the third
state variable of the Lorenzsystem in thecontrol loop and introducing the
controller to both the second and thethird state equations (seeEq.1). The
next section is devoted to the open loop analysis of the system. Using the
Harmonic Balance (HB) idea, the analytical predicted periodic solutions of
the systemhavebeenderived and theirstabilityfeatures havebeen studied.
In section 3 an analysis is done to predict the chaotic dynamics and nally
in section4, a MIMO DFC is used to stabilizean unstable periodic solution
of thesystem. Also it has shown thatthiscontrol structurecan beused for
stabilization oftheEPs of the system.
2 The open loop analysis
Consider thefollowing classical Lorenz chaoticsystem
8
<
:
x_ = x+y
y_ =x y zx
z_ = z+xy
(1)
The Lorenz equations have three parameters , and . To simplify
matters, most researchers have kept = 10 and = 8=3 while varying .
As shown in [14] byassuming f = x the system equations can be rewritten
in thefollowing form
1
+(1+
1
)
_
f+(1 )+fz=0
_z= z+f(
1
_
f+f)
(2)
Eq.2 puts in evidence the feedback structure of the system, as shown in
Fig.1 wherea linear subsystemis connected to a nonlinear one. Due to the
existenceofthedynamicalterm1=(s+)inthenonlinearsubsystemofFig.1,
it may be dicult to use the general approach originally proposed by Tesi
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Proceedings, 5th Chaotic Modeling and Simulation International
Conference, 12 – 15 June 2012, Athens Greece
Fig.1. Thefeedbackstructure of the Lorenzsystem.
and Genesio in 1992 to nd the periodic solutions of thesystem. So in this
paper we use the idea of the well-known HB method in thefollowing simple
manner.
2.1 Periodic solutions
At the rst step, it is assumed that these steady solutions can be approxi-
mated as
x=A+Bcos!t
(3)
Substituting Eq.3 in the rststate equation ofEq.1 results in
y=
1
_x+x=A+Bcos(!t)
1
B!sin(!t)
(4)
Then by substituting Eq.3 and Eq.4 in the third state equation of Eq.1,
eliminating the second harmonics which appear, and ignoring the transient
solution, weconclude the steadysolutionof z as
z=
3
80
(10A
2
+5B
2
+
8AB(160+3!
2
)cos(!t)
64+9!2
+
416AB!sin(!t)
64+9!2
) (5)
After substituting Eq.3 and Eq.5 in the second state equation of Eq.1 and
doingsimilarcalculations,weobtainan expressionfor ythatshouldbeequal
to Eq.4. Equalizing the related coecients results in the following three
equations forA
2
,B
2
and!
2
in terms of
A
2
=
2
1561512117
(527280000
p
3m 3845514915+4056000
p
3m
3
12168000
p
3
2
m+106221667541 9892726305
p
3m) (6)
B
2
=
4
425866941
(21884475789 2258100
p
3m
3
+4498028807
p
3m
319929891 293553000
p
3m+6774300
p
3
2
m) (7)
!
2
=
40
201
( 1113+39+13
p
3m)
(8)
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Proceedings, 5th Chaotic Modeling and Simulation International
Conference, 12 – 15 June 2012, Athens Greece
where m =
p
2123 130+32. The solution of the problem is possible,
whenthetermsA
2
,B
2
and!
2
arerealandpositive. Therefore, wecanderive
the domain of parameter space where there are admissible solutions. The
domain ofexistenceis
7:7693 24:7368
(9)
Theobtained results on periodic solutions (Predicted Limit Cycles) are ap-
proximate, dueto therstharmonicanalysiscarriedoutonthesystem. The
reliabilityofa PLC isbased on a strong attenuation(lteringhypothesis) of
the higher frequencycomponents2!, 3! , along theloop.
2.2 Stability analysis
ThesystemEPs are: C
=(
p
(( 1));
p
(( 1)); 1) that exist
for > 1 and C
0
= (0;0;0). In this region C
0
is a saddle and C
are
symmetric stable xed points. For 0 < 1 there exist just C
0
which is
astable node. We use the Loeb criterion to check the stability features of
PLCs. according to this criteria, in case the PLC be stable, the following
inequalitywill betrue[14]:
!
B
=
!=
B=
0
(10)
That istruein ourcasefor 15:1 .
3 The chaotic dynamic prediction
In this section the famous phenomenon of Homoclinic Orbit (HO) which
is one of the main routes to chaos in the most dynamic systems has been
analyzed and an approximate region ofparameter space is derived in which
thisphenomenonmayoccur. Asstatedin[15],theHOconditionsincludesthe
existenceofastablePLCandasaddletypeEP(dierentfromthatgenerating
the PLC) and the interaction between PLC and EPas B jE Aj, where
Edenotes the mentioned saddle EPand is equal to zero in theLorenzcase.
This inequality is valid for 7:77 15:06. Considering the regions of
PLC existence (Eq.9), PLC stabilization ( 15:1) and the region in which
the interaction condition is satised (7:77 15:06), we predict that
the HO phenomenon may occur at some values around = 15. Therefore
the Lorenz system may show chaotic behaviors. Numerical solutions show
thatHomoclinicbifurcationoccurs at =13:962 whichisnearthepredicted
value.
4 Chaos control
It is obvious that this system exhibits a chaotic behaviorin someregions of
itsparameter space, for exampleat =24:5. Whereas theeorts for stabi-
lization of this system with the use of a SISO DFC have notbeen successful
140
Proceedings, 5th Chaotic Modeling and Simulation International
Conference, 12 – 15 June 2012, Athens Greece
Fig.2.(a) MIMO Lure form of the Lorenzsystem. (b) The closed loop system.
uptonow,inthispaperweconsiderthesystemasa MIMO systemwiththe
LureformshowninFig.2a; whichLandndenoterespectivelythedual-input
dual-output linear andnonlinear parts of the system.
n=
zx
xy
(11)
L:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
2
4
x_
y_
z_
3
5
=
2
4
0
1 0
0 0
3
5
2
4
x
y
z
3
5
+
2
4
00
10
01
3
5
u
1
u
2
y
1
y
2
=
10 0
00 1
2
4
x
y
z
3
5
=)L(s)=
"
s2+(1+)s+(1 )
0
0
1
s+
#
(12)
Thegoal istodesigna MIMODFCto stabilizeanunstableperiodicsolution
of the system. The closed loop system is shown in Fig.2b. So the MIMO
DFC will bein the following form
U
=
k
11
k
12
k
21
k
22
x(t ) x(t)
z(t ) z(t)
(13)
Theaim is to determine the gain matrix and delay() of the controller, so
that the closed loop system has a periodic responsein the form of Eq.3 for
( =24:5). For simplicity we consider a simple casethat is in the following
form
U
=
u
1
u
2
=
0k
0k
x(t ) x(t)
z(t ) z(t)
(14)
So our suggested closed loop system isas follows
8
<
:
_x= x+y
_y=x y zx+k(z(t ) z(t))
_z= z+xy+k(z(t ) z(t))
(15)
21
Proceedings, 5th Chaotic Modeling and Simulation International
Conference, 12 – 15 June 2012, Athens Greece
Fig.3.(a) Controlsignal tendstozero. (b) The steadystate periodicresponses of
the closed loop system.
coincideswiththeperiodofthedesiredperiodicsolutionoftheclosedloop
system. Incasestabilizationbesuccessful, thecontrol signal will vanish and
there will not beany power dissipation in the feedback loop. So by setting
=2=! andusing theapproximatione e s 1 s, wetrytodeterminek.
OncemorelookingbacktoEq.3,afterdoingsomecalculationssimilartothose
of section (2.1)and substituting A, B and ! with their valuesat ( =24:5)
from Eqs. 6-8 (A =7:8685;B = 1:0405;! = 9:5104 rad=s) T = 0:6607 s),
the controller’s gain k = 2:5227 is obtained. The value obtained for delay
here(T =)isnearlyequal tothevalueobtainedforitin[7]thatwas0:67s.
Fig.3a and Fig.3b show the control signal and the steady state stable
periodicresponsesoftheclosedloopsystem. Thecontrol signal tendstozero
which meansthat thestabilization strategyhasbeen successful.
Fig.4 showsa zoom view ofthesteadyresponseoftherststatevariable
of thesystem(x(t)). It illustrates that thebias A, amplitudeB and period
Tofx areequalto thoseobtainedfromEqs.6-8at =24:5andconrmsthe
accuracy of theimplemented analytical approach. Anoticeablepoint about
Fig.4. The bias, amplitude and period of the statevariablex.
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