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Research Papers

Control of Continuous Time Chaotic Systems With Unknown Dynamics and Limitation on State Measurement

[+] Author and Article Information
Hojjat Kaveh

Department of Mechanical Engineering,
Sharif University of Technology,
P.O. Box 11155-9567,
Tehran 1458889694, Iran
e-mail: hojjatks@gmail.com

Hassan Salarieh

Department of Mechanical Engineering,
Sharif University of Technology,
P.O. Box 11155-9567,
Tehran 1458889694, Iran
e-mail: salarieh@sharif.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 29, 2018; final manuscript received November 2, 2018; published online November 28, 2018. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 14(1), 011007 (Nov 28, 2018) (5 pages) Paper No: CND-18-1096; doi: 10.1115/1.4041968 History: Received March 29, 2018; Revised November 02, 2018

This paper has dedicated to study the control of chaos when the system dynamics is unknown and there are some limitations on measuring states. There are many chaotic systems with these features occurring in many biological, economical and mechanical systems. The usual chaos control methods do not have the ability to present a systematic control method for these kinds of systems. To fulfill these strict conditions, we have employed Takens embedding theorem which guarantees the preservation of topological characteristics of the chaotic attractor under an embedding named “Takens transformation.” Takens transformation just needs time series of one of the measurable states. This transformation reconstructs a new chaotic attractor which is topologically similar to the unknown original attractor. After reconstructing a new attractor its governing dynamics has been identified. The measurable state of the original system which is one of the states of the reconstructed system has been controlled by delayed feedback method. Then the controlled measurable state induced a stable response to all of the states of the original system.

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Figures

Grahic Jump Location
Fig. 1

Steps toward controlling the original system

Grahic Jump Location
Fig. 2

Reconstructed chaotic attractor of the Lorenz system

Grahic Jump Location
Fig. 3

Periodic solution of the Lorenz system with k1=7.6, k2=−7.828

Grahic Jump Location
Fig. 4

Phase space of the Lorenz reconstructed system (a) and the original system (b) after applying controller with k1=7.6, k2=−7.828

Grahic Jump Location
Fig. 5

Phase space of the Chen reconstructed system (a) and the original system (b) after applying controller with k1=−2, k2=0

Grahic Jump Location
Fig. 6

Periodic solution of the Chen system with k1=−2, k2=0

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