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Research Papers

Nonfragile Fuzzy Output Feedback Synchronization of a New Chaotic System: Design and Implementation

[+] Author and Article Information
A. Azarang, M. Miri, S. Kamaei

Department of Communication
and Electronic Engineering,
School of Electrical and Computer Engineering,
Shiraz University,
Shiraz 71867-88773, Iran

M. H. Asemani

Department of Power and Control Engineering,
School of Electrical and Computer Engineering,
Shiraz University,
Shiraz 71867-88773, Iran
e-mail: asemani@shirazu.ac.ir

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 27, 2017; final manuscript received July 11, 2017; published online October 9, 2017. Assoc. Editor: Katrin Ellermann.

J. Comput. Nonlinear Dynam 13(1), 011008 (Oct 09, 2017) (11 pages) Paper No: CND-17-1091; doi: 10.1115/1.4037416 History: Received February 27, 2017; Revised July 11, 2017

A new three-dimensional (3D) chaotic system is proposed with four nonlinear terms which include two quadratic terms. To analyze the dynamical properties of the new system, mathematical tools such as Lyapunov exponents (LEs), Kaplan–York dimensions, observability constants, and bifurcation diagram have been exploited. The results of these calculations verify the specific features of the new system and further determine the effect of different system parameters on its dynamics. The proposed system has been experimentally implemented as an analog circuit which practically confirms its predicted chaotic behavior. Moreover, the problem of master–slave synchronization of the proposed chaotic system is considered. To solve this problem, we propose a new method for designing a nonfragile Takagi–Sugeno (T–S) fuzzy static output feedback synchronizing controller for a general chaotic T–S system and applied the method to the proposed system. Some practical advantages are achieved employing the new nonlinear controller as well as using system output data instead of the full-state data and considering gain variations because of the uncertainty in values of practical components used in implementation the controller. Then, the designed controller has been realized using analog devices to synchronize two circuits with the proposed chaotic dynamics. Experimental results show that the proposed nonfragile controller successfully synchronizes the chaotic circuits even with inexact analog devices.

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Figures

Grahic Jump Location
Fig. 1

Chaotic attractor of the proposed system in 3D phase space, for the parameters: (a, b, c, d, e, f, g, h, m, n) = (10, 10, 1.5, 3, 70, 1, 0.5, 0.5, 0.5, 5)

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Fig. 3 (a)

Bifurcation diagram and (b): LEs of the system as a function of d

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Fig. 4

Circuit diagram for Eq. (9). AD633: Analog multiplier, TL084 low-cost quad op-amps. R1 = 100 kΩ, R2 = 5 kΩ, R3 = 7.5 kΩ, R4 = 25 kΩ, R5 = 10 kΩ, R6 = 150 kΩ, R7 = 10 kΩ (potentiometer), R8 = 1 kΩ, R9 = 1.2 kΩ, R10 = 1.6 kΩ, R11 = 56 kΩ, R12 = 680 Ω, R13 = 9 kΩ, R14 = 12 kΩ, R15 = 40 kΩ, R16 = 20 kΩ, R17 = 250 kΩ, R18 = 1 MΩ, R19 = 3 kΩ, and C = 200 nF. In all circuits we have used supply voltages V+ = +9 V and V− = −9 V.

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Fig. 5

(a) 2D attractor of the state variable of the designed and implemented circuit, in xz phase plane and (b) attractors in xz phase plane, achieved from solving system (9)

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Fig. 6

Synchronization errors of x, y, and z components of the two chaotic systems

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Fig. 7

Control signal u(t) applied to slave system

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Fig. 8

(a) Block diagram of the designed controller in Eq. (48) and (b) Internal circuitry of block B1; TL084 low-cost quad op-amps, AD633 analog multiplier, R = 1 kΩ, R1 = 1.5 kΩ, R2 = R3 = 6.8 kΩ, R4 = R5 = 1 kΩ, R6 = R7 = 7.5 kΩ, R8 = 1.3 kΩ, R9 = R10 = 7.5 kΩ, R11 = R12 = 1.6 kΩ, R13 = R14 = 6.8 kΩ, R15 = R16 = 1 kΩ, R18 = R19 = 10 R17 = 10 kΩ, R21 = 19 R20 = 19 kΩ, R23 = R24 = 10 R22 = 10 kΩ, R26 = 29 R25 = 29 kΩ and R27 = R28 = R29 = 1 kΩ. In all circuits, we have used supply voltages V+ = +9 V and V− = −9 V.

Grahic Jump Location
Fig. 9

Circuit diagram for the slave system. AD633: Analog multiplier, TL082 and TL084 low-cost dual and quad op-amps, respectively. R1 = 100 kΩ, R2 = 5 kΩ, R3 = 7.5 kΩ, R4 = 25 kΩ, R5 = 10 kΩ, R6 = 150 kΩ, R7 = 10 kΩ (potentiometer), R8 = 1 kΩ, R9 = 1.2 kΩ, R10 = 1.6 kΩ, R11 = 56 kΩ, R12 = 680 Ω, R13 = 9 kΩ, R14 = 12 kΩ, R15 = 40 kΩ, R16 = 20 kΩ, R17 = 250 kΩ, R18 = 1 MΩ, R19 = 3 kΩ, R20 = 1.5 kΩ, and C = 200 nF. In all circuits, we have used supply voltages V+ = +9 V and V− = −9 V.

Grahic Jump Location
Fig. 10

Before applying the control signal to the slave system. (a) Xm(t) as a function of Xs(t) and (b) error between x components of the master and slave systems; e1(t) = Xm(t) − Xs(t).

Grahic Jump Location
Fig. 11

After applying control signal to the slave system. (a) Xm(t) as a function of Xs(t) and (b) error between x components of the master and slave systems; e1(t) = Xm(t) − Xs(t).

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Fig. 12

Measured synchronization errors of x, y, and z components of the two chaotic systems, dotted: data captured from the oscilloscope, solid: curve fitted by the interpolation

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