1. LOGO
PID Control of Multi-wing Attractors in Lorenz-like
Chaotic Nonlinear Dynamical Systems
Presented by:
Anish Acharya
Department of Instrumentation and Electronics
Engineering, Jadavpur University, Salt-Lake Campus,
LB-8, Sector 3, Kolkata-700098, India
ICCNT 2012, Paper ID: ICCCNT1108-9
2. LOGO
Overview of the Presentation
Need for chaotic control
Commonly available chaotic control methods
Brief Description of the Multi-wing Chaotic Attractors to be Controlled
PID Control of Multi-wing Chaotic Attractors
Simulations and results
Conclusion
Reference
3. LOGO
Need for chaotic control
nonlinear, highly initial condition sensitive
exhibits aperiodic oscillations in the time series of the state variables.
order in disorder.
Chaotic systems may cause trouble due to their unusual, unpredictable
behavior.
The prime objective is to suppress the chaotic oscillations completely or
reduce them to regular oscillations.
4. LOGO
Commonly available chaotic control methods
Open loop control methods, adaptive control methods, traditional linear
and non linear control methods, fuzzy
infinite number of unstable periodic orbits
continuous switching from one orbit to the other
unpredictable, random wandering of chaotic states over longer period of
time.
stabilization, by means of small system perturbations, of one of these
unstable periodic orbits.
5. LOGO
Multi-wing Chaotic Attractors to be Controlled
• Basics of Multi-wing Attractors:
Square /cross terms in state equation are replaced by:
• Chaotic Multi-wing Lu System
( ) ( ) ( )2
0
1
1 0.5sgn 0.5sgn
N
i i i
i
f x F x F x E x E
=
= − + − − + ∑
( )
1 for 0
sgn 0 for 0
1 for 0
x
x x
x
>
= =
− <
6. LOGOChaotic Multi-wing Lu System
• The double-wing Lu system is represented by:
• Typical parameter settings :
• Equilibrium points :
• State equations to be controlled:
x ax ay
y cy xz
z xy bz
=− +
= −
= −
&
&
&
36, 3, 20a b c= = =
( ) ( )0,0,0 ; , ,bc bc c± ±
( )
( )
1
x ax ay
y cy P xz u
z f x bz
=− +
= − +
= −
&
&
&
0 1 2 3 4
1 2 3 4
0.05, 100, 10, 12, 16.67, 18.18,
0.3, 0.45, 0.6, 0.75
P F F F F F
E E E E
= = = = = =
= = = =
8. LOGO
Chaotic Multi-wing Rucklidge System
• The double-wing Shimizu-Morioka system is represented by
• Typical parameter settings :
• Equilibrium points :
• The state equations :
• The suggested parameters for
2
x ax by yz
y x
z y z
=− + −
=
= −
&
&
&
2, 7.7a b= =
( ) ( )0,0,0 ; 0, ,b b±
( ) ( )
( )
1
x ax ay
y c a x cy P xz u
z f x bz
= − +
= − + − +
= −
&
&
&
0 1 2 3 1 2 30.5, 4, 9.23, 12, 18.18, 1.5, 2.25, 3.0P F F F F E E E= = = = = = = =
3N =
10. LOGO
Chaotic Multi-wing Sprott-1 System
The double-wing Sprott-1 system is represented by:
• equilibrium points :
• State equations:
• The suggested parameters for :
21
x yz
y x y
z x
=
= −
= −
&
&
&
( )1, 1,0± ±
( )1
x yz
y x y u
z f x
=
= − +
= −
&
&
&
4N =
0 1 2 3 4 1 2 3 41, 5, 5, 6.67, 8.89, 2, 3, 4, 5F F F F F E E E E= = = = = = = = =
12. LOGO
PID Control of Multi-wing Chaotic Attractors
Each of the above four multi-wing chaotic systems are to be controlled using a PID
controller
• This enforce the second state variable (y) to track the unit reference step signal(r)
• Integral of Time multiplied Absolute Error has been taken as the performance index
(J) for fast tracking of the second state.
.p i d
de
u K e K e dt K
dt
e r y
= + +
= −
∫
( ) ( ) ( )
0 0
J t e t dt t r t y t dt
∞ ∞
= = −∫ ∫
13. LOGO
Contd…
• Chaotic systems are governed by nonlinear differential equations
• GA based PID controller design with time domain optimization methods adopted to
control multi wing attractors
31. LOGO
CONCLUSION
GA based optimum PID controllers are designed to suppress chaotic
oscillations in few highly complex multi-wing Lorenz like chaotic systems.
The controller enforces fast tracking of the second state which also damps
chaotic oscillation in the other states and found to be robust enough for
different initial conditions for such typical nonlinear dynamical systems.
32. LOGO
REFERENCES
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Berlin, 2003.
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2005.
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Review Letters, vol. 64, no. 11, pp. 1196-1199, 1990.
4. K. Pyragas, “Continuous control of chaos by self-controlling feedback”, Physics
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1, pp. 167-175, Oct. 2005.
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