Abstract
This paper is concerned with the event-triggered L1-gain control of a class of nonlinear positive switched systems. First, an event-triggering condition in the form of 1-norm is presented for the systems. By virtue of the event-triggering strategy, the original system is transformed into an interval uncertain system. An event-triggered L1-gain controller is designed by decomposing the controller gain matrix into the sum of nonnegative and non-positive components. Under the design controller, the resulting closed-loop systems are positive and L1-gain stable. The obtained approach is developed for the systems subject to input saturation. All presented conditions are solvable in terms of linear programming. Finally, two examples are provided to verify the effectiveness of the design.
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Zhao X, Zhang L, and Shi P, Stability of a class of switched positive linear time-delay systems, International Journal of Robust and Nonlinear Control, 2013, 23(5): 578–589.
Farina L and Rinaldi S, Positive Linear Systems: Theory and Applications, John Wiley & Sons, New Jersey, 2011.
Lam J, Chen Y, Liu X, et al., Positive systems theory and applications, Vol. 480, Part of the Lecture Notes in Control and Information Sciences Book Series, Springer, Switzerland, 2019.
Shorten R, Wirth F, and Leith D, A positive systems model of TCP-like congestion control: Asymptotic results, IEEE/ACM Transactions on Networking, 2006, 14(3): 616–629.
Hernandez-Vargas E, Colaneri P, Middleton R, et al., Discrete-time control for switched positive systems with application to mitigating viral escape, International Journal of Robust and Nonlinear Control, 2011, 21(10): 1093–1111.
Caswell H, Construction, Analysis, and Interpretation, Sunderland, Sinauer, 2001.
Liu X, Stability analysis of switched positive systems: A switched linear copositive Lyapunov function method, IEEE Transactions on Circuits and Systems II: Express Briefs, 2009, 56(5): 414–418.
Fornasini E and Valcher M E, Stability and stabilizability criteria for discrete-time positive switched systems, IEEE Transactions on Automatic Control, 2011, 57(5): 1208–1221.
Xiang M and Xiang Z, Stability, L1-gain and control synthesis for positive switched systems with time-varying delay, Nonlinear Analysis: Hybrid Systems, 2013, 9: 9–17.
Wang J and Zhao J, Stabilisation of switched positive systems with actuator saturation, IET Control Theory & Applications, 2016, 10(6): 717–723.
Zappavigna A, Colaneri P, Geromel J C, et al., Dwell time analysis for continuous-time switched linear positive systems, Proceedings of the 2010 American Control Conference, 2010, 6256–6261.
Mason O and Shorten R, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Transactions on Automatic Control, 2007, 52(7): 1346–1349.
Huang J, Ma X, Che H, et al., Further result on interval observer design for discrete-time switched systems and application to circuit systems, IEEE Transactions on Circuits and Systems II: Express Briefs, 2019, DOI: https://doi.org/10.1109/TCSII.2019.2957945.
Colaneri P, Geromel J C, and Astolfi A, Stabilization of continuous-time switched nonlinear systems, Systems & Control Letters, 2008, 57(1): 95–103.
Zhang J, Han Z, Zhu F, et al., Absolute exponential stability and stabilization of switched nonlinear systems, Systems & Control Letters, 2014, 66: 51–57.
Zhang J and Raïssi T, Saturation control of switched nonlinear systems, Nonlinear Analysis: Hybrid Systems, 2019, 32: 320–336.
El-Farra N H, Mhaskar P, and Christofides P D, Output feedback control of switched nonlinear systems using multiple Lyapunov functions, Systems & Control Letters, 2005, 54(12): 1163–1182.
Ma L, Huo X, Zhao X, et al., Adaptive fuzzy tracking control for a class of uncertain switched nonlinear systems with multiple constraints: A small-gain approach, International Journal of Fuzzy Systems, 2019, 21(8): 2609–2624.
Ma L, Xu N, Zhao X, et al., Small-gain technique-based adaptive neural output-feedback fault-tolerant control of switched nonlinear systems with unmodeled dynamics, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020, DOI: https://doi.org/10.1109/TSMC.2020.2964822.
Wang D, Wang Z, Li G, et al., Distributed filtering for switched nonlinear positive systems with missing measurements over sensor networks, IEEE Sensors Journal, 2016, 16(12): 4940–4948.
Aleksandrov A Y, Chen Y, Platonov A V, et al., Stability analysis for a class of switched nonlinear systems, Automatica, 2011, 47(10): 2286–2291.
Mancilla-Aguilar J L, A condition for the stability of switched nonlinear systems, IEEE Transactions on Automatic Control, 2000, 45(11): 2077–2079.
Zhang J, Raïssi T, and Li S, Non-fragile saturation control of nonlinear positive Markov jump systems with time-varying delays, Nonlinear Dynamics, 2019, 97: 1495–1513.
Shaker H R and How J P, Stability analysis for class of switched nonlinear systems, Proceedings of the 2010 American Control Conference, 2010, 2517–2520.
Dorf R C, Farren M, and Phillips C, Adaptive sampling frequency for sampled-data control systems, IRE Transactions on Automatic Control, 1962, 7(1): 38–47.
Hu J, Wang Z, Liang J, et al., Event-triggered distributed state estimation with randomly occurring uncertainties and nonlinearities over sensor networks: A delay-fractioning approach, Journal of the Franklin Institute, 2015, 352(9): 3750–3763.
Ren H, Zong G, and Li T, Event-triggered finite-time control for networked switched linear systems with asynchronous switching, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2018, 48(11): 1874–1884.
Donkers M C F and Heemels W, Output-based event-triggered control with guaranteed \({{\cal L}_\infty }\)-gain and improved and decentralized event-triggering, IEEE Transactions on Automatic Control, 2011, 57(6): 1362–1376.
Eqtami A, Dimarogonas D V, and Kyriakopoulos K J, Event-triggered control for discrete-time systems, Proceedings of the 2010 American Control Conference, 2010, 4719–4724.
Qi Y, Zeng P, and Bao W, Event-triggered and self-triggered H∞ control of uncertain switched linear systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020, 50(4): 1442–1454.
Selivanov A and Fridman E, Event-triggered H∞ control: A switching approach, IEEE Transactions on Automatic Control, 2015, 61(10): 3221–3226.
Ma G and Pagilla P R, Periodic event-triggered dynamic output feedback control of switched systems, Nonlinear Analysis: Hybrid Systems, 2019, 31: 247–264.
Wang Y, Jia Z, and Zuo Z, Dynamic event-triggered and self-triggered output feedback control of networked switched linear systems, Neurocomputing, 2018, 314: 39–47.
Postoyan R, Tabuada P, Nešić D, et al., A framework for the event-triggered stabilization of nonlinear systems, IEEE Transactions on Automatic Control, 2014, 60(4): 982–996.
Huo X, Ma L, Zhao X, et al., Event-triggered adaptive fuzzy output feedback control of MIMO switched nonlinear systems with average dwell time, Applied Mathematics and Computation, 2020, 365: 124665.
Postoyan R, Anta A, Heemels W, et al., Periodic event-triggered control for nonlinear systems, The 52nd Conference on Decision and Control, 2013, 7397–7402.
Yin Y, Lin Z, Liu Y, et al., Event-triggered constrained control of positive systems with input saturation, International Journal of Robust and Nonlinear Control, 2018, 28(11): 3532–3542.
Liu L, Zhang J, Shao Y, et al., Event-triggered control of positive switched systems based on linear programming, IET Control Theory & Applications, 2019, 14(1): 145–155.
Hespanha J P and Morse A S, Stability of switched systems with average dwell-time, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, 3: 2655–2660.
Hu T, Lin Z, and Chen B M, Analysis and design for discrete-time linear systems subject to actuator saturation, Systems & Control Letters, 2002, 45(2): 97–112.
Chang X and Yang G, Nonfragile H∞ filtering of continuous-time fuzzy systems, IEEE Transactions on Signal Processing, 2010, 59(4): 1528–1538.
Xiong J, Chang X, and Yi X, Design of robust nonfragile fault detection filter for uncertain dynamic systems with quantization, Applied Mathematics and Computation, 2018, 338: 774–788.
Chang X, Liu Y, and Shen M, Resilient control design for lateral motion regulation of intelligent vehicle, IEEE/ASME Transactions on Mechatronics, 2019, 24(6): 2488–2497.
Chen G and Yang Y, Finite-time stability of switched positive linear systems, International Journal of Robust and Nonlinear Control, 2014, 24(1): 179–190.
Wu A and Zeng Z, Exponential stabilization of memristive neural networks with time delays, IEEE Transactions on Neural Networks and Learning Systems, 2012, 23(12): 1919–1929.
Zhang G, Shen Y, Yin Q, et al., Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays, Information Sciences, 2013, 232: 386–396.
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This paper was supported by the National Nature Science Foundation of China under Grant Nos. 61873314 and 61703132, the Fundamental Research Funds for the Provincial Universities of Zhejiang under Grant No. GK209907299001-007, the Natural Science Foundation of Zhejiang Province, China under Grant Nos. LY20F030008 and LY20F030011, and the Foundation of Zhejiang Provincial Education Department of China under Grant No. Y201942017.
This paper was recommended for publication by Editor ZHAO Yanlong.
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Zhang, J., Liu, L., Li, S. et al. Event-Triggered L1-Gain Control of Nonlinear Positive Switched Systems. J Syst Sci Complex 34, 873–898 (2021). https://doi.org/10.1007/s11424-020-9324-4
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DOI: https://doi.org/10.1007/s11424-020-9324-4