# American Institute of Mathematical Sciences

• Previous Article
Performance analysis of a discrete-time $Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption
• JIMO Home
• This Issue
• Next Article
Online and offline cooperation under buy-online, pick-up-in-store: Pricing and inventory decisions
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. [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. [3] B. R. Barmish, New Tools for Robustness of Linear Systems, New York, MacMillan, 1994. [4] G. C. Calafiorea, F. 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. [5] M. C. Campi, S. Garatti and M. Prandini, The scenario approach for systems and control design, Annual Reviews in Control, 33 (2009), 149-157. [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. [7] Y. Kang, Z. Li, Y. 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. [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. [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. [10] A. Prékopa, Stochastic Programming, Norwell, MA: Kluwer, 1995. doi: 10.1007/978-94-017-3087-7. [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, [12] P. Shi, E. 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. [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. [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. [15] Y. Yin, Y. Liu, K. 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. [16] Y. Yin, P. 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. [17] 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. [18] L. Zhu, Y. Yin, F. 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.

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. [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. [3] B. R. Barmish, New Tools for Robustness of Linear Systems, New York, MacMillan, 1994. [4] G. C. Calafiorea, F. 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. [5] M. C. Campi, S. Garatti and M. Prandini, The scenario approach for systems and control design, Annual Reviews in Control, 33 (2009), 149-157. [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. [7] Y. Kang, Z. Li, Y. 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. [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. [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. [10] A. Prékopa, Stochastic Programming, Norwell, MA: Kluwer, 1995. doi: 10.1007/978-94-017-3087-7. [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, [12] P. Shi, E. 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. [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. [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. [15] Y. Yin, Y. Liu, K. 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. [16] Y. Yin, P. 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. [17] 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. [18] L. Zhu, Y. Yin, F. 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.
State trajectory of a-posteriori Monte-Carlo analysis
 [1] Hussain Alazki, Alexander Poznyak. Robust output stabilization for a class of nonlinear uncertain stochastic systems under multiplicative and additive noises: The attractive ellipsoid method. Journal of Industrial & Management Optimization, 2016, 12 (1) : 169-186. doi: 10.3934/jimo.2016.12.169 [2] Yufei Sun, Grace Aw, Kok Lay Teo, Guanglu Zhou. Portfolio optimization using a new probabilistic risk measure. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1275-1283. doi: 10.3934/jimo.2015.11.1275 [3] Matthew H. Henry, Yacov Y. Haimes. Robust multiobjective dynamic programming: Minimax envelopes for efficient decisionmaking under scenario uncertainty. Journal of Industrial & Management Optimization, 2009, 5 (4) : 791-824. doi: 10.3934/jimo.2009.5.791 [4] John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843 [5] Ahmadreza Argha, Steven W. Su, Lin Ye, Branko G. Celler. Optimal sparse output feedback for networked systems with parametric uncertainties. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 283-295. doi: 10.3934/naco.2019019 [6] René Henrion, Christian Küchler, Werner Römisch. Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 363-384. doi: 10.3934/jimo.2008.4.363 [7] Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53 [8] Elena K. Kostousova. On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6149-6162. doi: 10.3934/dcds.2018153 [9] Xiang-Kai Sun, Xian-Jun Long, Hong-Yong Fu, Xiao-Bing Li. Some characterizations of robust optimal solutions for uncertain fractional optimization and applications. Journal of Industrial & Management Optimization, 2017, 13 (2) : 803-824. doi: 10.3934/jimo.2016047 [10] Lingshuang Kong, Changjun Yu, Kok Lay Teo, Chunhua Yang. Robust real-time optimization for blending operation of alumina production. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1149-1167. doi: 10.3934/jimo.2016066 [11] Alexey G. Mazko. Positivity, robust stability and comparison of dynamic systems. Conference Publications, 2011, 2011 (Special) : 1042-1051. doi: 10.3934/proc.2011.2011.1042 [12] Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273 [13] Alireza Goli, Hasan Khademi Zare, Reza Tavakkoli-Moghaddam, Ahmad Sadeghieh. Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 187-209. doi: 10.3934/naco.2019014 [14] Ying Ji, Shaojian Qu, Fuxing Chen. Environmental game modeling with uncertainties. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 989-1003. doi: 10.3934/dcdss.2019067 [15] Sebastian Hage-Packhäuser, Michael Dellnitz. Stabilization via symmetry switching in hybrid dynamical systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 239-263. doi: 10.3934/dcdsb.2011.16.239 [16] Kais Ammari, Eduard Feireisl, Serge Nicaise. Polynomial stabilization of some dissipative hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4371-4388. doi: 10.3934/dcds.2014.34.4371 [17] Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063 [18] Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004 [19] Valérie Dos Santos, Bernhard Maschke, Yann Le Gorrec. A Hamiltonian perspective to the stabilization of systems of two conservation laws. Networks & Heterogeneous Media, 2009, 4 (2) : 249-266. doi: 10.3934/nhm.2009.4.249 [20] Roberto Guglielmi. Indirect stabilization of hyperbolic systems through resolvent estimates. Evolution Equations & Control Theory, 2017, 6 (1) : 59-75. doi: 10.3934/eect.2017004

2017 Impact Factor: 0.994

## Tools

Article outline

Figures and Tables

[Back to Top]