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PID controller design for second order nonlinear uncertain systems

Abstract

Although the classical PID (proportional-integral-derivative) controller is most widely and successfully used in engineering systems which are typically nonlinear with various uncertainties, almost all the existing investigations on PID controller focus on linear systems. The aim of this paper is to present a theory on PID controller for nonlinear uncertain systems, by giving a simple and analytic design method for the PID parameters together with a mathematic proof for the global stability and asymptotic regulation of the closed-loop control systems. To be specific, we will construct a 3-dimensional manifold within which the three PID parameters can be chosen arbitrarily to globally stabilize a wide class of second order nonlinear uncertain dynamical systems, as long as some knowledge on the upper bound of the derivatives of the nonlinear uncertain function is available. We will also try to make the feedback gains as small as possible by investigating the necessity of the manifold from which the PID parameters are chosen, and to establish some necessary and sufficient conditions for global stabilization of several special classes of nonlinear uncertain systems.

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Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 11688101), and by National Center for Mathematics and Interdisciplinary Sciences, CAS. The second author would like to thank Prof. Yi HUANG for valuable discussions on control of nonlinear uncertain systems, and to thank Prof. Pengnian CHEN for his useful information on the Markus-Yamabe conjecture.

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Correspondence to Cheng Zhao or Lei Guo.

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Zhao, C., Guo, L. PID controller design for second order nonlinear uncertain systems. Sci. China Inf. Sci. 60, 022201 (2017). https://doi.org/10.1007/s11432-016-0879-3

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Keywords

  • PID controller
  • nonlinear uncertain systems
  • global stability
  • Lyapunov function
  • asymptotic regulation