Abstract
We investigate the solution and stability of continuous-time cross-dimensional linear systems (CCDLSs) with dimension bounded by V-addition and V-product. Using the integral iteration method, the solution to CCDLSs can be obtained. Based on the new algebraic expression of the solution and the Jordan decomposition method of matrix, a necessary and sufficient condition is derived for judging whether a CCDLS is asymptotically stable with a given initial state. This condition demonstrates a method for finding the domain of attraction and its relationships. Then, all the initial states that can be stabilized are studied, and a method for designing the corresponding controller is proposed. Two examples are presented to illustrate the validity of the theoretical results.
摘要
利用V-加法和V-乘法研究了维数有界的跨维数线性连续系统 (CCDLSs) 的解和稳定性. 使用积分迭代法, 得到CCDLSs的解. 基于解的代数表示以及矩阵的若尔当分解, 给出相应的充要条件判断一个CCDLS在给定初始状态后是否渐进稳定. 该条件提供了一种确定吸引域以及吸引域间关系的方法. 然后, 研究了所有可镇定的初始状态, 并提出相应控制器的设计方法. 最后, 给出两个例子说明理论结果的有效性.
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Qing-le ZHANG designed the research and drafted the manuscript. Qing-le ZHANG and Jun-e FENG processed the data. Biao WANG and Jun-e FENG helped organize the manuscript. Qing-le ZHANG and Jun-e FENG revised and finalized the paper.
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Qing-le ZHANG, Biao WANG, and Jun-e FENG declare that they have no conflict of interest.
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Project supported by the National Natural Science Foundation of China (Nos. 61773371 and 61877036) and the Natural Science Fund of Shandong Province, China (No. ZR2019MF002)
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Zhang, Ql., Wang, B. & Feng, Je. Solution and stability of continuous-time cross-dimensional linear systems. Front Inform Technol Electron Eng 22, 210–221 (2021). https://doi.org/10.1631/FITEE.1900504
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DOI: https://doi.org/10.1631/FITEE.1900504