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On positively invariant polyhedrons for continuous-time positive linear systems. (English) Zbl 1454.93115
Summary: This paper is concerned with the determination of positively invariant polyhedron for positive linear systems subject to external disturbances whose \((\infty,1)\)-norm or \((\infty,\infty)\)-norm are bounded by a prescribed constant. Necessary and sufficient conditions for the existence of a positively invariant polyhedron are derived in terms of a set of inequalities which can be solved by linear programming, and the link between Lyapunov stability and positively invariant polyhedron is also revealed.
93C28 Positive control/observation systems
93C05 Linear systems in control theory
93C73 Perturbations in control/observation systems
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
90C05 Linear programming
Full Text: DOI
[1] Article Number: 145. · Zbl 1422.93012
[2] Du, B.; Lam, J.; Shu, Z.; Chen, Y., On reachable sets for positive linear systems under constrained exogenous inputs, Automatica, 74, 230-237 (2016) · Zbl 1348.93043
[3] Oucheriah, S., Robust tracking and model following of uncertain dynamic delay systems by memoryless linear controllers, IEEE Trans. Automat. Control, 44, 7, 1473-1477 (1999) · Zbl 0955.93026
[4] Khalil, H. K., Nonlinear Systems (2002), Prentice-Hall: Prentice-Hall New Jersey
[5] Castelan, E. B.; Hennet, J. C., On invariant polyhedra of continuous-time linear systems, IEEE Trans. Automat. Control, 38, 11, 1680-1685 (1993) · Zbl 0790.93099
[6] Milani, B. E.A.; Dórea, C. E.T., On invariant polyhedra of continuous-time systems subject to additive disturbances, Automatica, 32, 5, 785-789 (1996) · Zbl 0851.93046
[7] Lapierre, L.; Zapata, R.; Lepinay, P., Combined path-following and obstacle avoidance control of a wheeled robot, Int. J. Robot. Res., 26, 4, 361-375 (2007)
[8] Ménec, S. L., Linear differential game with two pursuers and one evader, Adv. Dyn. Games, 11, 209-226 (2011) · Zbl 1218.91026
[9] Haimovich, H.; Seron, M. M., Bounds and invariant sets for a class of switching systems with delayed-state-dependent perturbations, Automatica, 49, 748-754 (2013) · Zbl 1268.93077
[10] Willems, J., Lyapunov functions for diagonally dominant systems, Automatica, 12, 519-523 (1976) · Zbl 0345.93040
[11] Kofman, E.; Haimovich, H.; Seron, M. M., A systematic method to obtain ultimate bounds for perturbed systems, Int. J. Control, 80, 2, 167-178 (2007) · Zbl 1140.93428
[12] Shen, J.; Zheng, W., Stability analysis of linear delay systems with cone invariance, Automatica, 53, 30-36 (2015) · Zbl 1371.93171
[13] Shen, J.; Lam, J., On the decay rate of discrete-time linear delay systems with cone invariance, IEEE Trans. Autom. Control, 62, 7, 3442-3447 (2017) · Zbl 1370.93228
[14] Mézo, T. L.; Jaulin, L.; Zerr, B., An interval approach to compute invariant sets, IEEE Trans. Autom. Control, 62, 8, 4236-4242 (2017) · Zbl 1373.93060
[15] Zhao, X.; Yin, Y.; Liu, L., Stability analysis and delay control for switched positive linear systems, IEEE Trans. Autom. Control, 63, 7, 2184-2190 (2018) · Zbl 1423.93334
[16] Lam, J.; Chen, Y.; Liu, X., Positive Systems: Theory and Applications (POSTA 2018) (2019), Springer Nature Switzerland AG
[17] Zhao, X.; Yin, Y.; Zheng, X., State-dependent switching control of switched positive fractional-order systems, ISA Trans., 62, 103-108 (2016)
[18] Farina, L.; Rinaldi, S., Positive Linear Systems: Theory and Applications (2000), Wiley, New York · Zbl 0988.93002
[19] Haddad, W. M.; Chellaboina, V.-S.; Hui, Q., Nonnegative and Compartmental Dynamic Systems (2010), Princeton University Press: Princeton University Press Princeton, NJ
[20] Bitsoris, G., Existence of positively invariant polyhedral sets for continuous-time linear systems, Control Theory Adv. Tech., 7, 3, 407-427 (1991)
[21] LaSalle, J. P.; Artstein, Z., The Stability of Dynamical Systems (1976), SIAM Editions, Philadelphia
[22] Bitsoris, G., On the stability of nonlinear systems, Int. J. Control, 38, 3, 699-711 (1983) · Zbl 0523.93047
[23] Dinh, T. N.; Mazenc, F.; Niculescu, S. I., Interval observer composed of observers for nonlinear systems, Proceedings of the European Control Conference, 660-665 (2014), Strasbourg, France
[24] Zhang, J.; Cai, X.; Zhang, W., Robust model predictive control with l_1-gain performance for positive systems, J. Frankl. Inst., 352, 7, 2831-2846 (2015) · Zbl 1395.93185
[25] Chen, Y.; Bo, Y.; Du, B., Positive L1-filter design for continuous-time positive Markov jump linear systems: full-order and reduced-order, IET Control Theory Appl., 13, 12, 1855-1862 (2019) · Zbl 1432.93337
[26] M. Fang, L. Wang, Z.G. Wu, Asynchronous stabilization of Boolean control networks with stochastic switched signals, IEEE Trans. Syst. Man Cybern. Syst. doi:10.1109/TSMC.2019.2913088.
[27] L. Wang, M. Fang, Z.G. Wu, et al. Necessary and sufficient conditions on pinning stabilization for stochastic Boolean networks, IEEE Trans. Cybern. doi:10.1109/TCYB.2019.2931051.
[28] Wang, L.; Fang, M.; Wu, Z. G., Mean square stability for Markov jump Boolean networks, Sci. China Inf. Sci., 63, 1, 112205 (2020)
[29] Liu, X., Stability analysis of a class of nonlinear positive switched systems with delays, Nonlinear Anal. Hybrid Syst., 16, 1-12 (2015) · Zbl 1310.93072
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