Event-Triggered Control of Positive Semi-Markovian Jump Systems Without/with Input Saturation

doi:10.1007/s12204-021-2335-z

Abstract

Abstract: This paper is concerned with the event-triggered control of positive semi-Markovian jump systems without/with input saturation. The considered systems are subject to a stochastic semi-Markovian process whose sojourn time is dependent on a non-exponential distribution. First, an event-triggering condition is introduced in a linear form for the systems. A class of event-triggered feedback controllers is proposed using matrix decomposition technique. By using a stochastic co-positive Lyapunov function, the systems’ positivity and stability are guaranteed. Then, the obtained results are developed for the systems with input saturation. A cone set is chosen as the attraction domain and the corresponding attraction domain gain matrix is designed in terms of standard linear programming approach. Finally, two numerical examples are provided to verify the validity and effectiveness of the presented theoretical findings.

Key words: event-triggered control, positive semi-Markovian jump systems, input saturation, matrix decomposition, linear programming

CLC Number:

TP13

ZHANG Suhuan (张素焕), ZHANG Junfeng∗ (张俊锋), LI Shuo (李烁), YU Shanen (余善恩). Event-Triggered Control of Positive Semi-Markovian Jump Systems Without/with Input Saturation[J]. J Shanghai Jiaotong Univ Sci, 2022, 27(5): 723-736.

References 32

[1]	FARINA L, RINALDI S. Positive linear systems: Theory and applications [M]. New York: John Wiley & Sons, 2000: 50.
[2]	LAM J, CHEN Y, LIU X, et al. Positive systems: theory and applications [M]. Cham: Springer, 2019: 480.
[3]	DE LEENHEER P, AEYELS D. Stabilization of positive linear systems [J]. Systems & Control Letters, 2001, 44(4): 259-271.
[4]	RAMI M A, TADEO F. Controller synthesis for positive linear systems with bounded controls [J]. IEEE Transactions on Circuits, and Systems II: Express Briefs, 2007, 54(2): 151-155.
[5]	SHORTEN R, WIRTH F, LEITH D. A positive systems model of TCP-like congestion control: Asymptotic results [J]. IEEE/ACM Transactions on Networking, 2006, 14(3): 616-629.
[6]	BLANCHINI F, COLANERI P, VALCHER M E. Copositive Lyapunov functions for the stabilization of positive switched systems [J]. IEEE Transactions on Automatic Control, 2012, 57(12): 3038-3050.
[7]	BENZAOUIA A,MESQUINE F, BENHAYOUNM, et al. Stabilization of continuous-time fractional positive systems with delays and asymmetric control bounds [J]. Journal of Dynamic Systems, Measurement, and Control, 2019, 141(5): 051008.
[8]	BOLZERN P, COLANERI P, DE NICOLAO G. Stochastic stability of positive Markov jump linear systems [J]. Automatica, 2014, 50(4): 1181-1187.
[9]	ZHANG J, HAN Z, ZHU F. Stochastic stability and stabilization of positive systems with Markovian jump parameters [J]. Nonlinear Analysis: Hybrid Systems, 2014, 12: 147-155.
[10]	ZHU S, HAN Q, ZHANG C. L1-stochastic stability and L1-gain performance of positive Markov jump linear systems with time-delays: Necessary and sufficient conditions [J]. IEEE Transactions on Automatic Control, 2017, 62(7): 3634-3639.
[11]	ZHU S, HAN Q, ZHANG C. Investigating the effects of time-delays on stochastic stability and designing l1- gain controllers for positive discrete-time Markov jump linear systems with time-delay [J]. Information Sciences, 2016, 355/356: 265-281.
[12]	HU T, LIN Z, CHEN B. An analysis and design method for linear systems subject to actuator saturation and disturbance [J]. Automatica, 2002, 38(2): 351-359.
[13]	HU T, LIN Z. Control systems with actuator saturation: Analysis and design [M]. New York: Springer Science + Business Media, 2001.
[14]	ZHANG J, DENG Z, WANG Y. Robust stability and stabilization of positive interval systems subject to actuator saturation [J]. Asian Journal of Control, 2014, 16(5): 1553-1560.
[15]	WANG J, ZHAO J. Stabilisation of switched positive systems with actuator saturation [J]. IET Control Theory & Applications, 2016, 10(6): 717-723.
[16]	PARK I S, KWON N K, PARK P. A linear programming approach for stabilization of positive Markovian jump systems with a saturated single input [J]. Nonlinear Analysis: Hybrid Systems, 2018, 29: 322-332.
[17]	ZHANG J, RA¨ISSI T. Saturation control of switched nonlinear systems [J]. Nonlinear Analysis: Hybrid Systems, 2019, 32: 320-336.
[18]	QIW, GAO X, KAO Y, et al. Stabilization for positive Markovian jump systems with actuator saturation [J]. Circuits, Systems, and Signal Processing, 2017, 36(1): 374-388.
[19]	ZHANG J, RA¨ISSI T, LI S. Non-fragile saturation control of nonlinear positive Markov jump systems with time-varying delays [J]. Nonlinear Dynamics, 2019, 97(2): 1495-1513.
[20]	RAKKIYAPPAN R, MAHESWARI K, VELMURUGAN G, et al. Event-triggered H∞ state estimation for semi-Markov jumping discrete-time neural networks with quantization [J]. Neural Networks, 2018, 105: 236-248.
[21]	ZHANG L, LIANG H, SUN Y, et al. Adaptive eventtriggered fault detection scheme for semi-Markovian jump systems with output quantization [J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2021, 51(4): 2370-2381.
[22]	WANG J, CHEN M, SHEN H. Event-triggered dissipative filtering for networked semi-Markov jump systems and its applications in a mass-spring system model [J]. Nonlinear Dynamics, 2017, 87(4): 2741-2753.
[23]	HU Z, MU X. Stabilization for switched stochastic systems with semi-Markovian switching signals and actuator saturation [J]. Information Sciences, 2019, 483: 419-431.
[24]	OGURA M, MARTIN C F. Stability analysis of positive semi-Markovian jump linear systems with state resets [J]. SIAM Journal on Control and Optimization, 2014, 52(3): 1809-1831.
[25]	LI L, QIW, CHEN X, et al. Stability analysis and control synthesis for positive semi-Markov jump systems with time-varying delay [J]. Applied Mathematics and Computation, 2018, 332: 363-375. [26] QI W, PARK J H, CHENG J, et al. Stochastic stability and L1-gain analysis for positive nonlinear semi- Markov jump systems with time-varying delay via TS fuzzy model approach [J]. Fuzzy Sets and Systems, 2019, 371: 110-122.
[27]	DORF R, FARREN M, PHILLIPS C. Adaptive sampling frequency for sampled-data control systems [J]. IRE Transactions on Automatic Control, 1962, 7(1): 38-47.
[28]	HEEMELSWP M H, JOHANSSON K H, TABUADA P. An introduction to event-triggered and selftriggered control [C]// 2012 IEEE 51st Conference on Decision and Control. Maui, HI, USA: IEEE, 2012: 3270-3285.
[29]	LI T, FU J. Event-triggered control of switched linear systems [J]. Journal of the Franklin Institute, 2017, 354(15): 6451-6462.
[30]	LI H, ZUO Z, WANG Y. Event-triggered control for Markovian jump systems with partially unknown transition probabilities and actuator saturation [J]. Journal of the Franklin Institute, 2016, 353(8): 1848-1861.
[31]	XIAO S, ZHANG Y, ZHANG B. Event-triggered network-based state observer design of positive systems [J]. Information Sciences, 2018, 469: 30-43.
[32]	YIN Y, LIN Z, LIU Y, et al. Event-triggered constrained control of positive systems with input saturation [J]. International Journal of Robust and Nonlinear Control, 2018, 28(11): 3532-3542.