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

doi:10.1007/s10483-022-2921-8

Applied Mathematics and Mechanics (English Edition) ›› 2022, Vol. 43 ›› Issue (11): 1747-1762.doi: https://doi.org/10.1007/s10483-022-2921-8

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

Zisong MEI, Zaihua WANG

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

收稿日期:2022-06-01 修回日期:2022-07-18 发布日期:2022-10-29
通讯作者: Zaihua WANG, E-mail: zhwang@nuaa.edu.cn
基金资助:
The National Natural Science Foundation of China (No. 12072370)

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

Zisong MEI, Zaihua WANG

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

Received:2022-06-01 Revised:2022-07-18 Published:2022-10-29
Contact: Zaihua WANG, E-mail: zhwang@nuaa.edu.cn
Supported by:
The National Natural Science Foundation of China (No. 12072370)

摘要/Abstract

摘要： 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.

关键词: human balance, inverted pendulum, proportional-derivative-acceleration (PDA) feedback, neutral delay differential equation (NDDE), multiplicity-induced dominancy

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.

Key words: human balance, inverted pendulum, proportional-derivative-acceleration (PDA) feedback, neutral delay differential equation (NDDE), multiplicity-induced dominancy

中图分类号:

93D15
34K20

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 (English Edition), 2022, 43(11): 1747-1762.

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

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

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