Abstract
In this note, a benchmark example system which is not stabilizable by a smooth state feedback controller is considered with the fully actuated system (FAS) approach. It is shown that a smooth controller exists which drives the trajectories starting from a large domain in the initial value space to the origin exponentially. Such a result brings about a generalization of Lyapunov asymptotical stability, which is termed as global exponential sub-stability. The region of attraction is allowed to be an unbounded open set of the initial values with closure containing the origin. This sub-stability result may be viewed to be superior to some local stability results in the Lyapunov sense because the region of attraction is much larger than any finite ball containing the origin and meanwhile the feasible trajectories are always driven to the origin exponentially. Based on this sub-stabilization result, globally asymptotically stabilizing controllers for the system can be provided in two general ways, one is through combination with existing globally stabilizing controllers, and the other is by using a pre-controller to first move an initial point which is not within the region of attraction into the region of attraction.
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Acknowledgements
The author is grateful to his Ph.D. students, Dongjun Wu, Weizhen Liu, Guangtai Tian, Qin Zhao, etc., for helping him with reference selection and proofreading, and to Professors Bin Zhou, Aiguo Wu and Mingzhe Hou for helpful discussions. Thanks also go to Professor Yiguang Hong for introducing this particular problem.
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This work has been partially supported by the Major Program of National Natural Science Foundation of China under Grant Nos. 61690210 and 61690212, the National Natural Science Foundation of China under Grant No. 61333003, and also by the Science Center Program of the National Natural Science Foundation of China under Grant No. 62188101.
This paper was recommended for publication by Guest Editor WU Ai-Guo.
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Duan, GR. Stabilization via Fully Actuated System Approach: A Case Study. J Syst Sci Complex 35, 731–747 (2022). https://doi.org/10.1007/s11424-022-2091-7
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DOI: https://doi.org/10.1007/s11424-022-2091-7