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Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback

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  • Department of Basic Courses, Army Engineering University, Nanjing 211101, China

Received date: 2022-06-01

  Revised date: 2022-07-18

  Online published: 2022-10-29

Supported by

The National Natural Science Foundation of China (No. 12072370)

Abstract

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.

Cite this article

Zisong MEI, Zaihua WANG . Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback[J]. Applied Mathematics and Mechanics, 2022 , 43(11) : 1747 -1762 . DOI: 10.1007/s10483-022-2921-8

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