This paper presents a time-invariant extremum seeking controller (ESC) for nonlinear autonomous systems with limit cycles. For this time-invariant ESC, we propose a method to prove the closed loop system has an asymptotically stable limit cycle. The method is based on a perturbation theorem for maps, and, unlike existing techniques that use averaging and singular perturbation tools, it is not limited to weakly nonlinear systems. We use a typical example system to show that our method does indeed establish asymptotic stability of the limit cycle with minimal amplitude. Utilizing the example, we provide a general guide for analytic computations that are required to apply our method. The corresponding Mathematica code is available as supplementary material.
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Limit Cycle Minimization by Time-Invariant Extremum Seeking Control
Conventional perturbation-based extremum seeking control (ESC) employs a slow time-dependent periodic signal to find an optimum of an unknown plant. To ensure stability of the overall system, the ESC parameters are selected such that there is sufficient time-scale separation between the plant and the ESC dynamics. This approach is suitable when the plant operates at a fixed time-scale. In case the plant slows down during operation, the time-scale separation can be violated. As a result, the stability and performance of the overall system can no longer be guaranteed. In this paper, we propose an ESC for periodic systems, where the external time-dependent dither signal in conventional ESC is replaced with the periodic signals present in the plant, thereby making ESC time-invariant in nature. The advantage of using a state-based dither is that it inherently contains the information about the rate of the rhythmic task under control. Thus, in addition to maintaining time-scale separation at different plant speeds, the adaptation speed of a time-invariant ESC automatically changes, without changing the ESC parameters. We illustrate the effectiveness of the proposed time-invariant ESC with a Van der Pol oscillator example and present a stability analysis using averaging and singular perturbation theory.
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- Award ID(s):
- 1728057
- NSF-PAR ID:
- 10109683
- Date Published:
- Journal Name:
- Proceedings of the ... American Control Conference
- Page Range / eLocation ID:
- 2359 to 2365
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.23919/ACC.2019.8815344
