Equilibrium strategies in a supply chain with capital constrained suppliers: The impact of external financing
A dynamical system method for solving the split convex feasibility problem
November 2021, 17(6): 3013-3026. doi: 10.3934/jimo.2020105
Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation
Yueyuan Zhang 1,, , Yanyan Yin 2, and Fei Liu 2,
School of Mechanical and Electric Engineering, Soochow University, Suzhou 215131, China
Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA 6845, Australia, Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation,Jiangnan University, Wuxi 214122, China
* Corresponding author: Yueyuan Zhang
Received August 2019 Revised December 2019 Published November 2021 Early access June 2020
Fund Project: This work was partially supported by The National Natural Science Foundation of China (61773183)
This paper investigates the control synthesis for discrete-time semi-Markov jump systems with nonlinear input. Observer-based controllers are designed in this paper to achieve a better performance and robustness. The nonlinear input caused by actuator saturation is considered as a group of linear controllers in the convex hull. Moreover, the elapse time and mode dependent Lyapunov functions are investigated and sufficient conditions are derived to guarantee the $ H_\infty $ performance index. The largest domain of attraction is estimated as the existing saturation in the system. Finally, a numerical example is utilized to verify the effectiveness and feasibility of the developed strategy.
Keywords: Discrete-time semi-Markov jump system, H∞ performance index, nonlinear input.
Citation: Yueyuan Zhang, Yanyan Yin, Fei Liu. Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3013-3026. doi: 10.3934/jimo.2020105
Y. Altinok and D. Kolcak, An application of the semi-Markov model for earthquake occurrences in North Anatolia, Turkey, Journal of the Balkan Geophysical Society, 2 (1999), 90-99. Google Scholar
V. S. Barbu, M. Boussemart and N. Limnios, Discrete-time semi-Markov model for reliability and survival analysis, Communications in Statistics Theory and Methods, 33 (2004), 2833-2868. doi: 10.1081/STA-200037923. Google Scholar
V. S. Barbu and N. Limnios, Semi-Markov Chains and Hidden semi-Markov Models Toward Applications: Their Use in Reliability and DNA Analysis, Lecture Notes in Statistics, 191. Springer, New York, 2008. Google Scholar
P. Bolzern, P. Colaneri and G. De Nicolao, Stochastic stability of positive Markov jump linear systems, Automatica, 50 (2014), 1181-1187. doi: 10.1016/j.automatica.2014.02.016. Google Scholar
V. S. Barbu and N. Limnios, Empirical estimation for discrete-time semi-Markov processes with applications in reliability, Nonparametric Statistics, 18 (2006), 483-498. doi: 10.1080/10485250701261913. Google Scholar
Y. Y. Cao and Z. Lin, Stability analysis of discrete-time systems with actuator saturation by a saturation-dependent Lyapunov function, Automatica, 39 (2003), 1235-1241. doi: 10.1016/S0005-1098(03)00072-4. Google Scholar
H. Chen, M. M. Ma, H. Wang, Z. Y. Liu and Z. X. Cai, Moving Horizon H∞ Tracking Control of Wheeled Mobile Robots With Actuator Saturation, IEEE Transactions on Control Systems Technology, 17 (2009), 449-457. Google Scholar
B. S. Chen and S. S. Wang, The stability of feedback control with nonlinear saturating actuator: Time domain approach, IEEE Transactions on Automatic Control, 33 (1988), 483-487. doi: 10.1109/9.1234. Google Scholar
O. L. V Costa, E. Assumpção Filho, E. K. Boukas and R. Marques, Constrained quadratic state feedback control of discrete-time Markovian jump linear systems, Automatica, 35 (1999), 617-626. doi: 10.1016/S0005-1098(98)00202-7. Google Scholar
G. D'Amico, M. Guillen and R. Manca, Full backward non-homogeneous semi-Markov processes for disability insurance models: A Catalunya real data application, Insurance: Mathematics and Economics, 45 (2009), 173-179. doi: 10.1016/j.insmatheco.2009.05.010. Google Scholar
G. D'Amico, F. Petroni and F. Prattico, Wind speed modeled as an indexed semi-Markov process, Environmetrics, 24 (2013), 367-376. doi: 10.1002/env.2215. Google Scholar
F. Garelli, P. Camocardi and R. J. Mantz, Variable structure strategy to avoid amplitude and rate saturation in pitch control of a wind turbine, International Journal of Hydrogen Energy, 35 (2010), 5869-5875. doi: 10.1016/j.ijhydene.2009.12.124. Google Scholar
W. Gao and R. R. Selmic, Neural network control of a class of nonlinear systems with actuator saturation, IEEE Transactions on Neural Networks, 17 (2006), 147-156. doi: 10.1109/TNN.2005.863416. Google Scholar
Z. Hou, J. Luo, P. Shi. Peng and S. K. Nguang, Stochastic stability of Ito differential equations with semi-Markovian jump parameters, IEEE Transactions on Automatic Control, 51 (2006), 1383-1387. doi: 10.1109/TAC.2006.878746. Google Scholar
W. M. H. Heemels, J. Daafouz and G. Millerioux, Observer-Based Control of Discrete-Time LPV Systems With Uncertain Parameters, IEEE Transactions on Automatic Control, 55 (2010), 2130-2135. doi: 10.1109/TAC.2010.2051072. Google Scholar
T. Hu, Z. Lin and B. M. Chen, An analysis and design method for linear systems subject to actuator saturation and disturbance, Automatica, 38 (2002), 351-359. doi: 10.1016/S0005-1098(01)00209-6. Google Scholar
H. Huang, D. Li, Z. Lin and Y. Xi, An improved robust model predictive control design in the presence of actuator saturation, Automatica, 47 (2011), 861-864. doi: 10.1016/j.automatica.2011.01.045. Google Scholar
F. Li, L. Wu, P. Shi and C. C. Lim, State estimation and sliding mode control for {semi-Markovian} jump systems with mismatched uncertainties, Automatica, 51 (2015), 385-393. doi: 10.1016/j.automatica.2014.10.065. Google Scholar
F. Li, P. Shi, C. C. Lim and L. Wu, Fault detection filtering for nonhomogeneous Markovian jump systems via a fuzzy approach, IEEE Transactions on Fuzzy Systems, 26 (2018), 131-141. doi: 10.1109/TFUZZ.2016.2641022. Google Scholar
C. H. Lien, Robust observer-based control of systems with state perturbations via LMI approach, IEEE Transactions on Automatic Control, 49 (2004), 1365-1370. doi: 10.1109/TAC.2004.832660. Google Scholar
Z. Ning, L. Zhang and W. X. Zheng, Observer-Based Stabilization of Nonhomogeneous semi-Markov Jump Linear Systems With Mode-Switching Delays, IEEE Transactions on Automatic Control, 64 (2019), 2029-2036. doi: 10.1109/TAC.2018.2863655. Google Scholar
J. Raouf and E. Boukas, Observer-based controller design for linear singular systems with Markovian switching, 2004 43rd IEEE Conference on Decision and Control (CDC), 4 (2004), 3619-3624. Google Scholar
H. Shen, F. Li, S. Xu and V. Sreeram, Slow state variables feedback stabilization for semi-Markov jump systems with singular perturbations, IEEE Transactions on Automatic Control, 63 (2018), 2709-2714. doi: 10.1109/TAC.2017.2774006. Google Scholar
O. Thomas and J. Sobanjo, Comparison of Markov chain and semi-Markov models for crack deterioration on flexible pavements, Journal of Infrastructure Systems, 19 (2013), 186-195. doi: 10.1061/(ASCE)IS.1943-555X.0000112. Google Scholar
T. Yang, L. Zhang, V. Sreeram, A. N. Vargas, T. Hayat and B. Ahmad, Time-varying filter design for semi-Markov jump linear systems with intermittent transmission, International Journal of Robust and Nonlinear Control, 27 (2017), 4035-4049. doi: 10.1002/rnc.3779. Google Scholar
T. Yang, L. Zhang and H. K. Lam, H∞ control of semi-Markov jump nonlinear systems under σ-error mean square stability, International Journal of Systems Science, 48 (2017), 2291-2299. Google Scholar
Y. Yin, P. Shi, F. Liu, K. L. Teo and C. C. Lim, Robust filtering for nonlinear nonhomogeneous Markov jump systems by fuzzy approximation approach, IEEE Transactions on Cybernetics, 45 (2015), 1706-1716. doi: 10.1109/TCYB.2014.2358680. Google Scholar
Y. Yin, P. Shi, F. Liu and K. L. Teo, Observer-based H∞ control on nonhomogeneous Markov jump systems with nonlinear input, International Journal of Robust and Nonlinear Control, 24 (2014), 1903-1924. doi: 10.1002/rnc.2974. Google Scholar
L. Zhang, Y. Leng and P. Colaneri, Stability and stabilization of discrete-time semi-Markov jump linear systems via semi-Markov kernel approach, IEEE Transactions on Automatic Control, 61 (2016), 503-508. Google Scholar
Y. Zhang, C. C. Lim and F. Liu, Robust mixed H2/H∞ model predictive control for Markov jump systems with partially uncertain transition probabilities, Journal of the Franklin Institute, 355 (2018), 3423-3437. doi: 10.1016/j.jfranklin.2018.01.035. Google Scholar
B. Zhou, W. X. Zheng and G. R. Duan, Stability and stabilization of discrete-time periodic linear systems with actuator saturation, Automatica, 47 (2011), 1813-1820. doi: 10.1016/j.automatica.2011.04.015. Google Scholar
Z. Zuo, D. W. Ho and Y. Wang, Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation, Automatica, 46 (2010), 569-576. doi: 10.1016/j.automatica.2010.01.024. Google Scholar
Figure 1. The response of the jump mode
Figure 2. The response of the system state
Figure 3. The estimation of the domain attraction for different modes
Table 1. The error $ \sigma $ for different maximum sojourn time in different mode
Maximum sojourn time $ T^i_{max} $ error $ \sigma $
$ T^1_{max} $ $ T^2_{max} $ $ T^3_{max} $ $ \sigma $
3 4 3 ln(0.032)=-3.442
4 4 3 ln(0.1048)=-2.2557
10 10 10 ln(1)=0
Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170
Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, 2022, 1 (1) : 137-160. doi: 10.3934/fmf.2021005
Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1803-1811. doi: 10.3934/dcdss.2020106
Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6131-6154. doi: 10.3934/dcdsb.2021010
Veena Goswami, Gopinath Panda. Synchronized abandonment in discrete-time renewal input queues with vacations. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021163
Qingling Zhang, Guoliang Wang, Wanquan Liu, Yi Zhang. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1197-1211. doi: 10.3934/dcdsb.2011.16.1197
Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111
Qiuli Liu, Xiaolong Zou. A risk minimization problem for finite horizon semi-Markov decision processes with loss rates. Journal of Dynamics & Games, 2018, 5 (2) : 143-163. doi: 10.3934/jdg.2018009
Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102
Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109
Yun Kang. Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2123-2142. doi: 10.3934/dcdsb.2013.18.2123
Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207
Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 275-288. doi: 10.3934/naco.2015.5.275
Jaydeep Swarnakar. Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021007
Samuel N. Cohen. Uncertainty and filtering of hidden Markov models in discrete time. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 4-. doi: 10.1186/s41546-020-00046-x
Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525
Liqiang Jin, Yanyan Yin, Kok Lay Teo, Fei Liu. Event-triggered mixed $ H_\infty $ and passive control for Markov jump systems with bounded inputs. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1343-1355. doi: 10.3934/jimo.2020024
Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066
Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264
Yueyuan Zhang Yanyan Yin Fei Liu