Practical stability of switched homogeneous positive nonlinear systems: max-separable Lyapunov function method. (English) Zbl 1510.93131
Summary: The practical stability problem of switched homogeneous positive nonlinear systems (SHPNS) is addressed in this study, which includes two instances in terms of continuous-time and discrete-time. Sufficient conditions are presented by using the max-separable Lyapunov function (MSLF) approach, such that each solution of SHPNS is practically stable. The distinction between the existing results and the obtained results is that ours are not only relatively concise but also easily verifiable, and the theoretical results are also extended to a more general case without restricting the systems to be positive. Eventually, a pair of examples are proposed to explain the approach’s validity.
MSC:
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
| 37C75 | Stability theory for smooth dynamical systems |
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
| 93C28 | Positive control/observation systems |
Keywords:
practical stability; switched homogeneous positive systems; max-separable Lyapunov function; average dwell timeReferences:
| [1] | Luenberger, D., Positive linear systems, Introduction to Dynamic Systems: Theory, Models, and Applications (1979), Wiley: Wiley New York · Zbl 0458.93001 |
| [2] | Smith, H., Monotone Dynamical Systems, an Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, vol. 41 (1995), American Mathematical Society: American Mathematical Society Providence, Rhode Island, USA · Zbl 0821.34003 |
| [3] | Rantzer, A., Distributed control of positive systems, Proceedings of the 50th Conference on Decision and Control, 6608-6611 (2011), IEEE: IEEE Orlando, FL, USA |
| [4] | Sun, Y. G.; Tian, Y. Z.; Xie, X. J., Stabilization of positive switched linear systems and its application in consensus of multi-agent systems, IEEE Trans. Autom. Control, 62, 6608-6613 (2017) · Zbl 1390.93646 |
| [5] | Liu, X. W.; Yu, W. S.; Wang, L., Stability analysis for continuous-time positive systems with time-varying delays, IEEE Trans. Autom. Control, 55, 1024-1028 (2010) · Zbl 1368.93600 |
| [6] | Feyzmahdavian, H.; Charalambous, T.; Johansson, M., Exponential stability of homogeneous positive systems of degree one with time-varying delays, IEEE Trans. Autom. Control, 59, 1594-1599 (2014) · Zbl 1360.93596 |
| [7] | Feyzmahdavian, H. R.; Charalambous, T.; Johansson, M., Asymptotic stability and decay rates of homogeneous positive systems with bounded and unbounded delays, SIAM J. Control Optim., 52, 2623-2650 (2014) · Zbl 1320.34101 |
| [8] | Yang, H.; Zhang, Y., Exponential stability analysis for discrete-time homogeneous impulsive positive delay systems of degree one, J. Frankl. Inst., 357, 2295-2329 (2020) · Zbl 1451.93326 |
| [9] | Ding, X. Y.; Shu, L.; Wang, Z. H., On stability for switched linear positive systems, Math. Comput. Model., 53, 1044-1055 (2011) · Zbl 1217.93149 |
| [10] | Ding, X. Y.; Shu, L.; Liu, X., On linear copositive Lyapunov functions for switched positive systems, J. Frankl. Inst., 348, 2099-2107 (2011) · Zbl 1231.93058 |
| [11] | Pastravanu, O.; Matcovschi, M. H.; Voicu, M., Qualitative analysis results for arbitrarily switching positive systems, IFAC Proc. Vol., 44, 1326-1331 (2011) |
| [12] | Sun, Y. G.; Wu, Z. R., On the existence of linear copositive Lyapunov functions for 3-dimensional switched positive linear systems, J. Frankl. Inst., 350, 1379-1387 (2013) · Zbl 1293.93424 |
| [13] | Xiang, M.; Xiang, Z. R., Exponential stability of discrete-time switched linear positive systems with time-delay, Appl. Math. Comput., 230, 193-199 (2014) · Zbl 1410.39031 |
| [14] | Li, S.; Xiang, Z. R.; Zhang, J. F., Exponential stability analysis for singular switched positive systems under dwell-time constraints, J. Frankl. Inst., 357, 13834-13871 (2020) · Zbl 1454.93240 |
| [15] | Zhao, X. D.; Zhang, L. X.; Shi, P.; Liu, M., Stability of switched positive linear systems with average dwell time switching, Automatica, 48, 1132-1137 (2012) · Zbl 1244.93129 |
| [16] | Zhao, X. D.; Zhang, L. X.; Shi, P., Stability of a class of switched positive linear time-delay systems, Int. J. Robust Nonlinear Control, 23, 578-589 (2013) · Zbl 1284.93208 |
| [17] | Xu, Y.; Dong, J. G.; Lu, R. Q.; Xie, L. H., Stability of continuous-time positive switched linear systems: a weak common copositive Lyapunov functions approach, Automatica, 97, 278-285 (2018) · Zbl 1406.93286 |
| [18] | Zhang, J.; Sun, Y. G., Practical exponential stability of discrete-time switched linear positive systems with impulse and all modes unstable, Appl. Math. Comput., 409, 126408 (2021) · Zbl 1510.93284 |
| [19] | Li, S.; Xiang, Z. R., Stabilisation of a class of positive switched nonlinear systems under asynchronous switching, Int. J. Syst. Sci., 48, 1537-1547 (2017) · Zbl 1362.93128 |
| [20] | Liu, X. W., Stability analysis of a class of nonlinear positive switched systems with delays, Nonlinear Anal., 16, 1-12 (2015) · Zbl 1310.93072 |
| [21] | Zhang, J. F.; Han, Z. Z.; Zhu, F. B.; Zhao, X. D., Absolute exponential stability and stabilization of switched nonlinear systems, Syst. Control Lett., 66, 51-57 (2014) · Zbl 1288.93075 |
| [22] | Dong, J. G., Stability of switched positive nonlinear systems, Int. J. Robust Nonlinear Control, 26, 3118-3129 (2016) · Zbl 1346.93341 |
| [23] | Li, Y. N.; Sun, Y. G.; Meng, F. W.; Tian, Y. Z., Exponential stabilization of switched time-varying systems with delays and disturbances, Appl. Math. Comput., 324, 131-140 (2018) · Zbl 1426.34101 |
| [24] | Tian, Y. Z.; Sun, Y. G., Exponential stability of switched nonlinear time-varying systems with mixed delays: comparison principle, J. Frankl. Inst., 357, 6918-6931 (2020) · Zbl 1447.93300 |
| [25] | Kawano, Y.; Besselink, B.; Cao, M., Contraction analysis of monotone systems via separable functions, IEEE Trans. Autom. Control, 65, 3486-3501 (2020) · Zbl 1533.93616 |
| [26] | Yang, H. T.; Zhang, Y., Impulsive control of continuous-time homogeneous positive delay systems of degree one, Int. J. Robust Nonlinear Control, 29, 3341-3362 (2019) · Zbl 1426.93278 |
| [27] | Zou, Y.; Meng, Z. Y.; Meng, D. Y., On exponential stability of switched homogeneous positive systems of degree one, Automatica, 103, 302-309 (2019) · Zbl 1415.93224 |
| [28] | Zhang, N.; Sun, Y. G.; Meng, F. W., State bounding for switched homogeneous positive nonlinear systems with exogenous input, Nonlinear Anal., 29, 363-372 (2018) · Zbl 1388.93049 |
| [29] | Sun, Y. G.; Tian, Y. Z., Polynomial stability of positive switching homogeneous systems with different degrees, Appl. Math. Comput., 414, 126699 (2022) · Zbl 1510.93135 |
| [30] | Chen, G. P.; Deng, F. Q.; Yang, Y., Practical finite-time stability of switched nonlinear time-varying systems based on initial state-dependent dwell time methods, Nonlinear Anal., 41, 101031 (2021) · Zbl 1478.93594 |
| [31] | Rantzer, A., Optimizing positively dominated systems, Proceedings of the 51st Conference on Decision and Control, 272-277 (2012), IEEE: IEEE Maui, HI, USA |
| [32] | Wu, X. H.; Mu, X. W., Practical scaled consensus for nonlinear multiagent systems with input time delay via a new distributed integral-type event-triggered scheme, Nonlinear Anal., 40, 100995 (2021) · Zbl 1478.93646 |
| [33] | Umbría, F.; Gordillo, F.; Salas, F., A controller for practical stability of capacitor voltages in a five-level diode-clamped converter, Eur. J. Control, 28, 56-68 (2016) · Zbl 1336.93123 |
| [34] | Zhang, H.; Liu, J.; Xu, S. Y., Practical stability and event-triggered load frequency control of networked power systems, IEEE Trans. Syst. Man Cybern. (2022) |
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.
