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Research Papers

A New Property of Noninvasive Control Methods Applied to Stabilize Unstable Periodic Orbits

[+] Author and Article Information
Saeed Montazeri

Faculty of Mechanical Engineering,
Shahid Rajaee Teacher Training University,
Tehran 1876773467, Iran
e-mail: saeed.montazeri@outlook.com

Ali Rahmani Hanzaki

Assistant Professor
Faculty of Mechanical Engineering,
Shahid Rajaee Teacher Training University,
Tehran 1678815811, Iran
e-mail: a.rahmani@srttu.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 10, 2017; final manuscript received March 9, 2018; published online April 2, 2018. Assoc. Editor: Eihab Abdel-Rahman.

J. Comput. Nonlinear Dynam 13(5), 051007 (Apr 02, 2018) (6 pages) Paper No: CND-17-1450; doi: 10.1115/1.4039628 History: Received October 10, 2017; Revised March 09, 2018

This paper is devoted to present a theorem on a new property of noninvasive control methods applied to stabilize unstable periodic orbits (UPOs). This property is related to the optimal energy consumption of the controller in the presence of noise. The approach of parameter optimization is applied to study optimal energy consumption of the controller. Throughout this paper, the problem of energy consumption of the controller is studied when the system state is close to the UPO, and fluctuates around it because of the presence of noise.

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Figures

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Fig. 1

(a) The resistor–inductor circuit and (b) the spring–damper system

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Fig. 2

An abject on frictionless surface. The forces acting on the object are the force field F=x−x3, the damping force FD, and the excitation force f.

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Fig. 3

(a) The unstable periodic orbit of the system defined by Eq. (21) and (b) the control input. Because the signal vanishes, the control law is noninvasive.

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Fig. 4

The energy consumption of the controller versus the feedback gain for Duffing oscillator of Eq. (21). The values of w in Eq. (22) are displayed with dots. It is noted that T=2×104 s, D=0.005, and strength of noise ε(t) is 0.001 db. The optimal feedback gain estimated from the curve fitting is about kop≈0.17.

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Fig. 5

The variations of F1 versus k for Duffing oscillator (21). The values of F1 in Eq. (23) are displayed with dots. Note that T=2×104 s, D=0.005, and strength of noise ε(t) is 0.001 db. The value of delay time is τ=2 and the values of k are located in the interval 0.1<k<0.4. The curve represents fourth-order polynomial curve fitting. The optimal feedback gain estimated from the curve fitting is about kop≈0.17 as expected.

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Fig. 6

The variations of ∫pdt versus k. It is obvious that the values in vertical axis are negligible in comparison with the values in Fig. 4.

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Fig. 7

The variations of w versus k for Duffing oscillator (21) for a different physical realization, whose power expression is p=Fcx1 in this case and all other parameters (e.g., the integration time T, D, …) remain unchanged

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Fig. 8

The SSE values for different curve orders in Fig. 4

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Fig. 9

The SSE values for different curve orders in Fig. 5

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Fig. 10

A mass–spring–damper system

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Fig. 11

The variations of w versus k2 for mass–spring–damper system (24). Values of w in Eq. (25) are displayed with dots. Note that T=15,000s, D=0.03 and strength of noise ζ(t) is 0.001 db. The optimal feedback gain estimated from the curve fitting is about kop≈1.05.

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Fig. 12

The variations of F1 versus k2 for mass–spring–damper system (24). The curve represents the fourth-order polynomial curve fitting. The optimal feedback gain estimated from the curve fitting is about kop≈0.76. It differs from the value kop=1.05 in Fig. 11 as it is expected.

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Fig. 13

The variations of ∫pdt versus k2. As it is obvious, the values in vertical axis are not negligible in comparison with the values in Fig. 11.

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Fig. 14

The variations of w versus k2 for mass–spring–damper system (24) with different physical realization. The power expression is p=Fx1 in this case and all other parameters (e.g., the integration time T, D, …) remain unchanged.

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