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July  2019, 15(3): 1447-1453. doi: 10.3934/jimo.2018103

Probabilistic control of Markov jump systems by scenario optimization approach

a. 

School of Textile and Clothing, Key Laboratory of Eco-Textile (Ministry of Education), Jiangnan University, Wuxi 214122, China

b. 

Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth WA6845, Australia

c. 

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

* Corresponding author: Yanyan YIN

Received  November 2017 Revised  January 2018 Published  July 2018

Fund Project: The first author is supported by NSFC grant 61503155.

This paper addresses the new problem of probabilistic robust stabilization for uncertain stochastic systems by using scenario optimization approach, where the uncertainties are not assumed to be norm-bounded. State feedback controllers are designed to guarantee that the closed-loop system is robust probabilistic stable. The problem of designing the controller gains is formulated and solved as linear matrix inequality (LMI) constraints. Simulation results are presented to illustrate the correctness and usefulness of the controllers designed.

Citation: Yanqing Liu, Yanyan Yin, Kok Lay Teo, Song Wang, Fei Liu. Probabilistic control of Markov jump systems by scenario optimization approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1447-1453. doi: 10.3934/jimo.2018103
References:
[1]

S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, vol. 15, Studies in Applied Mathematics, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[2]

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

[3]

B. R. Barmish, New Tools for Robustness of Linear Systems, New York, MacMillan, 1994. Google Scholar

[4]

G. C. CalafioreaF. Dabbeneb and R. Tempo, Research on probabilistic methods for control system design, Automatica J. IFAC, 47 (2011), 1279-1293.  doi: 10.1016/j.automatica.2011.02.029.  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.   Google Scholar

[6]

O. L. V. Costa, M. D. Fragoso and R. P. Marques, Discrete-time Markovian Jump Linear Systems, London: Springer-Verlag, 2005. doi: 10.1007/b138575.  Google Scholar

[7]

Y. KangZ. LiY. Dong and H. Xi, Markovian based fault-tolerant control for wheeled mobile manipulators, IEEE Trans on Control System Technology, 20 (2012), 266-276.  doi: 10.1109/TCST.2011.2109062.  Google Scholar

[8]

N. M. Krasovskii and E. A. Lidskii, Analytical design of controllers in systems with random attributes, Automation and Remote Control, 22 (1961), 1141-1146.   Google Scholar

[9]

A. Nemirovski, Several NP-hard problems arising in robust stability analysis, Mathematics of Control, Signals and Systems, 6 (1993), 99-105.  doi: 10.1007/BF01211741.  Google Scholar

[10]

A. Prékopa, Stochastic Programming, Norwell, MA: Kluwer, 1995. doi: 10.1007/978-94-017-3087-7.  Google Scholar

[11]

Y. Shi, Y. Yin, S. Wang, Y. Liu and F. Liu, Distributed leader-following consensus of nonlinear multi-agent systems with nonlinear input dynamics, Neurocomputing, Google Scholar

[12]

P. ShiE. K. Boukas and R. Agarwal, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Trans on Automatic Control, 44 (1999), 1592-1597.  doi: 10.1109/9.780431.  Google Scholar

[13]

Y. Yin, Z. Lin, Y. Liu and K. L. Teo, Event-triggered constrained control of positive systems with input saturation, International Journal of Robust and Nonlinear Control, 2018. doi: 10.1002/rnc.4097.  Google Scholar

[14]

Y. Yin and Z. Lin, Constrained control of uncertain nonhomogeneous Markovian jump systems, International Journal of Robust and Nonlinear Control, 27 (2017), 3937-3950.   Google Scholar

[15]

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 and Nonlinear Control, 28 (2018), 144-153.  doi: 10.1002/rnc.3858.  Google Scholar

[16]

Y. YinP. Shi and F. Liu, Gain-scheduled robust fault detection on time-delay stochastic nonlinear systems, IEEE Trans on Industrial Electronics, 58 (2011), 4908-4916.  doi: 10.1109/TIE.2010.2103537.  Google Scholar

[17]

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

[18]

L. ZhuY. YinF. Liu and S. Wang, $H_∞$ filtering for uncertain periodic Markov jump systems with periodic and partly unknown information, Circuits, Systems, and Signal Processing, (2018), 1-15.  doi: 10.1007/s00034-018-0759-y.  Google Scholar

show all references

References:
[1]

S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, vol. 15, Studies in Applied Mathematics, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[2]

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

[3]

B. R. Barmish, New Tools for Robustness of Linear Systems, New York, MacMillan, 1994. Google Scholar

[4]

G. C. CalafioreaF. Dabbeneb and R. Tempo, Research on probabilistic methods for control system design, Automatica J. IFAC, 47 (2011), 1279-1293.  doi: 10.1016/j.automatica.2011.02.029.  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.   Google Scholar

[6]

O. L. V. Costa, M. D. Fragoso and R. P. Marques, Discrete-time Markovian Jump Linear Systems, London: Springer-Verlag, 2005. doi: 10.1007/b138575.  Google Scholar

[7]

Y. KangZ. LiY. Dong and H. Xi, Markovian based fault-tolerant control for wheeled mobile manipulators, IEEE Trans on Control System Technology, 20 (2012), 266-276.  doi: 10.1109/TCST.2011.2109062.  Google Scholar

[8]

N. M. Krasovskii and E. A. Lidskii, Analytical design of controllers in systems with random attributes, Automation and Remote Control, 22 (1961), 1141-1146.   Google Scholar

[9]

A. Nemirovski, Several NP-hard problems arising in robust stability analysis, Mathematics of Control, Signals and Systems, 6 (1993), 99-105.  doi: 10.1007/BF01211741.  Google Scholar

[10]

A. Prékopa, Stochastic Programming, Norwell, MA: Kluwer, 1995. doi: 10.1007/978-94-017-3087-7.  Google Scholar

[11]

Y. Shi, Y. Yin, S. Wang, Y. Liu and F. Liu, Distributed leader-following consensus of nonlinear multi-agent systems with nonlinear input dynamics, Neurocomputing, Google Scholar

[12]

P. ShiE. K. Boukas and R. Agarwal, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Trans on Automatic Control, 44 (1999), 1592-1597.  doi: 10.1109/9.780431.  Google Scholar

[13]

Y. Yin, Z. Lin, Y. Liu and K. L. Teo, Event-triggered constrained control of positive systems with input saturation, International Journal of Robust and Nonlinear Control, 2018. doi: 10.1002/rnc.4097.  Google Scholar

[14]

Y. Yin and Z. Lin, Constrained control of uncertain nonhomogeneous Markovian jump systems, International Journal of Robust and Nonlinear Control, 27 (2017), 3937-3950.   Google Scholar

[15]

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 and Nonlinear Control, 28 (2018), 144-153.  doi: 10.1002/rnc.3858.  Google Scholar

[16]

Y. YinP. Shi and F. Liu, Gain-scheduled robust fault detection on time-delay stochastic nonlinear systems, IEEE Trans on Industrial Electronics, 58 (2011), 4908-4916.  doi: 10.1109/TIE.2010.2103537.  Google Scholar

[17]

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

[18]

L. ZhuY. YinF. Liu and S. Wang, $H_∞$ filtering for uncertain periodic Markov jump systems with periodic and partly unknown information, Circuits, Systems, and Signal Processing, (2018), 1-15.  doi: 10.1007/s00034-018-0759-y.  Google Scholar

Figure 1.  State trajectory of a-posteriori Monte-Carlo analysis
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