Solution and stability of continuous-time cross-dimensional linear systems

Qing-le ZHANG; Biao WANG; Jun-e FENG

doi:10.1631/FITEE.1900504

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Solution and stability of continuous-time cross-dimensional linear systems

Regular Papers | Updated：2022-05-19

- Solution and stability of continuous-time cross-dimensional linear systems
- 跨维数线性连续系统的解和稳定性
- Frontiers of Information Technology & Electronic Engineering Vol. 22, Issue 2, Pages: 210-221(2021)
- Affiliations：
  
  School of Mathematics, Shandong University, Jinan 250100, China
- Author bio：
  
  [ "Qing-le ZHANG, E-mail: zqlcfc03@163.com" ]
  [ "Biao WANG, E-mail: wangbiao9956@163.com" ]
  Jun-e FENG, E-mail: fengjune@sdu.edu.cn
- Funds：
- DOI：10.1631/FITEE.1900504
  CLC： O231
- Received：19 September 2019，
  
  Revised：2020-;8-19，
  
  Published Online：06 October 2020，
  
  Published：2021-02
- Accepted：
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Qing-le ZHANG, Biao WANG, Jun-e FENG. Solution and stability of continuous-time cross-dimensional linear systems[J]. Frontiers of Information Technology & Electronic Engineering, 2021, 22(2): 210-221.
DOI：

Qing-le ZHANG, Biao WANG, Jun-e FENG. Solution and stability of continuous-time cross-dimensional linear systems[J]. Frontiers of Information Technology & Electronic Engineering, 2021, 22(2): 210-221. DOI： 10.1631/FITEE.1900504.

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。

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.

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references

DZ Cheng . . On finite potential games . . Automatica , , 2014 . . 50 ( ( 7 ): ): 1793 - - 1801 . . DOI: 10.1016/j.automatica.2014.05.005 http://doi.org/10.1016/j.automatica.2014.05.005 . .

DZ Cheng . . From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic Systems . . Elsevier, Amsterdam, the Netherlands , , 2019 . . .

DZ Cheng , , , HS Qi , , , Y Zhao . . Analysis and control of Boolean networks: a semi-tensor product approach . . Acta Autom Sin , , 2011 . . 37 ( ( 5 ): ): 529 - - 540 . . DOI: 10.3724/SP.J.1004.2011.00529 http://doi.org/10.3724/SP.J.1004.2011.00529 . .

DZ Cheng , , , ZQ Liu , , , HS Qi . . Cross-dimensional linear systems . . 2017 . . https://arxiv.org/abs/1710.03530 https://arxiv.org/abs/1710.03530 , , . .

DZ Cheng , , , HS Qi , , , ZQ Liu . . Linear system on dimension-varying state space . . IEEE 14 th Int Conf on Control and Automation , , 2018 . . p. 112 - - 117 . . . .

JE Feng , , , QL Zhang , , , JL Zhao . . Cheng's projection and its application in model reduction . . J Liaocheng Univ (Nat Sci Ed) , , 2019a . . 32 ( ( 2 ): ): 1 - - 7 . . DOI: 10.19728/j.issn1672-6634.2019.02.001 http://doi.org/10.19728/j.issn1672-6634.2019.02.001 . .

JE Feng , , , B Wang , , , YY Yu . . On dimensions of linear discrete dimension-unbounded systems . . Int J Contr Autom Syst , , 2019b . . 18 ( ( X ): ): 1 - - 7 . . DOI: 10.1007/s12555-019-0147-9 http://doi.org/10.1007/s12555-019-0147-9 . .

HT Li , , , XY Ding . . A control Lyapunov function approach to feedback stabilization of logical control networks . . SIAM J Contr Optim , , 2019 . . 57 ( ( 2 ): ): 810 - - 831 . . DOI: 10.1137/18M1170443 http://doi.org/10.1137/18M1170443 . .

XD Li , , , JH Shen , , , R Rakkiyappan . . Persistent impulsive effects on stability of functional differential equations with finite or infinite delay . . Appl Math Comput , , 2018 . . 329 14 - - 22 . . DOI: 10.1016/j.amc.2018.01.036 http://doi.org/10.1016/j.amc.2018.01.036 . .

YL Li , , , HT Li , , , XY Ding . . Set stability of switched delayed logical networks with application to finite-field consensus . . Automatica , , 2020 . . 113 108768 DOI: 10.1016/j.automatica.2019.108768 http://doi.org/10.1016/j.automatica.2019.108768 . .

Y Liu , , , BW Li , , , JQ Lu , , , 等 . . Pinning control for the disturbance decoupling problem of Boolean networks . . IEEE Trans Autom Contr , , 2017 . . 62 ( ( 12 ): ): 6595 - - 6601 . . DOI: 10.1109/TAC.2017.2715181 http://doi.org/10.1109/TAC.2017.2715181 . .

JQ Lu , , , J Zhong , , , C Huang , , , 等 . . On pinning controllability of Boolean control networks . . IEEE Trans Autom Contr , , 2016 . . 61 ( ( 6 ): ): 1658 - - 1663 . . DOI: 10.1109/TAC.2015.2478123 http://doi.org/10.1109/TAC.2015.2478123 . .

J Pan , , , H Yang , , , B Jiang . . Modeling and control of spacecraft formation based on impulsive switching with variable dimensions . . Comput Simul , , 2014 . . 6 ( ( 31 ): ): 124 - - 128 . . DOI: 10.3969/j.issn.1006-9348.2014.06.028 http://doi.org/10.3969/j.issn.1006-9348.2014.06.028 . .

B Wang , , , JE Feng , , , M Meng . . Matrix approach to model matching of composite asynchronous sequential machines . . IET Contr Theory Appl , , 2017 . . 11 ( ( 13 ): ): 2122 - - 2130 . . DOI: 10.1049/iet-cta.2016.1651 http://doi.org/10.1049/iet-cta.2016.1651 . .

B Wang , , , JE Feng , , , M Meng . . Model matching of switched asynchronous sequential machines via matrix approach . . Int J Contr , , 2019 . . 92 ( ( 10 ): ): 2430 - - 2440 . . DOI: 10.1080/00207179.2018.1441552 http://doi.org/10.1080/00207179.2018.1441552 . .

ZC Wang , , , GL Chen , , , HZ Ba . . Stability analysis of nonlinear switched systems with sampled-data controllers . . Appl Math Comput , , 2019 . . 357 297 - - 309 . . DOI: 10.1016/j.amc.2019.04.006 http://doi.org/10.1016/j.amc.2019.04.006 . .

YH Wu , , , TL Shen . . A finite convergence criterion for the discounted optimal control of stochastic logical networks . . IEEE Trans Autom Contr , , 2018a . . 63 ( ( 1 ): ): 262 - - 268 . . DOI: 10.1109/TAC.2017.2720730 http://doi.org/10.1109/TAC.2017.2720730 . .

YH Wu , , , TL Shen . . Policy iteration algorithm for optimal control of stochastic logical dynamical systems . . IEEE Trans Neur Netw Learn Syst , , 2018b . . 29 ( ( 5 ): ): 2031 - - 2036 . . DOI: 10.1109/TNNLS.2017.2661863 http://doi.org/10.1109/TNNLS.2017.2661863 . .

H Yang , , , B Jiang , , , V Cocquempot . . Stabilization of Switched Nonlinear Systems with Unstable Modes . . Springer, Switzerland , , 2014 . . DOI: 10.1007/978-3-319-07884-7 http://doi.org/10.1007/978-3-319-07884-7 . .

XY Yang , , , XD Li , , , Q Xi , , , 等 . . Review of stability and stabilization for impulsive delayed systems . . Math Biosci Eng , , 2018 . . 15 ( ( 6 ): ): 1495 - - 1515 . . DOI: 10.3934/mbe.2018069 http://doi.org/10.3934/mbe.2018069 . .

KZ Zhang , , , KH Johansson . . Long-term behavior of cross-dimensional linear dynamical systems . . Proc 37 th Chinese Control Conf , , 2018 . . p. 158 - - 163 . . DOI: 10.23919/ChiCC.2018.8482746 http://doi.org/10.23919/ChiCC.2018.8482746 . .

Y Zhang , , , T Zhou . . Controllability analysis for a networked dynamic system with autonomous subsystems . . IEEE Trans Autom Contr , , 2017 . . 62 ( ( 7 ): ): 3408 - - 3415 . . DOI: 10.1109/TAC.2016.2612831 http://doi.org/10.1109/TAC.2016.2612831 . .

GD Zhao , , , YZ Wang . . Formulation and optimization control of a class of networked evolutionary games with switched topologies . . Nonl Anal Hybr Syst , , 2016 . . 22 98 - - 107 . . DOI: 10.1016/j.nahs.2016.03.009 http://doi.org/10.1016/j.nahs.2016.03.009 . .

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