doi: 10.3934/jimo.2021070
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Design of probabilistic $ l_2-l_\infty $ filter for uncertain Markov jump systems with partial information of the transition probabilities

1. 

Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, China

2. 

School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Western Australia, 6102, Australia

3. 

School of Mathematical Sciences, Sunway University, Kuala Lumpur 47500, Malaysia

4. 

Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China

* Corresponding author: Fei Liu

Received  December 2020 Revised  February 2021 Early access April 2021

Fund Project: The first author is supported in part by the National Natural Science Foundation of China under grant nos. 61773011, 61773183, the Ministry of Education of China under the 111 Project B12018 and Curtin Fellowship

In this paper, the problem of $ l_2-l_\infty $ probabilistic filtering for uncertain Markov jump systems with partial information of the transition probabilities is studied, where the uncertainties are caused by randomly changing interior parameters. Combining the original system and the filtering system, an augmented error system is proposed. Some concepts of probability theory are introduced to handle the uncertainties. Due to the complicated structure of real practical systems, only partial information on the transition probabilities are available. In this paper, by using Lyapunov functional method and probability theory, linear matrix inequalities (LMIs) type of sufficient conditions are derived. Based on these sufficient conditions, a probability filter is constructed such that the augmented error system with partial information of the transition probabilities is stochastically stable with a given confidence level and satisfying an $ l_2-l_\infty $ performance index. Furthermore, the gain matrices of the filter are obtained through the introduction of slack matrices. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Citation: Liqiang Jin, Yanqing Liu, Yanyan Yin, Kok Lay Teo, Fei Liu. Design of probabilistic $ l_2-l_\infty $ filter for uncertain Markov jump systems with partial information of the transition probabilities. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021070
References:
[1]

N. AbbassiD. BenboudjemaS. Derrode and W. Pieczynski, Optimal filter approximations in conditionally Gaussian pairwise Markov switching models, IEEE Transactions on Automatic Control, 60 (2015), 1104-1109.  doi: 10.1109/TAC.2014.2340591.  Google Scholar

[2]

S. Aberkane and V. Dragan, $H_\infty$ filtering of periodic Markovian jump systems: Application to filtering with communication constraints, Automatica, 48 (2012), 3151-3156.  doi: 10.1016/j.automatica.2012.08.040.  Google Scholar

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N. AgrawalA. KumarV. Bajaj and G. K. Singh, Design of bandpass and bandstop infinite impulse response filters using fractional derivative, IEEE Transactions on Industrial Electronics, 66 (2019), 1285-1295.  doi: 10.1109/TIE.2018.2831184.  Google Scholar

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M. C. CampiS. Garatti and M. Prandini, The scenario approach for systems and control design, Annual Reviews in Control, 33 (2009), 149-157.  doi: 10.1016/j.arcontrol.2009.07.001.  Google Scholar

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H. D. ChoiC. K. AhnH. R. Karimi and M. T. Lim, Filtering of discrete-time switched neural networks ensuring exponential dissipative and $l_2-l_{\infty}$ performances, IEEE Transactions on Cybernetics, 47 (2017), 3195-3207.  doi: 10.1109/TCYB.2017.2655725.  Google Scholar

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H. H. Dam and W.-K. Ling, Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank, Journal of Industrial and Management Optimization, 15 (2019), 97-112.  doi: 10.3934/jimo.2018034.  Google Scholar

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S. DongC. L. P. ChenM. Fang and Z.-G. Wu, Dissipativity-based asynchronous fuzzy sliding mode control for T–S fuzzy hidden Markov jump systems, IEEE Transactions on Cybernetics, 50 (2020), 4020-4030.  doi: 10.1109/TAC.2017.2776747.  Google Scholar

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S. DongZ.-G. WuY.-J. PanH. Su and Y. Liu, Hidden-Markov-model-based asynchronous filter design of nonlinear Markov jump systems in continuous-time domain, IEEE Transactions on Cybernetics, 49 (2019), 2294-2304.  doi: 10.1109/TCYB.2018.2824799.  Google Scholar

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F. LiP. ShiC. C. Lim and L. Wu, Fault detection filtering for nonhomogeneous Markovian jump systems via fuzzy approach, IEEE Transactions on Fuzzy Systems, 26 (2018), 131-141.  doi: 10.1109/TFUZZ.2016.2641022.  Google Scholar

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R. Li and J. Cao, Finite-time stability analysis for Markovian jump memristive neural networks with partly unknown transition probabilities, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 2924-2935.  doi: 10.1109/TNNLS.2016.2609148.  Google Scholar

[13]

F. LiS. XuH. Shen and Q. Ma, Passivity-based control for hidden Markov jump systems with singular perturbations and partially unknown probabilities, IEEE Transactions on Automatic Control, 65 (2020), 3701-3706.  doi: 10.1109/TAC.2019.2953461.  Google Scholar

[14]

Y. LiuY. YinK. L. TeoS. Wang and F. Liu, Probabilistic control of Markov jump systems by scenario optimization approach, Journal of Industrial and Management Optimization, 15 (2019), 1447-1453.  doi: 10.3934/jimo.2018103.  Google Scholar

[15]

L. LinQ. WangB. He and X. Peng, Evaluation of fault diagnosability for nonlinear uncertain systems with multiple faults occurring simultaneously, Journal of Systems Engineering and Electronics, 31 (2020), 643-646.  doi: 10.23919/JSEE.2020.000039.  Google Scholar

[16]

Y. Z. LunA. D'Innocenzo and M. D. Di Benedetto, Robust stability of polytopic time-inhomogeneous Markov jump linear systems, Automatica, 105 (2019), 286-297.  doi: 10.1016/j.automatica.2019.03.031.  Google Scholar

[17]

C. F. MoraisJ. M. PalmaP. L. D. Peres and R. C. L. F. Oliveira, An LMI approach for $H_2$ and $H_\infty$ reduced-order filtering of uncertain discrete-time Markov and Bernoulli jump linear systems, Automatica, 95 (2018), 463-471.  doi: 10.1016/j.automatica.2018.06.014.  Google Scholar

[18]

J. WangS. MaC. Zhang and M. Fu, $H_\infty$ state estimation via asynchronous filtering for descriptor Markov jump systems with packet losses, Signal Processing, 154 (2019), 159-167.  doi: 10.1016/j.sigpro.2018.09.003.  Google Scholar

[19]

G. WangR. Xue and J. Wang, A distributed maximum correntropy Kalman filter, Signal Processing, 160 (2019), 247-251.  doi: 10.1016/j.sigpro.2019.02.030.  Google Scholar

[20]

Y. WeiJ. QiuP. Shi and H.-K. Lam, A new design of H-infinity piecewise filtering for discrete-time nonlinear time-varying delay systems via T–S fuzzy affine models, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2034-2047.  doi: 10.1109/TSMC.2016.2598785.  Google Scholar

[21]

S. Xing and F. Deng, Delay-dependent $H_\infty$ filtering for discrete singular Markov jump systems with Wiener process and partly unknown transition probabilities, Journal of the Franklin Institute, 355 (2018), 6062-6082.  doi: 10.1016/j.jfranklin.2018.05.061.  Google Scholar

[22]

Y. YinP. ShiF. LiuK. 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

[23]

Y. YinY. LiuK. L. Teo and S. Wang, Event-triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach, International Journal of Robust Nonlinear Control, 28 (2018), 144-153.  doi: 10.1002/rnc.3858.  Google Scholar

[24]

Y. YinP. ShiF. Liu and K. L. Teo, Observer-based $H_\infty$ 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

[25]

K. YinD. YangJ. Liu and H. Li, Positive $l_1$-gain asynchronous filter design of positive Markov jump systems, Journal of the Franklin Institute, 357 (2020), 11072-11093.  doi: 10.1016/j.jfranklin.2020.08.033.  Google Scholar

[26]

Y. YinJ. ShiF. Liu and Y. Liu, Robust fault detection of singular Markov jump systems with partially unknown information, Information Sciences, 537 (2020), 368-379.  doi: 10.1016/j.ins.2020.05.069.  Google Scholar

[27]

L. ZhangB. Cai and Y. Shi, Stabilization of hidden semi-Markov jump systems: Emission probability approach, Automatica, 101 (2019), 87-95.  doi: 10.1016/j.automatica.2018.11.027.  Google Scholar

[28]

Y. ZhangY. OuX. Wu and Y. Zhou, Resilient dissipative dynamic output feedback control for uncertain Markov jump Lur'e systems with time-varying delays, Nonlinear Analysis: Hybrid Systems, 24 (2017), 13-27.  doi: 10.1016/j.nahs.2016.11.002.  Google Scholar

show all references

References:
[1]

N. AbbassiD. BenboudjemaS. Derrode and W. Pieczynski, Optimal filter approximations in conditionally Gaussian pairwise Markov switching models, IEEE Transactions on Automatic Control, 60 (2015), 1104-1109.  doi: 10.1109/TAC.2014.2340591.  Google Scholar

[2]

S. Aberkane and V. Dragan, $H_\infty$ filtering of periodic Markovian jump systems: Application to filtering with communication constraints, Automatica, 48 (2012), 3151-3156.  doi: 10.1016/j.automatica.2012.08.040.  Google Scholar

[3]

N. AgrawalA. KumarV. Bajaj and G. K. Singh, Design of bandpass and bandstop infinite impulse response filters using fractional derivative, IEEE Transactions on Industrial Electronics, 66 (2019), 1285-1295.  doi: 10.1109/TIE.2018.2831184.  Google Scholar

[4]

G. C. Calafiore and M. C. Campi, The scenario approach to robust control design, IEEE Transaction on Automatic Control, 51 (2006), 742-753.  doi: 10.1109/TAC.2006.875041.  Google Scholar

[5]

M. C. CampiS. Garatti and M. Prandini, The scenario approach for systems and control design, Annual Reviews in Control, 33 (2009), 149-157.  doi: 10.1016/j.arcontrol.2009.07.001.  Google Scholar

[6]

X.-H. ChangZ.-M. Li and J. H. Park, Fuzzy generalized $H_2$ filtering for nonlinear discrete-time systems with measurement quantization, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 2419-2430.  doi: 10.1109/TSMC.1972.5408561.  Google Scholar

[7]

H. D. ChoiC. K. AhnH. R. Karimi and M. T. Lim, Filtering of discrete-time switched neural networks ensuring exponential dissipative and $l_2-l_{\infty}$ performances, IEEE Transactions on Cybernetics, 47 (2017), 3195-3207.  doi: 10.1109/TCYB.2017.2655725.  Google Scholar

[8]

H. H. Dam and W.-K. Ling, Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank, Journal of Industrial and Management Optimization, 15 (2019), 97-112.  doi: 10.3934/jimo.2018034.  Google Scholar

[9]

S. DongC. L. P. ChenM. Fang and Z.-G. Wu, Dissipativity-based asynchronous fuzzy sliding mode control for T–S fuzzy hidden Markov jump systems, IEEE Transactions on Cybernetics, 50 (2020), 4020-4030.  doi: 10.1109/TAC.2017.2776747.  Google Scholar

[10]

S. DongZ.-G. WuY.-J. PanH. Su and Y. Liu, Hidden-Markov-model-based asynchronous filter design of nonlinear Markov jump systems in continuous-time domain, IEEE Transactions on Cybernetics, 49 (2019), 2294-2304.  doi: 10.1109/TCYB.2018.2824799.  Google Scholar

[11]

F. LiP. ShiC. C. Lim and L. Wu, Fault detection filtering for nonhomogeneous Markovian jump systems via fuzzy approach, IEEE Transactions on Fuzzy Systems, 26 (2018), 131-141.  doi: 10.1109/TFUZZ.2016.2641022.  Google Scholar

[12]

R. Li and J. Cao, Finite-time stability analysis for Markovian jump memristive neural networks with partly unknown transition probabilities, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 2924-2935.  doi: 10.1109/TNNLS.2016.2609148.  Google Scholar

[13]

F. LiS. XuH. Shen and Q. Ma, Passivity-based control for hidden Markov jump systems with singular perturbations and partially unknown probabilities, IEEE Transactions on Automatic Control, 65 (2020), 3701-3706.  doi: 10.1109/TAC.2019.2953461.  Google Scholar

[14]

Y. LiuY. YinK. L. TeoS. Wang and F. Liu, Probabilistic control of Markov jump systems by scenario optimization approach, Journal of Industrial and Management Optimization, 15 (2019), 1447-1453.  doi: 10.3934/jimo.2018103.  Google Scholar

[15]

L. LinQ. WangB. He and X. Peng, Evaluation of fault diagnosability for nonlinear uncertain systems with multiple faults occurring simultaneously, Journal of Systems Engineering and Electronics, 31 (2020), 643-646.  doi: 10.23919/JSEE.2020.000039.  Google Scholar

[16]

Y. Z. LunA. D'Innocenzo and M. D. Di Benedetto, Robust stability of polytopic time-inhomogeneous Markov jump linear systems, Automatica, 105 (2019), 286-297.  doi: 10.1016/j.automatica.2019.03.031.  Google Scholar

[17]

C. F. MoraisJ. M. PalmaP. L. D. Peres and R. C. L. F. Oliveira, An LMI approach for $H_2$ and $H_\infty$ reduced-order filtering of uncertain discrete-time Markov and Bernoulli jump linear systems, Automatica, 95 (2018), 463-471.  doi: 10.1016/j.automatica.2018.06.014.  Google Scholar

[18]

J. WangS. MaC. Zhang and M. Fu, $H_\infty$ state estimation via asynchronous filtering for descriptor Markov jump systems with packet losses, Signal Processing, 154 (2019), 159-167.  doi: 10.1016/j.sigpro.2018.09.003.  Google Scholar

[19]

G. WangR. Xue and J. Wang, A distributed maximum correntropy Kalman filter, Signal Processing, 160 (2019), 247-251.  doi: 10.1016/j.sigpro.2019.02.030.  Google Scholar

[20]

Y. WeiJ. QiuP. Shi and H.-K. Lam, A new design of H-infinity piecewise filtering for discrete-time nonlinear time-varying delay systems via T–S fuzzy affine models, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2034-2047.  doi: 10.1109/TSMC.2016.2598785.  Google Scholar

[21]

S. Xing and F. Deng, Delay-dependent $H_\infty$ filtering for discrete singular Markov jump systems with Wiener process and partly unknown transition probabilities, Journal of the Franklin Institute, 355 (2018), 6062-6082.  doi: 10.1016/j.jfranklin.2018.05.061.  Google Scholar

[22]

Y. YinP. ShiF. LiuK. 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

[23]

Y. YinY. LiuK. L. Teo and S. Wang, Event-triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach, International Journal of Robust Nonlinear Control, 28 (2018), 144-153.  doi: 10.1002/rnc.3858.  Google Scholar

[24]

Y. YinP. ShiF. Liu and K. L. Teo, Observer-based $H_\infty$ 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

[25]

K. YinD. YangJ. Liu and H. Li, Positive $l_1$-gain asynchronous filter design of positive Markov jump systems, Journal of the Franklin Institute, 357 (2020), 11072-11093.  doi: 10.1016/j.jfranklin.2020.08.033.  Google Scholar

[26]

Y. YinJ. ShiF. Liu and Y. Liu, Robust fault detection of singular Markov jump systems with partially unknown information, Information Sciences, 537 (2020), 368-379.  doi: 10.1016/j.ins.2020.05.069.  Google Scholar

[27]

L. ZhangB. Cai and Y. Shi, Stabilization of hidden semi-Markov jump systems: Emission probability approach, Automatica, 101 (2019), 87-95.  doi: 10.1016/j.automatica.2018.11.027.  Google Scholar

[28]

Y. ZhangY. OuX. Wu and Y. Zhou, Resilient dissipative dynamic output feedback control for uncertain Markov jump Lur'e systems with time-varying delays, Nonlinear Analysis: Hybrid Systems, 24 (2017), 13-27.  doi: 10.1016/j.nahs.2016.11.002.  Google Scholar

Figure 1.  The system mode trajectory
Figure 2.  The system states curve
Figure 3.  The filtering system state
Figure 4.  The error response curve
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