Stability Test of Networked Systems with Multiple Delays

2b2335bb-3b99-4146-9185-612cd4746b4220211120101325897naun:naunmdt@crossref.orgMDT DepositInternational Journal of Applied Mathematics and Informatics2074-127810.46300/91014http://www.naun.org/cms.action?id=3064319202131920211510.46300/91014.2021.15https://www.naun.org/cms.action?id=23306Stability Test of Networked Systems with Multiple DelaysYangXiaoInstitute of Information Science Beijing Jiaotong University Beijing 100044, ChinaJinfengKouInstitute of Information Science Beijing Jiaotong University Beijing 100044, ChinaIn this paper, we propose a sufficient stability condition for networked systems with multiple delays based on the 2-D polynomials and 2-D Hurwitz-Schur stability. The main advantage of the new stability condition is that it is applicable to the general case of networked systems with multiple, incommensurate delays yet numerically tractable. The characteristic polynomials of networked systems are mapping into 2-D hybrid polynomials, then to test the Hurwitz-Schur stability can. determine the networked systems, examples including system simulations verify the validity of the proposed test algorithms.112020211120202110210717https://www.naun.org/main/UPress/ami/2021/a342013-017(2021).pdf10.46300/91014.2021.15.17https://www.naun.org/main/UPress/ami/2021/a342013-017(2021).pdf10.1016/s1004-4132(07)60020-6Y. Xiao, “2-D algebr aic test fo r robust stabilit y of time-delay systems with interval parameters”, Journal of S ystems Engineering and Electronics, Vol.17, No.4, pp.802-810, 2006. Y. Xiao, “2-D algebr aic test for robust stability of quasipolynomials with in terval parameters”, Asian Journal of Control, Vol.8, No.2, pp.174–179, 2006. 10.1109/iscas.2001.921199Y. Xiao, “Hurwitz-Schur Stability Test of Interval Bivariate Polynomials”, Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS2001), 2001, pp.II-829-832. 10.1109/9.362855V. L. Kharitonov and A. P. Zhabko, "Robust Stability of TimeDelay Systems", IEEE Trans. Automatic Control, Vol.39, No.12, pp.2388-2397, 1994. V. B. Kolmanovskii, S. Nicule scu, K. Gu, “Delay Effects on Stability: A Survey”, Proc. of 38th Conference on Decision and Control, Phoenix, Arizona, USA, 1999, pp. 1993-1998. 10.1109/9.553692Y. Sun, J. Hsieh, and H. Ya ng, “On t he Stability of Unc ertain Systems with Multiple Time-Varying Delays”, IEEE Trans. Automatic Control, Vol.42, No.1,1997, pp.101-105. J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993. 10.1109/9.975503E. Fridman an d U. Shaked , “New bounded real lemma representations for time-delay systems and th eir applications,” IEEE Trans. Automatic Contr., vol. 46 , pp. 1973–1979, Dec. 2001. 10.1137/1.9781611970777S. Boyd, L. El Ghaoui, and E. Feron et al., Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. 10.23919/ecc.1997.7082683S. I. Niculescu, C. E. de Souza, J. M. Dion, and L. Dugard, “Robust stability and stabilization of un certain linear systems with state delay: single delay case (I),” in Proc.IFAC Symp. Rob ust Control Design, Riode Jane iro, Brazil, Sept. 1994, pp. 469– 474. 10.1016/0167-6911(94)90071-xJ. H. Su, “Further results on the robust stability of linear systems with a singl e time delay,” System Control Le tt., vol. 23 , pp. 375–379, 1994. 10.1016/s1474-6670(17)58225-7X. Li and C. E. de Souza, “ Criteria for rob ust stability of uncertain linear systems with tim e-varying state delays,” in IFAC 13thWorld Congr., vol. 1 , San Francisco, CA, 1996, pp . 137–142. 10.1109/9.920802J. H. Kim, “Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty,” IEEE Trans. Automatic Control, vol. 46, No. 5, pp. 789–792, 2001. 10.1109/9.983390D. Yue and S. Won, “An improvement on ‘Delay and its timederivative dependent robust stabil ity of time-delayed linear systems with uncertainty’,” IEEE Trans. Automatic Control ., vol. 47, No. 2, pp. 407–408, 2002. 10.1080/00207170210123833J. Zhang, P. Tsiotras, and C. Knopse, “Stability analysis of LPV time delayed systems,” Int. J .Control, vol. 75 , pp. 538–558, 2002. 10.1080/002071799221172V. Kolmanovskii, S.-I. Niculescu, and J. P. R ichard, “On the Lyapunov–Krasovskii functions for stability analysis of linear delay systems,” Int. J . Control, vol. 72, pp. 374–384, 1999. 10.1109/tac.1980.1102482E .W. Kamen, “On the rel ationship between zero crit eria for two-variable polynomials and as ymptotic stability of delay differential equations”, IEEE Trans. Automatic Control, vol.25, no.5, pp.983-984. 1980. 10.1016/0022-247x(85)90068-xJ. K. Hale, E. F. Infante, and F. S. P. Tsen, “Stability in linear delay equations,” J. Math. Anal. Appl., vol. 105 , pp. 533–555, 1985. 10.1109/tac.2008.923686J. Chen, S. I. Niculescu and P. L. Fu, “Robust Stability of QuasiPolynomials: Frequency-Sweeping Conditions and Vert ex Tests”, IEEE Transactions on Automatic Control, Vol. 53, No. 5, pp.1219- 1234, 2008. 10.1109/9.29423M. Fu, A.W. Olbrot, M. P. Polis, “Robust stabilit y for timedelay systems: the edg e theorem and gr aphical tests”, IEEE Trans. Automatic Control, Vol.34, No.8, pp.613-82 0, 1989. 10.1109/tac.2004.837589J. A. Marshall, M. E. Broucke, and B. A. Fran cis, “Formations of vehicles in cyclic pursuit,” IEEE Transactions on Automatic Control, vol. 49, pp. 1963–1974, 2004. 10.1007/978-3-642-18482-6_26J. D. Birdwell, J. Chiasson, Z. Tang, C. Abdallah, M. M. Hay at, and T. Wang, “Dynamic time delay models for load balancing. parti: Deterministic models,” in Advances in Time-Delay Systems, S.-I.Niculescu and K. Gu, Eds. Berlin: Springer, 2004, pp. 355–370. 10.1109/cdc.1999.833346Y. Xiao, R. Unbehauen and X. Y. Du, “Robust Hurwitz stability of conditions of polytopes of bivariate poly nomials”, Proceedings of 38th IEEE Conference on Decision and Control, Phoenix, Arizona, 1999, pp. 5034-5035. 10.1109/cdc.1999.833346Y. Xiao, R. Un behauen, X. Du , "Robust Hurwitz Stab ility of Conditions of Polytopes of Bivariate Polynomials", Proceedings of 38th I EEE Conference on Decision and Control, 1999 , pp.5034-5035. 10.1109/81.703840Y. Xiao and R. Unbehauen, "New Stability Test Algorithm for Two-Dimensional Digital Filters", IEEE Trans. Circuits and Systems I, Vol.45, No.7, pp.739-740, 1998. 10.1109/tassp.1976.1162836D. L. A. Davis, “Correct proof of Huang's theorem on stability”, IEEE Trans. on Acoustics, Speech, and Signal Processing, vol.24, no.10, pp.425-426, 1976.