Finite-time state bounding of homogeneous nonlinear positive systems with disturbance. (English) Zbl 1480.93373
Summary: This paper deals with the finite-time state bounding problem of homogeneous nonlinear positive systems with disturbance. Based on a technique used in positive systems, explicit conditions are established such that all solutions of homogeneous nonlinear positive systems with degree \(0<p<1\) converge to a ball in finite time. The approach used in this paper is different from the usual Lyapunov-Krasovskii functional method. We also extend the main result to the general nonlinear time-varying systems. Finally, two numerical examples are given to show the effectiveness of our results.
MSC:
| 93D40 | Finite-time stability |
| 93C28 | Positive control/observation systems |
| 93C10 | Nonlinear systems in control theory |
References:
| [1] | Farina, L.; Rinaldi, S., Positive Linear Systems: Theory and Applications (2000), Wiley: Wiley New York, NY, USA · Zbl 0988.93002 |
| [2] | Kaczorek, T., Positive 1D and 2D Systems (2002), Springer-Verlag: Springer-Verlag New York, NY, USA: · Zbl 1005.68175 |
| [3] | Liu, J.; Lam, J.; Shu, Z., Positivity-preserving consensus of homogeneous multiagent systems, IEEE Trans. Autom. Control, 65, 2724-2729 (2020) · Zbl 1533.93728 |
| [4] | Shen, J.; Liu, J.; Cui, Y., An exact characterization of the \(l_1 / l_-\) index of positive systems and its application to fault detection filter design, IEEE Trans. Circt. Syst. II (2020) |
| [5] | Liu, J.; Lam, J.; Xie, X., Generalized lead-lag \(h_\infty\) compensators for MIMO linear systems, IEEE Trans. Syst. Man Cybern., 99, 1-11 (2020) |
| [6] | Cui, Y.; Yang, N.; Liu, J., A novel approach for positive edge consensus of nodal networks, J. Frankl. Inst., 357, 4349-4362 (2020) · Zbl 1437.93118 |
| [7] | Liu, J.; Lam, J.; Wang, Y.; Cui, Y.; Shu, Z., Robust and nonfragile consensus of positive multiagent systems via observer-based output-feedback protocols, Int. J. Robust Nonlinear Control, 30, 5386-5403 (2020) · Zbl 1465.93199 |
| [8] | Chang, X.; Yang, G., Nonfragile \(h_\infty\) filtering of continuous-time fuzzy systems, IEEE Trans. Signal Process., 59, 1528-1538 (2011) · Zbl 1391.93134 |
| [9] | Ma, L.; Zong, G.; Zhao, X.; Huo, X., Observed-based adaptive finite-time tracking control for a class of nonstrict-feedback nonlinear systems with input saturation, J. Frankl. Inst., 357, 11518-11544 (2020) · Zbl 1450.93062 |
| [10] | Haddad, W. M.; Chellaboina, V., Stability theory for nonnegative and compartmental dynamical systems with time delay, Syst. Control Lett., 51, 355-361 (2004) · Zbl 1157.34352 |
| [11] | Yin, Y.; Zong, G.; Zhao, X., Improved stability criteria for switched positive linear systems with average dwell time switching, J. Frankl. Inst., 354, 3472-3484 (2017) · Zbl 1364.93693 |
| [12] | Mason, O.; Verwoerd, M., Observations on the stability properties of cooperative system, Syst. Control Lett., 58, 461-467 (2009) · Zbl 1161.93021 |
| [13] | Li, S.; Xiang, Z.; Karimi, H. R., Stability and l1-gain controller design for positive switched systems with mixed time-varying delays, Appl. Math. Comput., 222, 507-518 (2013) · Zbl 1328.93215 |
| [14] | Bokharaie, V. S.; Mason, O.; Verwoerd, M., D-stability and delay independent stability of homogeneous cooperative systems, IEEE Trans. Autom. Control, 55, 2882-2885 (2010) · Zbl 1368.93589 |
| [15] | Bokharaie, V. S.; Mason, O.; Wirth, F., Stability and positivity of equilibria for subhomogeneous cooperative systems, Nonlinear Anal., 74, 6416-6426 (2011) · Zbl 1235.34164 |
| [16] | Shen, J.; Lam, J., Decay rate constrained stability analysis for positive systems with discrete and distributed delays, Syst. Sci. Control Eng., 2, 7-12 (2014) |
| [17] | Li, Y.; Sun, Y.; Meng, F., Exponential stabilization of switched time-varying systems with delays and disturbances, Appl. Math. Comput., 324, 131-140 (2018) · Zbl 1426.34101 |
| [18] | Li, Y.; Sun, Y.; Meng, F., New criteria for exponential stability of switched time-varying systems with delays and nonlinear disturbances, Nonlinear Anal. Hybrid Syst., 26, 284-291 (2017) · Zbl 1373.93271 |
| [19] | Tian, Y.; Cai, Y.; Sun, Y.; Gao, H., Finite-time stability for impulsive switched delay systems with nonlinear disturbances, J. Frankl. Inst., 353, 3578-3597 (2016) · Zbl 1347.93198 |
| [20] | Liu, X.; Yu, W.; Wang, L., Stability analysis for continuous-time positive systems with time-varying delays, IEEE Trans. Autom. Control, 55, 1024-1028 (2010) · Zbl 1368.93600 |
| [21] | Feyzmahdavian, H. R.; 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 |
| [22] | Sun, Y.; Tian, Y.; Xie, X. J., Stabilization of positive switched linear systems and its application in consensus of multiagent systems, IEEE Trans. Autom. Control, 62, 6608-6613 (2017) · Zbl 1390.93646 |
| [23] | Sun, Y.; Wu, Z.; Meng, F., Common weak linear copositive Lyapunov functions for positive switched linear systems, Complexity, 2018, 1-7 (2018) · Zbl 1398.93309 |
| [24] | Zhang, N.; Sun, Y.; Meng, F., State bounding for switched homogeneous positive nonlinear systems with exogenous input, Nonlinear Anal. Hybrid Syst., 29, 363-372 (2018) · Zbl 1388.93049 |
| [25] | Zhu, X.; Sun, Y.; Xie, X., State bounding for a class of nonlinear time-varying systems with delay and disturbance, J. Frankl. Inst., 355, 8213-8224 (2018) · Zbl 1398.93143 |
| [26] | Fridman, E.; Shaked, U., On reachable sets for linear systems with delay and bounded peak inputs, Automatica, 39, 2005-2010 (2003) · Zbl 1037.93042 |
| [27] | Kim, J. H., Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances, Automatica, 44, 2940-2943 (2008) · Zbl 1152.93309 |
| [28] | Zuo, Z.; Ho, D. W.C.; Wang, Y., Reachable set bounding for delayed systems with polytopic uncertainties: the maximal Lyapunov-Krasovskii functional approach, Automatica, 46, 949-952 (2010) · Zbl 1191.93127 |
| [29] | Kwon, O. M.; Lee, S. M.; Park, J. H., On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances, Inf. Sci., 181, 3735-3748 (2011) · Zbl 1243.93015 |
| [30] | Nam, P. T.; Pathirana, P. N., Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays, Automatica, 47, 1838-1841 (2011) · Zbl 1226.93024 |
| [31] | That, N. D.; Nam, P. T.; Ha, Q. P., Reachable set bounding for linear discrete-time systems with delays and bounded disturbances, J. Optim. Theory Appl., 157, 96-107 (2013) · Zbl 1264.93020 |
| [32] | Zuo, Z.; Che, Y.; Wang, Y.; Ho, D. W.C.; Chen, M. Z.Q.; Li, H., A note on reachable set bounding for delayed systems with polytopic uncertainties, J. Frankl. Inst., 350, 1827-1835 (2013) · Zbl 1392.93004 |
| [33] | Zhang, B.; Lam, J.; Xu, S., Reachable set estimation and controller design for distributed delay systems with bounded disturbances, J. Frankl. Inst., 351, 3068-3088 (2014) · Zbl 1290.93020 |
| [34] | Shen, T.; Petersen, I. R., An ultimate state bound for a class of linear systems with delay, Automatica, 87, 447-449 (2018) · Zbl 1378.93056 |
| [35] | Hien, L. V.; Trinh, H. M., A new approach to state bounding for linear time-varying system with delay and bounded disturbances, Automatica, 50, 1735-1738 (2014) · Zbl 1296.93046 |
| [36] | Zhang, J.; Sun, Y., Reachable set estimation for switched nonlinear positive systems with impulse and time delay, Int. J. Robust Nonlinear Control, 30, 3332-3343 (2020) · Zbl 1466.93015 |
| [37] | Nam, P. T.; Pathirana, P. N.; Trinh, H., Reachable set bounding for nonlinear perturbed time-delay systems: the smallest bound, Appl. Math. Lett., 43, 68-71 (2015) · Zbl 1310.93029 |
| [38] | Zhang, N.; Sun, Y.; Zhao, P., State bounding for homogeneous positive systems of degree one with time-varying delay and exogenous input, J. Frankl. Inst., 354, 2893-2904 (2017) · Zbl 1364.93361 |
| [39] | Sun, Y.; Meng, F., Reachable set estimation for a class of nonlinear time-varying systems, Complexity, 2017, 1-6 (2017) · Zbl 1373.93061 |
| [40] | Zhu, X.; Sun, Y., Reachable set bounding for homogeneous nonlinear systems with delay and disturbance, Complexity, 2019, 1-6 (2019) · Zbl 1421.93109 |
| [41] | Dong, J. G., On the decay rates of homogeneous positive systems of any degree with time-varying delays, IEEE Trans. Autom. Control, 60, 2983-2988 (2015) · Zbl 1360.93494 |
| [42] | Smith, H., Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Am. Math. Soc. (1995) · Zbl 0821.34003 |
| [43] | Hale, J. H.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York: · Zbl 0787.34002 |
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