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Research Papers

Robust Generation of Limit Cycles in Nonlinear Systems: Application on Two Mechanical Systems

[+] Author and Article Information
Ali Reza Hakimi

Department of Electrical and
Electronic Engineering,
Shiraz University of Technology,
Modares Boulevard,
P.O. Box 71555-313,
Shiraz, Iran
e-mail: a.hakimi@sutech.ac.ir

Tahereh Binazadeh

Department of Electrical and
Electronic Engineering,
Shiraz University of Technology,
Modares Boulevard,
P.O. Box 71555-313,
Shiraz, Iran
e-mail: binazadeh@sutech.ac.ir

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 28, 2015; final manuscript received October 20, 2016; published online February 1, 2017. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 12(4), 041013 (Feb 01, 2017) (8 pages) Paper No: CND-15-1261; doi: 10.1115/1.4035190 History: Received August 28, 2015; Revised October 20, 2016

This paper studies inducing robust stable oscillations in nonlinear systems of any order. This goal is achieved through creating stable limit cycles in the closed-loop system. For this purpose, the Lyapunov stability theorem which is suitable for stability analysis of the limit cycles is used. In this approach, the Lyapunov function candidate should have zero value for all the points of the limit cycle and be positive in the other points in the vicinity of it. The proposed robust controller consists of a nominal control law with an additional term that guarantees the robust performance. It is proved that the designed controller results in creating the desirable stable limit cycle in the phase trajectories of the uncertain closed-loop system and leads to induce stable oscillations in the system's output. Additionally, in order to show the applicability of the proposed method, it is applied on two practical systems: a time-periodic microelectromechanical system (MEMS) with parametric errors and a single-link flexible joint robot in the presence of external disturbances. Computer simulations show the effective robust performance of the proposed controllers in generating the robust output oscillations.

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Figures

Grahic Jump Location
Fig. 1

Convergence of closed-loop system trajectories to the desired limit cycle in Example 1

Grahic Jump Location
Fig. 2

The time response of the actual and the desired output in Example 1

Grahic Jump Location
Fig. 3

Time response of the robust control input (52)

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Fig. 4

The mechanical structure of single-link flexible joint robot [24]

Grahic Jump Location
Fig. 5

Convergence of closed-loop system trajectories to the desired limit cycle in Example 2

Grahic Jump Location
Fig. 6

The time response of the actual and the desired output in Example 2

Grahic Jump Location
Fig. 7

Time response of the robust control input (60)

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