Chaos control in lateral oscillations of spinning disks via nonlinear feedback

Salarieh, Hassan; Sadeghian, Hoda; Merat, Kaveh

doi:10.1016/j.nonrwa.2008.09.006

Cited by 10 publications

(8 citation statements)

References 17 publications

(24 reference statements)

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“…One can obtain the state‐space form of the system equations as

({,, \begin{array}{l} {\overset{false˙}{x}}_{1} = x_{2}, \\ {\overset{false˙}{x}}_{2} = - ((),, ω^{2} - Ω^{2}) x_{1} - γ x_{1} ((),, x_{1}^{2} + x_{3}^{2}) + 2 normalΩ x_{4}, \\ {\overset{false˙}{x}}_{3} = x_{4}, \\ {\overset{false˙}{x}}_{4} = - ((),, ω^{2} - Ω^{2}) x_{3} - γ x_{3} ((),, x_{1}^{2} + x_{3}^{2}) - 2 normalΩ x_{2} + F + u, \end{array})

where x 1 = η ,

x_{2} = \overset{η}{true}

, x 3 = ξ , and

x_{4} = \overset{ξ}{true}

. Salarieh et al obtained the first UPO of the chaotic system (5) via the harmonic balance method for system parameters ω = 1, Ω = 0.8, γ = 1, and F = 4 as

({,, \begin{array}{l} η_{ref} = sin ((),, normalΩ t) true {true\sum}_{k = 0}^{\infty} a_{2 k + 1} cos [(2 k + 1) normalΩ t] - cos ((),, normalΩ t) true {true\sum}_{k = 0}^{\infty} {((),, - 1)}^{k} a_{2 k + 1} sin [(2 k + 1) normalΩ t], \\ ξ_{ref} = cos ((),, normalΩ t) true true\sum \end{array})

…”

Section: Chaotic Spinning Disks Modelmentioning

confidence: 99%

“…Salarieh et al [38,39] obtained the first UPO of the chaotic system (5) via the harmonic balance method for system parameters ω = 1, Ω = 0.8, γ = 1, and F = 4 as where k is the index of the summation and the coefficients a 2k + 1 are specified in Table I.…”

Section: Chaotic Spinning Disks Modelmentioning

confidence: 99%

“…It is worth noting that in the proposed observer‐based control algorithm, we assume all states are not available. However, in , as the only existing works in stabilizing UPOs of chaotic spinning disk model in the literature, complete state information is assumed for the feedback to be used in the control signal.…”

Section: Chaotic Spinning Disks Modelmentioning

confidence: 99%

See 2 more Smart Citations

NON‐PDC Observer‐Based T‐S Fuzzy Tracking Controller Design and its Application in CHAOS Control

Asemani

Vatankhah

2017

Asian Journal of Control

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In many mechanical devices with chaotic behavior, stabilizing unstable periodic orbits (UPOs) of the system has positive effects in the lifetime and effectiveness of these devices. In this study, a new non‐parallel distributed compensation (non‐PDC) observer‐based tracking controller is presented for Takagi–Sugeno fuzzy systems to control the chaotic behavior of such systems. Asymptotic stability synthesis of the closed‐loop system is investigated using a fuzzy Lyapunov function to derive less conservative conditions than common quadratic Lyapunov function‐based approaches. To tackle the main drawback of the fuzzy Lyapunov‐based approaches, which assume some upper bounds on the derivatives of the fuzzy grade functions, we propose a new procedure by considering a constraint on the control signal. The new design conditions are given in the form of linear matrix inequalities (LMIs). The proposed control structure is applied to spinning disks in which chaos phenomena appear in lateral vibration. Simulation results are given to show the applicability of the proposed tracker to the UPO problem.

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“…One can obtain the state‐space form of the system equations as

({,, \begin{array}{l} {\overset{false˙}{x}}_{1} = x_{2}, \\ {\overset{false˙}{x}}_{2} = - ((),, ω^{2} - Ω^{2}) x_{1} - γ x_{1} ((),, x_{1}^{2} + x_{3}^{2}) + 2 normalΩ x_{4}, \\ {\overset{false˙}{x}}_{3} = x_{4}, \\ {\overset{false˙}{x}}_{4} = - ((),, ω^{2} - Ω^{2}) x_{3} - γ x_{3} ((),, x_{1}^{2} + x_{3}^{2}) - 2 normalΩ x_{2} + F + u, \end{array})

where x 1 = η ,

x_{2} = \overset{η}{true}

, x 3 = ξ , and

x_{4} = \overset{ξ}{true}

. Salarieh et al obtained the first UPO of the chaotic system (5) via the harmonic balance method for system parameters ω = 1, Ω = 0.8, γ = 1, and F = 4 as

({,, \begin{array}{l} η_{ref} = sin ((),, normalΩ t) true {true\sum}_{k = 0}^{\infty} a_{2 k + 1} cos [(2 k + 1) normalΩ t] - cos ((),, normalΩ t) true {true\sum}_{k = 0}^{\infty} {((),, - 1)}^{k} a_{2 k + 1} sin [(2 k + 1) normalΩ t], \\ ξ_{ref} = cos ((),, normalΩ t) true true\sum \end{array})

…”

Section: Chaotic Spinning Disks Modelmentioning

confidence: 99%

Section: Chaotic Spinning Disks Modelmentioning

confidence: 99%

See 1 more Smart Citation

NON‐PDC Observer‐Based T‐S Fuzzy Tracking Controller Design and its Application in CHAOS Control

Asemani

Vatankhah

2017

Asian Journal of Control

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show abstract

“…In the so called study, the feedback gain was obtained according to a minimum entropy algorithm. In addition, Salarieh et al. (2009) have controlled the chaos in lateral oscillation of a spinning disk using nonlinear feedback.…”

Section: Introductionmentioning

confidence: 99%

Chaos control in lateral oscillations of spinning disk via linear optimal control of discrete systems

Chekan

Nojoumian

Merat

et al. 2016

Journal of Vibration and Control

Self Cite

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Chaos control of a spinning disk having transverse vibration is considered in this article by stabilizing the system on its corresponding unstable periodic orbit (UPO). At the first step, the system continuous-time dynamic equations are quantized by utilizing a proper Poincare map. Then, using the regression method a linear description from the obtained cross points is achieved around the corresponding fixed point of the target UPO. Finally, through solving the Riccati equation, an optimal controller is introduced which stabilizes the system on its unstable fixed point. At the end, the effectiveness of the proposed control method is examined through numerical simulations.

show abstract

“…Since Ott, Grebogi, and Yorke [11] firstly proposed the method of chaos control in 1990, chaos control has attracted increasing attention in recent years, and lots of successful experiments have been reported [12][13][14][15][16]. Such as, Wang and his coworkers presented a nonlinear control law to achieve finite-time chaos control based on the finite-time stability theory in the cascade-connected system [17].…”

Section: Introductionmentioning

confidence: 99%

Analysis and control of a hyperchaotic system with only one nonlinear term

2011

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In order to promote the development of chaos in nonlinear systems, and explore more convenient controllers for the engineering application, a four-dimensional nonlinear dynamic system with only one nonlinear term was constructed and its complex dynamic characteristics were analyzed, including the phase trajectory map, Lyapunov exponents, and so on. Furthermore, the recursive backstepping method was proposed to design a different controller; the hyperchaotic system was controlled to an equilibrium point and a periodic orbit. Theoretical analysis is in agreement with simulation results. The results show that the recursive backstepping control method can wipe off chaos, and make the hyperchaotic system achieve stable states. The control process is a smooth transition, and the transition time is short.

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Chaos control in lateral oscillations of spinning disks via nonlinear feedback

Cited by 10 publications

References 17 publications

NON‐PDC Observer‐Based T‐S Fuzzy Tracking Controller Design and its Application in CHAOS Control

NON‐PDC Observer‐Based T‐S Fuzzy Tracking Controller Design and its Application in CHAOS Control

Chaos control in lateral oscillations of spinning disk via linear optimal control of discrete systems

Analysis and control of a hyperchaotic system with only one nonlinear term

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