Chaos Control
Part II
Amir massoud Farahmand
SoloGen@SoloGen.net
Review
• Why Chaos control?!
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–
–
–
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THE BEGINNING WAS CHAOS!
Chaos is Fascinating!
Chaos is Everywhere!
Chaos is Important!
Chaos is a new paradigm shift in science!
Review II
What is it?!
• Nonlinear dynamics
• Deterministic but looks stochastic
• Sensitive to initial conditions (positive Bol
(Lyapunov) exponents)
• Strange attractors
• Dense set of unstable periodic orbits
(UPO)
• Continuous spectrum
Review III
Chaos Control: Goals
• Stabilizing Fixed points
• Stabilizing Unstable Periodic Orbits
• Synchronizing of two chaotic
dynamics
• Anti-control of chaos
• Bifurcation control
Review IV
Chaos Control: Methods
• Linearization of Poincare Map
– OGY (Ott-Grebogi-York)
• Time Delayed Feedback Control
• Impulsive Control
– OPF (Occasional Proportional Feedback)
• Open-loop Control
• Conventional control methods
Chaos Control
Conventional control
• Back-stepping
– A. Harb, A. Zaher, and M. Zohdy, “Nonlinear recursive
chaos control,” ACC2002.
• Frequency domain methods
– Circle-like criterion to ensure L2 stability of a
T-periodic solution subject to the family of Tperiodic forcing inputs.
– M. Basso, R. Genesio, and L. Giovanardi, A. Tesi,
“Frequency domain methods for chaos control,” 2000.
Chaos Control
Conventional + Chaotic
• Taking advantage of inherit properties of chaotic
systems
• Periodic Chaotic systems are dense (according to
Devaney definition)
• Waiting for the sufficient time, every point of
the attractor will be visited.
• If we are sufficiently close to the goal, turn-on
the conventional controller, else do nothing!
– T. Vincent, “Utilizing chaos in control system design,” 2000.
Chaos Control
Conventional + Chaotic
• Henon map
• Stabilizing to the unstable fixed point
• Locally optimal LQR design
•
Farahmand, Jabehdar, “Stabilizing Chaotic Systems with Small
Control Signal”, unpublished
x1 (k  1)  1.4 x12  x2  1  u
x2 (k  1)  0.3x1

 Kx(k )
u (k )  

0
x(k)- x*  
otherwise
Figure 1 Henon map
Chaos Control
Conventional + Chaotic
threshold = 1.0
threshold = 1.0
control effort
states
x1
0.5
x2
0
-0.5
0
50
k
threshold = 0.1
100
x1
0
x2
control effort
states
1
-1
90
0
80
-0.5
70
-1
-1.5
2
-2
100
0.5
0
50
k
threshold = 0.1
100
50
k
100
60
50
0.03
40
0.02
30
0.01
20
0
10
-0.01
0
Settling time
1
-0.02
0
50
k
Figure 2. Sample response for two different attraction threshold
100
0
0.1
0.2
0.3
0.4
0.5
theta
0.6
Figure 5. Settling time for different theta
0.7
0.8
0.9
1
0.35
0.7
0.3
0.6
0.25
0.5
energy of u(t)
peak of u(t)
Chaos Control
Conventional + Chaotic
0.2
0.15
0.4
0.3
0.2
0.1
0.1
0.05
0
0
0
0.1
0.2
0.3
0.4
0.5
theta
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
theta
0.6
Figure 4. Controlling energy for different theta
Figure 3. Peak of control signal for different theta
0.7
0.8
0.9
1
Chaos Control
• Impulsive control of periodically forced chaotic
system
• Z. Guan, G. Chen, T. Ueta, “On impulsive control of
periodically forced pendulum system,” IEEE T-AC, 2000.
Anti-Control of Chaos
Definitions and Applications (I)
• Anti-control of chaos (Chaotification)
is
– Making a non-chaotic system, chaotic.
– Enhancing chaotic properties of a
chaotic system.
Anti-Control of Chaos
Definition and Applications (II)
• Stability is the main focus of traditional
control theory.
• There are some situations that chaotic
behavior is desirable
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Brain and heart regulation
Liquid mixing
Secure communication
Small control (Chaotification of non-chaotic system 
chaos control method (small control)  conventional
methods )
Anti-Control of Chaos
Discrete case (I)
Suppose we have a LTI system. If we change its
dynamic with a proper feedback such that it
1.
2.
is bounded
has positive Lyapunov exponent
then we may have made it chaotic.
We may use Marotto theorem to prove the
existence of chaos in the sense of Li and Yorke.
X. Wang and G. Chen, “Chaotification via arbitrarily small
feedback controls: theory, methods, and applications,”
2000.
Anti-Control of Chaos
Discrete case (II)
Anti-Control of Chaos
Discrete case (III)
Anti-Control of Chaos
Continuous case (I)
• Approximating a continuous system by its
time-delayed version (Discrete map).
• Making a discrete dynamics chaotic is
easy.
• It has not been proved yet!
• X. Wang, G. Chen, X. Yu, “Anticontrol of chaos in continuous-
time systems via time-delayed feedback,” 2000.
Anti-Control of Chaos
Continuous case (II)
Synchronization
(I)
• Carrier Clock, Secure communication,
Power systems and …
Si : x i  Fi ( x1, x2 ,...,xk , t ), i  1,...,k
• Formulation:
Qi (1 x1,..., k xk , t )  0,
i  1,...,k
lim Qi (  1 x1 ,...,  k xk , t )  0,
t 
Q( x1, x2 )  x1(t )  x2 (t )
• Synchronization
– Unidirectional (Model Reference Control)
– Mutual
i  1,...,k
Synchronization
(II)
• Linear coupling
Synchronization
(III)
• Drive-Response concept of Pecora-Carroll
• L.M. Pecora and T.L. Carol, “Synchronization in
chaotic systems,” 1990.
Synchronization of
Semipassive systems (I)
• A. Pogromsky, “Synchronization and adaptive
synchronization in semipassive systems,” 1997.
• Semipassive Systems
t
,
V x(t ), t   V x(t0 ), t0    u ( ), y ( )  H ( x( )) d
t0
  0; x    H ( x)  0
Isidori normal form
• Control Signal
 yi  a( zi , yi )  bi ( zi , yi )ui
i  1,2

 zi  q( zi , yi )
u1   y1  y2 
u2  u1
Synchronization of
Semipassive systems (II)
• Lemma: Suppose that previous systems
are semipassive with radially unbounded
continuous storage function. Then all
solutions of the coupled system with
following control exist on infinite time
interval and are bounded.
 ( y)   ( y)
y  ( y)  0
T
Synchronization of
Semipassive systems (III)
• Theorem I: Assume that
– A1. The functions q, a, b are continuous and locally
Lipschitz
– A2.The system is semipassive
– A3.There exist C2-smooth PD function V0 and … that
V0 ( z1  z2 )  q( z1 , y1 )  q( z2 , y1 )   
T
z1  z2
– A4.The matrix b1+b2 is PD:
b1 ( z1, y1 )  b2 ( z2 , y2 )  I m ,   0
– A5.
yT ( y)   y
2
then there exist …  that goal of synchronization is
achieved.
2
Synchronization of
Semipassive systems (IV)
• Lorenz system (Turbulent dynamics of the
thermally induced fluid convection in the atmosphere)
100
100
80
0
60
40
control effort
error(dB)
-100
-200
-300
20
0
-20
-40
-400
-60
-500
-80
-600
0
5
10
15
20
25
-100
0
5
10
T
Figure 1. error and control signal for linearly
coupled system
15
T
20
25