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.

跨维数线性连续系统的解和稳定性

[1]Cheng DZ, 2014. On finite potential games. Automatica, 50(7):1793-1801.

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

[3]Cheng DZ, Qi HS, Zhao Y, 2011. Analysis and control of Boolean networks: a semi-tensor product approach. Acta Autom Sin, 37(5):529-540 (in Chinese).

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

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

[6]Feng JE, Zhang QL, Zhao JL, 2019a. Cheng’s projection and its application in model reduction. J Liaocheng Univ (Nat Sci Ed), 32(2):1-7 (in Chinese).

[7]Feng JE, Wang B, Yu YY, 2019b. On dimensions of linear discrete dimension-unbounded systems. Int J Contr Autom Syst, 18(X):1-7.

[8]Li HT, Ding XY, 2019. A control Lyapunov function approach to feedback stabilization of logical control networks. SIAM J Contr Optim, 57(2):810-831.

[9]Li XD, Shen JH, Rakkiyappan R, 2018. Persistent impulsive effects on stability of functional differential equations with finite or infinite delay. Appl Math Comput, 329:14-22.

[10]Li YL, Li HT, Ding XY, 2020. Set stability of switched delayed logical networks with application to finite-field consensus. Automatica, 113:108768.

[11]Liu Y, Li BW, Lu JQ, et al., 2017. Pinning control for the disturbance decoupling problem of Boolean networks. IEEE Trans Autom Contr, 62(12):6595-6601.

[12]Lu JQ, Zhong J, Huang C, et al., 2016. On pinning controllability of Boolean control networks. IEEE Trans Autom Contr, 61(6):1658-1663.

[13]Pan J, Yang H, Jiang B, 2014. Modeling and control of spacecraft formation based on impulsive switching with variable dimensions. Comput Simul, 6(31):124-128 (in Chinese).

[14]Wang B, Feng JE, Meng M, 2017. Matrix approach to model matching of composite asynchronous sequential machines. IET Contr Theory Appl, 11(13):2122-2130.

[15]Wang B, Feng JE, Meng M, 2019. Model matching of switched asynchronous sequential machines via matrix approach. Int J Contr, 92(10):2430-2440.

[16]Wang ZC, Chen GL, Ba HZ, 2019. Stability analysis of nonlinear switched systems with sampled-data controllers. Appl Math Comput, 357:297-309.

[17]Wu YH, Shen TL, 2018a. A finite convergence criterion for the discounted optimal control of stochastic logical networks. IEEE Trans Autom Contr, 63(1):262-268.

[18]Wu YH, Shen TL, 2018b. Policy iteration algorithm for optimal control of stochastic logical dynamical systems. IEEE Trans Neur Netw Learn Syst, 29(5):2031-2036.

[19]Yang H, Jiang B, Cocquempot V, 2014. Stabilization of Switched Nonlinear Systems with Unstable Modes. Springer, Switzerland.

[20]Yang XY, Li XD, Xi Q, et al., 2018. Review of stability and stabilization for impulsive delayed systems. Math Biosci Eng, 15(6):1495-1515.

[21]Zhang KZ, Johansson KH, 2018. Long-term behavior of cross-dimensional linear dynamical systems. Proc 37^th Chinese Control Conf, p.158-163.

[22]Zhang Y, Zhou T, 2017. Controllability analysis for a networked dynamic system with autonomous subsystems. IEEE Trans Autom Contr, 62(7):3408-3415.

[23]Zhao GD, Wang YZ, 2016. Formulation and optimization control of a class of networked evolutionary games with switched topologies. Nonl Anal Hybr Syst, 22:98-107.