Solution and stability of continuous-time cross-dimensional linear systems
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Frontiers of Information Technology & Electronic Engineering www.jzus.zju.edu.cn; engineering.cae.cn; www.springerlink.com ISSN 2095-9184 (print); ISSN 2095-9230 (online) E-mail: [email protected]
Solution and stability of continuous-time cross-dimensional linear systems∗ Qing-le ZHANG, Biao WANG, Jun-e FENG‡ School of Mathematics, Shandong University, Jinan 250100, China E-mail: [email protected]; [email protected]; [email protected] Received Sept. 19, 2019; Revision accepted June 15, 2020; Crosschecked Aug. 19, 2020
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. Key words: Cross-dimensional; V-addition; V-product; Asymptotic stability; Stabilization https://doi.org/10.1631/FITEE.1900504 CLC number: O231
1 Introduction The cross-dimensional system, also known as the dimension varying system, is an extension of the classical linear system. It is often used to study complex systems such as departure and joining of spacecrafts, vehicle clutch systems, and modeling of biological systems (Pan et al., 2014). Because of the change of state dimension, a cross-dimensional system is usually regarded as a switching system with more general state jump behaviors. Therefore, switching is usually used to deal with crossdimensional systems (Yang H et al., 2014). Since some states with different dimensions may be closely related or completely independent of states with other dimensions, the factors affecting the dimen‡ *
Corresponding author
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) ORCID: Qing-le ZHANG, https://orcid.org/0000-0002-48016672; Jun-e FENG, https://orcid.org/0000-0003-3881-3042 c Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2020
sion change need to be considered when building the model. This provides the switching system, to some extent, with the ability to reflect some properties of the cross-dimensional systems. However, this method does not fully consider the dynamics of the system in the process of dimension change. The period of state transition may occur quite frequently in practice, so the dynamics in this process cannot be ignored. For this reason, it is necessary to establish a theory to describe and study cross-dime
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