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Model-free continuation of periodic orbits in certain nonlinear systems using continuous-time adaptive control

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Abstract

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package coco demonstrates successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.

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Data availability

Datasets generated and analyzed during this study are available upon request from the authors. Matlab scripts sufficient to generate this data will be posted to an open-source archive.

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Funding

This work is supported by Agriculture and Food Research Initiative Competitive Grant no. 2014-67021-22109 from the USDA National Institute of Food and Agriculture. Part of the editing of this paper was performed while the second author served at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Champaign, IL, USA

    Yang Li & Harry Dankowicz

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  1. Yang Li
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The authors contributed equally to the conception and design of this research, and to the writing of the manuscript. Implementation of algorithms in code and generation of numerical results was performed by Yang Li. Both authors read and approved the final manuscript.

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Correspondence to Harry Dankowicz.

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This work is supported by Agriculture and Food Research Initiative Competitive Grant no. 2014-67021-22109 from the USDA National Institute of Food and Agriculture.

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Li, Y., Dankowicz, H. Model-free continuation of periodic orbits in certain nonlinear systems using continuous-time adaptive control. Nonlinear Dyn 111, 4945–4957 (2023). https://doi.org/10.1007/s11071-022-08059-1

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