Transcript Nonlinear recursive chaos control
Chaos Control Part II
Amir massoud Farahmand [email protected]
Review • Why Chaos control?!
– 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 T periodic 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Control Signal”, unpublished x
1 (
k x
2 (
k
1 ) 1 ) 1 .
4
x
1 2 0 .
3
x
1
x
2 1
u u
(
k
)
Kx
(
k
0 ) x(k) x * otherwise Figure 1 Henon map
Chaos Control Conventional + Chaotic threshold = 1.0
threshold = 1.0
1 0.5
0 x1 x2 0.5
0 -0.5
-1 -0.5
0 50 k threshold = 0.1
100 -1.5
0 50 k threshold = 0.1
2 1 0 -1 -2 0 50 k x1 x2 100 0.03
0.02
0.01
0 -0.01
-0.02
0 50 k Figure 2. Sample response for two different attraction threshold 100 100 100 90 80 70 60 50 40 30 20 10 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
Chaos Control Conventional + Chaotic 0.35
0.3
0.25
0.2
0.15
0.1
0.05
0 0 0.1
0.2
0.3
0.4
0.5
theta 0.6
Figure 3. Peak of control signal for different theta 0.7
0.8
0.9
1 0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0 0.1
0.2
0.3
0.4
0.5
theta 0.6
Figure 4. Controlling energy 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 – Brain and heart regulation – Liquid mixing – Secure communication – Small control
(Chaotification of non-chaotic system
chaos control method (small control) methods )
conventional
Anti-Control of Chaos Discrete case (I) dynamic with a proper feedback such that it 1.
is bounded 2. has positive Lyapunov exponent then we may have made it chaotic.
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 … • Formulation:
S i
: i
F i
(
x
1 ,
x
2 ,...,
x k
,
t
) , i 1,..., k
Q i
( 1
x
1 ,...,
k x k
,
t
) 0 , i 1,..., k
t
lim
Q i
( 1
x
1 ,...,
k x k
,
t
) 0 , i 1,..., k
Q
(
x
1 ,
x
2 )
x
1 (
t
)
x
2 (
t
) • Synchronization – Unidirectional (Model Reference Control) – Mutual
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
V
x
(
t
),
t V x
(
t
0 ),
t
0
t t
0
u
( ),
y
( )
H
(
x
( ))
d
0 ; x
H
(
x
) 0 Isidori normal form
z
i i
a
(
z i
,
y i
)
q
(
z i
,
y i
)
b i
(
z i
,
y i
)
u i
i 1,2 • Control Signal
u
1
y
1
y
2
u
2
u
1
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 T
(
y
) 0
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
V
0 (
z
1
z
2 )
q
(
z
1 ,
y
1 )
q
(
z
2 ,
y
1 )
z
1
z
2 2 – A4.The matrix b1+b2 is PD:
b
1 (
z
1 ,
y
1 )
b
2 (
z
2 ,
y
2 )
I m
, 0 – A5.
y T
(
y
) then there exist … achieved.
y
2 that goal of synchronization is
Synchronization of Semipassive systems (IV) • Lorenz system ( Turbulent dynamics of the thermally induced fluid convection in the atmosphere) 100 0 -100 -200 -300 -400 -500 -600 0 5 10 15 20 25 100 80 60 40 20 0 -20 -40 -60 -80 -100 0 5 10 T
Figure 1. error and control signal for linearly coupled system
T 15 20 25