×

On exponential stability of switched homogeneous positive systems of degree one. (English) Zbl 1415.93224

Summary: In this paper, we investigate the stability problem of switched homogeneous positive systems of degree one. We first show that the positiveness of the considered system is guaranteed when each switched subsystem is cooperative. Then, an integral excitation condition is formulated to ensure the exponential stability of the switched homogeneous positive system of degree one. In particular, the proposed condition allows that some or even all switched subsystems are only Lyapunov stable rather than asymptotically stable. In the second part of this paper, we further consider the cases of sub-homogeneous systems, time-varying delay systems and discrete-time systems. In the sub-homogeneous case, we first show that the previous integral excitation condition is also sufficient such that the exponential stability is achieved. We also derive an exponential stability result by relaxing the cooperative and integral excitation conditions. In the time-varying delay case, a delay-independent exponential stability result is shown under the extended integral excitation condition. Finally, in the discrete-time case, based on the fact that each switched subsystem is order-preserving instead of cooperative, the exponential stability of the considered system is shown under a weak excitation condition.

MSC:

93D20 Asymptotic stability in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aeyels, D.; Leenheer, P. D., Extension of the perron-frobenius theorem to homogeneous systems, SIAM Journal on Control and Optimization, 41, 2, 563-582 (2002) · Zbl 1043.34034
[2] Alonso, H.; Rocha, P., A general stability test for switched positive systems based on a multidimensional system analysis, IEEE Transanctions on Automatic Control, 55, 11, 1660-2664 (2010) · Zbl 1368.93588
[3] Bokharaie, V. S.; Mason, O., On delay-independent stability of a class of nonlinear positive time-delay systems, IEEE Transactions on Automatic Control, 59, 7, 1974-1977 (2014) · Zbl 1360.93592
[4] Danskin, J. M., The theory of max-min, with applications, SIAM Journal of Applied Mathematics, 14, 4, 641-664 (1966) · Zbl 0144.43301
[5] Dong, J., On the decay rates of homogeneous positive systems of any degree with time-varying delays, IEEE Transactions on Automatic Control, 60, 11, 2983-2988 (2015) · Zbl 1360.93494
[6] Dong, J., Stability of switched positive nonlinear systems, International Journal of Robust and Nonlinear Control, 26, 3118-3129 (2016) · Zbl 1346.93341
[7] Driver, R. D., Existence and stability of solutions of a delay-differential system (1992), Springer-Verlag
[8] Fainshil, L.; Margaliot, M.; Chigansky, P., On the stability of positive linear switched systems under arbitrary switching laws, IEEE Transactions on Automatic Control, 54, 4, 897-899 (2009) · Zbl 1367.93431
[9] Feyzmahdavian, H. R., Besselink, B., & Johansson, H. Stability analysis of monotone systems via max- separable lyapunov functions. arXiv:160707966v1; Feyzmahdavian, H. R., Besselink, B., & Johansson, H. Stability analysis of monotone systems via max- separable lyapunov functions. arXiv:160707966v1 · Zbl 1390.93603
[10] Feyzmahdavian, H. R.; Charalambous, T.; Johansson, H., Asymptotic stability and decay rates of homogeneous positive systems with bounded and unbounded delays, SIAM Journal on Control and Optimization, 52, 4, 2623-2650 (2014) · Zbl 1320.34101
[11] Filippov, A. F., Differential equations with discontinuous righthand sides (1988), Kluwer Academic Publishers: Kluwer Academic Publishers Norwell, MA
[12] Guo, F.; Wen, C.; Mao, J.; Li, G.; Song, Y., A distributed hierarchical algorithm for multi-cluster constrained optimization, Automatica, 77, 230-238 (2017) · Zbl 1355.93006
[13] Gurvits, L.; Shorten, R.; Mason, O., On the stability of switched positive linear systems, IEEE Transactions on Automatic Control, 52, 6, 1099-1103 (2007) · Zbl 1366.93436
[14] Khalil, H., Nonlinear systems (2002), New Jersey: Prentice-Hall
[15] Knorn, F.; Mason, O.; Shorten, R., On linear co-positive Lyapunov functions for sets of linear positive systems, Automatica, 45, 8, 1943-1947 (2009) · Zbl 1185.93122
[16] Leenheer, P. D.; Angeli, D.; Sontag, E. D., Monotone chemical reaction networks, Journal of Mathematical Chemistry, 41, 3, 295-314 (2007) · Zbl 1117.80309
[17] Liu, X., Stability analysis of switched positive systems: A switched linear copositive Lyapunov function method, IEEE Transactions on Circuits and Systems II: Express Briefs, 56, 5, 414-418 (2009)
[18] Liu, X.; Dang, C., Stability analysis of positive switched linear systems with delays, IEEE Transactions on Automatic Control, 56, 7, 1684-1690 (2011) · Zbl 1368.93599
[19] Mason, O.; Verwoerd, M., Observations on the stability properties of cooperative systems, Systems & Control Letters, 58, 6, 461-467 (2009) · Zbl 1161.93021
[20] Meng, Z.; Xia, W.; Johansson, K.; Hirche, S., Stability of positive switched linear systems: weak excitation and robustness to time-varying delay, IEEE Transactions on Automatic Control, 62, 1, 399-405 (2017) · Zbl 1359.93352
[21] Meng, Z.; Yang, T.; Li, G.; Ren, W.; Wu, D., Synchronization of coupled dynamical systems: tolerance to weak connectivity and arbitrarily bounded time-varying delays, IEEE Transactions on Automatic Control, 63, 6, 1791-1797 (2018) · Zbl 1395.93062
[22] Shorten, R.; Wirth, F.; Leith, D., A positive systems model of TCPlike congestion control: Asymptotic results, IEEE/ACM Transactions on Networking, 14, 3, 616-629 (2006)
[23] Smith, H., Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems (1995), American Mathematical Society · Zbl 0821.34003
[24] Sontag, E. D., Molecular systems biology and control: a qualitative-quantitative approach, Proceedings of IEEE Conference on Decision and Control and European Control Conference, 2314-2319 (2005)
[25] Sun, Y., Stability analysis of positive switched systems via joint linear copositive Lyapunov functions, Nonlinear Analysis. Hybrid Systems, 19, 146-152 (2016) · Zbl 1329.93120
[26] Tian, D.; Liu, S., Exponential stability of switched positive homogeneous systems, (Hindawi (2017)), Article 4326028 pp., 1-8 · Zbl 1380.93212
[27] Tian, D.; Liu, S., Stability analysis for a class of switched positive nonlinear systems under dwell-time constraint, Advances in Difference Equations, 95, 1-13 (2018) · Zbl 1445.93030
[28] Zhao, X.; Zhang, L.; Shi, P., Stability of a class of switched positive linear time-delay systems, International Joutnal of Robust and Nonlinear Control, 23, 5, 578-589 (2013) · Zbl 1284.93208
[29] Zou, Y.; Zhou, Z.; Dong, X.; Meng, Z., Distributed formation control for multiple vertical takeoff and landing UAVs with switching topologies, IEEE/ASME Transactions on Mechatronics, 23, 4, 1750-1761 (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.