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On multiple limit cycles in sliding-mode control systems via a generalized describing function approach

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Abstract

This paper proposes a generalized describing function (DF) approach to analyze a class of frequency-dependent nonlinearities which are hard to deal with by traditional DF approaches. The proposed approach can intuitively predict the number, stability, and parameters of limit cycles by graphical illustrations of amplitude–phase characteristics. When applied to a 2-relative- degree sampling output feedback system, multiple limit cycles are found in the sliding-mode control. Both behaviors of complex limit cycles and influences of initial conditions on limit cycle oscillations are analyzed. Results show that each limit cycle has its own attraction region which can be estimated by the chattering amplitudes of the state variables, and initial conditions in different attraction regions may induce multiple limit cycles in the same control system. The undesired limit cycles can be eliminated by parameter tuning, such as control gains and sampling periods, to obtain a globally stable one. Simulations and hardware experiments further confirm the validity of the generalized DF approach.

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Acknowledgments

The authors gratefully acknowledge the financial support of National Natural Science Foundation of China under Grant No. 51175397.

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Authors and Affiliations

  1. School of Computer and Information Science, Southwest University, Chongqing, 400715, China

    Yu Shen

  2. Key Lab of Electronic Equipment Structure Design of Ministry of Education, Xidian University, Xi’an, 710071, China

    Yuan-ying Qiu

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  1. Yu Shen
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  2. Yuan-ying Qiu
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Correspondence to Yu Shen.

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Shen, Y., Qiu, Yy. On multiple limit cycles in sliding-mode control systems via a generalized describing function approach. Nonlinear Dyn 82, 819–834 (2015). https://doi.org/10.1007/s11071-015-2197-8

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  • DOI: https://doi.org/10.1007/s11071-015-2197-8

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