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Research Papers

Control of Discrete Time Chaotic Systems via Combination of Linear and Nonlinear Dynamic Programming

[+] Author and Article Information
Kaveh Merat

Department of Mechanical Engineering,
Sharif University of Technology,
PO Box 11155-9567,
Tehran, Iran
e-mail: k_merat@mech.sharif.edu

Jafar Abbaszadeh Chekan

Department of Mechanical Engineering,
Sharif University of Technology,
PO Box 11155-9567,
Tehran, Iran
e-mail: jafar.abbaszadeh@gmail.com

Hassan Salarieh

Department of Mechanical Engineering,
Sharif University of Technology,
PO Box 11155-9567,
Tehran, Iran
e-mail: salarieh@sharif.edu

Aria Alasty

Department of Mechanical Engineering,
Sharif University of Technology,
PO Box 11155-9567,
Tehran, Iran
e-mail: aalasti@sharif.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 18, 2013; final manuscript received May 19, 2014; published online September 12, 2014. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 10(1), 011008 (Sep 12, 2014) (8 pages) Paper No: CND-13-1143; doi: 10.1115/1.4027716 History: Received June 18, 2013; Revised May 19, 2014

In this article by introducing and subsequently applying the Min–Max method, chaos has been suppressed in discrete time systems. By using this nonlinear technique, the chaotic behavior of Behrens–Feichtinger model is stabilized on its first and second-order unstable fixed points (UFP) in presence and absence of noise signal. In this step, a comparison has also been carried out among the proposed Min–Max controller and the Pyragas delayed feedback control method. Next, to reduce the computation required for controller design, the clustering method has been introduced as a quantization method in the Min–Max control approach. To improve the performance of the acquired controller through clustering method obtained with the Min–Max method, a linear optimal controller is also introduced and combined with the previously discussed nonlinear control law. The resultant combined controller has been applied on the Henon map and through comparison with both Pyragas controller, and the linear optimal controller alone, its advantages are discussed.

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References

Figures

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Fig. 1

Chaotic attractor of the Behrens–Feichtinger system

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Fig. 8

The state variables and control signal for the Behrens–Feichtinger model obtained through stabilizing its second-order UFP via Pyragas controller (in the presence of noise signal)

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Fig. 6

The control signal versus state variables for the Behrens–Feichtinger model stabilizing its second-order UFP (in the presence of noise signal)

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Fig. 7

The state variables and control signal for the Behrens–Feichtinger model obtained through stabilizing its second-order UFP via Min–Max method (in the presence of noise signal)

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Fig. 2

Value of control signal versus state variables for the Behrens–Feichtinger model obtained through stabilizing its first-order UFP via Min–Max control method (without noise presence)

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Fig. 3

The state variables and control signal for the Behrens–Feichtinger model obtained through stabilizing its first-order UFP via Min–Max method (without noise presence)

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Fig. 4

Value of control signal versus state variables for the Behrens–Feichtinger model obtained through stabilizing its first-order UFP via the Min–Max control method (in the presence of noise signal)

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Fig. 5

The state variables and control signal for the Behrens–Feichtinger model obtained through stabilizing its first-order UFP via Min–Max method (in the presence of noise signal)

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Fig. 11

The state variables and control signal for controlling the Henon map via the combined controller. The controller has been activated after 50 steps (marked by dashed line), and linear controller has been active after a few steps (marked by dotted-dashed line).

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Fig. 9

The computed cluster centers for the Henon map

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Fig. 10

Value of control signal versus cluster centers for stabilizing the first-order UFP of the Henon system without noise parameter

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Fig. 12

The state variables and control signal for control of Henon map via linear optimal controller. The controller has been prepared after 50 steps, whereas it has been activated after the step marked with dotted-dashed line.

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