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July  2022, 18(4): 2335-2349. doi: 10.3934/jimo.2021070

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 Published  July 2022 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 and Management Optimization, 2022, 18 (4) : 2335-2349. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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