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.
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Citation: MEI, Z. S. and WANG, Z. H. Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback. Applied Mathematics and Mechanics (English Edition), 43(11), 1747–1762 (2022) https://doi.org/10.1007/s10483-022-2921-8
Project supported by the National Natural Science Foundation of China (No. 12072370)
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Mei, Z., Wang, Z. Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback. Appl. Math. Mech.-Engl. Ed. 43, 1747–1762 (2022). https://doi.org/10.1007/s10483-022-2921-8
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DOI: https://doi.org/10.1007/s10483-022-2921-8
Key words
- human balance
- inverted pendulum
- proportional-derivative-acceleration (PDA) feedback
- neutral delay differential equation (NDDE)
- multiplicity-induced dominancy