American Institute of Mathematical Sciences

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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,

1. 

School of Mechanical and Electric Engineering, Soochow University, Suzhou 215131, China
2. 

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.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Yueyuan Zhang, Yanyan Yin, Fei Liu. Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3013-3026. doi: 10.3934/jimo.2020105
References:
[1]

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. 

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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. 

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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.

[16]

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.

[17]

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.

[18]

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.

[19]

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.

[20]

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.

[21]

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.

[22]

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. 

[23]

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.

[24]

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.

[25]

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.

[26]

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. 

[27]

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.

[28]

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.

[29]

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. 

[30]

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.

[31]

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.

[32]

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.

show all references

References:
[1]

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. 

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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. 

[8]

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.

[9]

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.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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.

[16]

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.

[17]

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.

[18]

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.

[19]

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.

[20]

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.

[21]

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.

[22]

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. 

[23]

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.

[24]

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.

[25]

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.

[26]

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. 

[27]

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.

[28]

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.

[29]

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. 

[30]

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.

[31]

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.

[32]

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.

Figure 1.  The response of the jump mode
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Figure 2.  The response of the system state
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Figure 3.  The estimation of the domain attraction for different modes
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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
4 4 4 ln(0.1061)=-2.2434
6 6 6 ln(0.4859)=-0.7218
8 8 8 ln(0.9028)=-0.1023
10 10 10 ln(1)=0
Table Options

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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
4 4 4 ln(0.1061)=-2.2434
6 6 6 ln(0.4859)=-0.7218
8 8 8 ln(0.9028)=-0.1023
10 10 10 ln(1)=0
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