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Stochastic Stability, ℒ1-gain and Control Synthesis for Positive Semi-Markov Jump Systems

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  • Control Theory and Applications
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Abstract

This paper treats the problems of stochastic stability, ℒ1-gain and control synthesis for positive semi-Markov jump systems (S-MJSs). The system under consideration involves semi-Markov stochastic process related to Weibull distribution. The main motivation for this paper is that the positive condition sometimes needs to be considered in S-MJSs and the controller design methods in the existing works have some conservation. To deal with these problems, some sufficient conditions for stochastic stability of positive S-MJSs are established by implying the linear co-positive Lyapunov function. Then, some sufficient conditions for ℒ1-gain constraint are also presented, upon which, a state feedback controller is designed by decomposing the controller gain matrix such that the resulting closed-loop system is positive and stochastically stable with ℒ1-gain performance in the form of standard linear programming (LP). The advantages of the new framework lie in the following facts: (1) the weak infinitesimal operator is derived for S-MJSs under the constraint of positive condition and (2) the less conservative stabilizing controller is designed to achieve the desired control performance. Finally, numerical examples are given to demonstrate the validity of the main results.

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References

  1. L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley, New York, 2000.

    Book  MATH  Google Scholar 

  2. T. Kaczorek, Positive 1D and 2D Systems, Springer, London, 2002.

    Book  MATH  Google Scholar 

  3. P. Li, J. Lam, and Z. Shu, “ℋ∞ positive filtering for positive linear discrete-time systems: an augmentation approach,” IEEE Transactions on Automatic Control, vol. 55, no. 10, pp. 2337–2342, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  4. X. M. Chen, J. Lam, P. Li, and Z. Shu, “ℓ1-induced norm and controller synthesis of positive systems,” Automatica, vol. 49, no. 5, pp. 1377–1385, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Xiang, Z. R. Xiang, and H. R. Karimi, “Stabilization of positive switched systems with time-varying delays under asynchronous switching,” Int. Journal of Control, Automation, and Systems, vol. 12, no. 5, pp. 939–947, 2014.

    Article  Google Scholar 

  6. S. Li, Z. R. Xiang, and H. R. Karimi, “Mixed ℓ_/ℓ 1 fault detection observer design for positive switched systems with time-varying delay via delta operator approach,” Int. Journal of Control, Automation, and Systems, vol. 12, no. 4, pp. 709–721, 2014.

    Article  Google Scholar 

  7. J. F. Zhang and Z. Z. Han, “Robust stabilization of switched positive linear systems with uncertainties,” International Journal of Control, Automation, and Systems, vol. 11, no. 1, pp. 41–47, 2013.

    Article  Google Scholar 

  8. M. A. Rami and F. Tadeo, “Controller synthesis for positive linear systems with bounded controls,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 2, pp. 151–155, 2007.

    Article  Google Scholar 

  9. J. F. Zhang, X. D. Zhao, and R. D. Zhang, “An improved approach to controller design of positive systems using controller gain decomposition,” Journal of the Franklin Institute, vol. 354, no. 3, pp. 1356–1373, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. D. Zong, H. L. Ren, and L. L. Hou, “Finite-time stability of interconnected impulsive switched systems,” IET Control Theory and Applications, vol. 10, no. 6, pp. 648–654, 2016.

    Article  MathSciNet  Google Scholar 

  11. X. P. Xie, D. Yue, H. G. Zhang, and C. Peng, “Control synthesis of discrete-Time T-S fuzzy systems: reducing the conservatism whilst alleviating the computational burden,” IEEE Transactions on Cybernetics, vol. 47, no. 9, pp. 2480–2491, 2017.

    Article  Google Scholar 

  12. X. P. Xie, D. Yue, and C. Peng, “Multi-instant observer design of discrete-time fuzzy systems: a ranking-based switching approach,” IEEE Transactions on Fuzzy Systems, vol. 25, no. 5, pp. 1281–1292, 2017.

    Article  Google Scholar 

  13. X. T. Wu, Y. Tang, J. D. Cao, and X. R. Mao, “Stability analysis for continuous-time switched systems with stochastic switching signals,” IEEE Transactions on Automatic Control, https://doi.org/10.1109/TAC.2017.2779882.

  14. X. T. Wu, Y. Tang, and W. B. Zhang, “Stability analysis of stochastic delayed systems with an application to multiagent systems,” IEEE Transactions on Automatic Control, vol. 21, no. 12, pp. 4143–4149, 2016.

    Article  Google Scholar 

  15. Y. Tang, X. Xing, H. R. Karimi, L. Kocarev, and J. Kurths, “Tracking control of networked multi-agent systems under new characterizations of impulses and its applications in robotic systems,” IEEE Transactions on Industrial Electronics, vol. 63, no. 2, pp. 1299–1307, 2016.

    Article  Google Scholar 

  16. Y. Tang, H. J. Gao, J. Q. Lu, and J. Kurths, “Pinning distributed synchronization of stochastic dynamical networks: a mixed optimization approach,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 10, pp. 1804–1815, 2014.

    Article  Google Scholar 

  17. Q. Zhou, H. Y. Li, L. J. Wang, and R. Q. Lu, “Prescribed performance observer-based adaptive fuzzy control for nonstrict-feedback stochastic nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, https://doi.org/10.1109/TSMC.2017.2738155.

  18. L. J. Wang, M. Basin, H. Y. Li, and R. Q. Lu, “Observer-based composite adaptive fuzzy control for nonstrict-feedback systems with actuator failures,” IEEE Transactions on Fuzzy Systems, https://doi.org/10.1109/TFUZZ.2017.2774185.

  19. H. L. Ren, G. D. Zong, and T. S. Li, “Eventtriggered finite-time control for networked switched linear systems with asynchronous switching,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, https://doi.org/10.1109/TSMC.2017.2789186.

  20. T. Li, G. Li, and Q. Zhao, “Adaptive fault-tolerant stochastic shape control with application to particle distribution control,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 45, no. 12, pp. 1592–1604, 2015.

    Article  MathSciNet  Google Scholar 

  21. F. R. Lei, X. F. Xu, and T. Li, “Attitude tracking control for Mars entry vehicle via T-S model with time-varying input delay,” Nonlinear Dynamics, vol. 85, no. 3, pp. 1749–1764, 2016.

    Article  MATH  Google Scholar 

  22. L. H. Zhang, W. H. Qi, Y. G. Kao, X. W. Gao, and L. J. Zhao, “New results on finite-time stabilization for stochastic systems with time-varying delay,” International Journal of Control, Automation, and Systems, vol. 62, no. 8 pp. 649–658, 2018.

    Article  Google Scholar 

  23. W. H. Qi, Y. G. Kao, X. W. Gao, and Y. L. Wei, “Controller design for time-delay system with stochastic disturbance and actuator saturation via a new criterion,” Applied Mathematics and Computation, 2018, vol. 320, no. 1, pp. 535–546.

    Article  MathSciNet  Google Scholar 

  24. W. P. Blair and D. D. Sworder, “Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria,” International Journal of Control, vol. 21, no. 5, pp. 833–844, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. E. O. Svensson and N. Williams, “Optimal monetary policy under uncertainty: a Markov jump linear-quadratic approach,” Federal Reserve of St. Louis Review, vol. 90, pp. 275–293, 2008.

    Google Scholar 

  26. L. J. Shen and U. Buscher, “Solving the serial batching problem in job shop manufacturing systems,” European Journal of Operational Research, vol. 221, no. 1, pp. 14–26, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  27. V. A. Ugrinovskii and H. R. Pota, “Decentralized control of power systems via robust control of uncertain Markov jump parameter systems,” International Journal of Control, vol. 78, no. 9, pp. 662–677, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  28. K. You, M. Fu, and L. Xie, “Mean square stability for Kalman filtering with Markovian packet losses,” Automatica, vol. 47, no. 12, pp. 2647–2657, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  29. W. H. Qi, J. H. Park, J. Cheng, Y. G. Kao, and X. W. Gao, “Anti-windup design for stochastic Markovian switching systems with mode-dependent time-varying delays and saturation nonlinearity,” Nonlinear Analysis: Hybrid Systems, vol. 26, pp. 201–211, 2017.

    MathSciNet  MATH  Google Scholar 

  30. W. H. Qi, Y. G. Kao, and X. W. Gao, “Further results on finite-time stabilization for stochastic Markovian jump systems with time-varying delay,” International Journal of Systems Science, vol. 48, no. 14, pp. 2967–2975, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  31. W. H. Qi, Y. G. Kao, and X. W. Gao, “Passivity and passification for stochastic systems with Markovian switching and generally uncertain transition rates,” International Journal of Control, Automation, and Systems, vol. 15, no. 5, pp. 2161–2173, 2017.

    Article  Google Scholar 

  32. Y. K. Li, H. B. Sun, G. D. Zong, and L. L. Hou, “Composite anti-disturbance resilient control for Markovian jump nonlinear systems with partly unknown transition probabilities and multiple disturbances,” International Journal of Robust and Nonlinear Control, vol. 27, no. 14, pp. 2323–2337, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. P. He, “Fault detection filter design for a class of nonlinear Markovian jumping systems with mode-dependent time-varying delays,” Nonlinear Dynamics, vol. 91, no. 3, pp. 1871–1884, 2018.

    Article  MATH  Google Scholar 

  34. H. B. Sun, Y. K. Li, G. D. Zong, and L. L. Hou, “Disturbance attenuation and rejection for stochastic Markovian jump system with partially known transition probabilities,” Automatica, vol. 89, pp. 349–357, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  35. P. Bolzern, P. Colaneri, and N. G. De, “Stochastic stability of positive Markov jump linear systems,” Automatica, vol. 50, no. 4, pp. 1181–1187, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Lian, J. Liu, and Y. Zhuang, “Mean stability of positive Markov jump linear systems with homogeneous and switching transition probabilities,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 62, no. 8, pp. 801–805, 2015.

    Article  Google Scholar 

  37. J. F. Zhang, Z. Z. Han, and F. B. Zhu, “Stochastic stability and stabilization of positive systems with Markovian jump parameters,” Nonlinear Analysis: Hybrid Systems, vol. 12, pp. 147–155, 2014.

    MathSciNet  MATH  Google Scholar 

  38. S. Li and Z. R. Xiang, “Stochastic stability analysis and L¥-gain controller design for positive Markov jump systems with time-varying delays,” Nonlinear Analysis: Hybrid Systems, vol. 22, pp. 31–42, 2016.

    MathSciNet  MATH  Google Scholar 

  39. W. H. Qi, J. H. Park, J. Cheng, Y. G. Kao, and Y. G. Kao, “Exponential stability and ℒ1-gain analysis for positive time-delay Markovian jump systems with switching transition rates subject to average dwell time,” Information Sciences, vol. 428, pp. 224–234, 2018.

    Article  MathSciNet  Google Scholar 

  40. Y. F. Guo, “Stabilization of positive Markov jump systems,” Journal of the Franklin Institute, vol. 353, pp. 3428–3440, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  41. W. H. Qi and X.W. Gao, “State feedback controller design for singular positive Markovian jump systems with partly known transition rates,” Applied Mathematics Letters, vol. 46, pp. 111–116, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  42. W. H. Qi and X. W. Gao, “ ℒ1 control for positive Markovian jump systems with time-varying delays and partly known transition rates,” Circuits, Systems and Signal Processing, vol. 34, no. 8, pp. 2711–2716, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. F. Zhang, X. D. Zhao, F. B. Zhu, and Z. Z. Han, “ℒ1/ℓ 1-Gain analysis and synthesis of Markovian jump positive systems with time delay,” ISA Transactions, vol. 63, pp. 93–102, 2016.

    Article  Google Scholar 

  44. W. H. Qi and X. W. Gao, “ℒ1 control for positive Markovian jump systems with partly known transition rates,” International Journal of Control, Automation, and Systems, vol. 15, no. 1, pp. 2174–2181, 2017.

    Article  Google Scholar 

  45. S. Q. Zhu, Q. L. Han, and C. H. Zhang, “ℓ 1-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: a linear programming approach,” Automatica, vol. 50, no. 8, pp. 2098–2107, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  46. W. H. Qi and X. W. Gao, “Positive ℒ1-gain filter design for positive continuous-time Markovian jump systems with partly known transition rates,” International Journal of Control, Automation, and Systems, vol. 14, no. 6, pp. 1413–1420, 2016.

    Article  Google Scholar 

  47. E. P. C. Kao, “A semi-Markovian population model with application to hospital planning,” IEEE Trans. on Systems, Man and Cybernetics, vol. 3, no. 4, pp. 327–336, 1973.

    Article  MATH  Google Scholar 

  48. T. N. Mudge and H. B. Al-Sadoun, “A semi-Markov model for the performance of multiple-bus systems,” IEEE Transactions on Computers, vol. 34, no. 10, pp. 934–942, 1985.

    Article  Google Scholar 

  49. P. C. G. Vassiliou, “Fuzzy semi-Markov migration process in credit risk,” Fuzzy Sets and Systems, vol. 223, pp. 39–58, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  50. Z. T. Hou, J.W. Luo, P. Shi, and S. K. Nguang, “Stochastic stability of Ito differential equations with semi-Markovian jump parameters,” IEEE Transactions on Automatic Control, vol. 51, no. 8, pp. 1383–1387, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Huang and Y. Shi, “Stochastic stability and robust stabilization of semi-Markov jump linear systems,” International Journal of Robust and Nonlinear Control, vol. 23, no. 18, pp. 2028–2043, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  52. F. B. Li, L. G. Wu, P. Shi, and C. C. Lim, “State estimation and sliding mode control for semi-Markovian jump systems with mismatched uncertainties,” Automatica, vol. 51, pp. 385–393, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  53. W. H. Qi, J. H. Park, J. Cheng, and Y. G. Kao, “Robust stabilization for nonlinear time-delay semi-Markovian jump systems via sliding mode control,” IET Control Theory and Applications, vol. 11, no. 10, pp. 1504–1513, 2017.

    Article  MathSciNet  Google Scholar 

  54. X. H. Liu, X. H. Yu, X. J. Zhou, and H. S. Xi, “Finite-time ℋ∞ control for linear systems with semi-Markovian switching,” Nonlinear Dynamics, vol. 85, pp. 2297–2308, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  55. Y. L. Wei, J. H. Park, J. B. Qiu, L. G. Wu, and H. Y. Jung, “Sliding mode control for semi-Markovian jump systems via output feedback,” Automatica, vol. 81, pp. 133–141, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  56. H. Shen, L. Su, and J. H. Park, “Reliable mixed ℋ∞/passive control for T-S fuzzy delayed systems based on a semi-Markov jump model approach,” Fuzzy Sets and Systems, vol. 314, pp. 79–98, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  57. H. Shen, Z. G. Wu, and J. H. Park, “Reliable mixed passive and ℋ∞ filtering for semi-Markov jump systems with randomly occurring uncertainties and sensor failures,” International Journal of Robust and Nonlinear Control, vol. 25, no. 17, pp. 3231–3251, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  58. H. Shen, J. H. Park, Z. G. Wu, and Z. Q. Zhang, “Finite-time ℋ∞ synchronization for complex networks with semi-Markov jump topology,” Communications in Nonlinear Science and Numerical Simulation, vol. 24, no. 1–3, pp. 40–51, 2015.

    Article  MathSciNet  Google Scholar 

  59. W. H. Qi, G. D. Zong, and H. R. Karimi, “Observer-based adaptive SMC for nonlinear uncertain singular semi-Markov jump systems with applications to DC motor,” IEEE Transactions on Circuits and Systems I: Regular Papers, https://doi.org/10.1109/TCSI.2018.2797257.

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Authors and Affiliations

  1. School of Engineering, Qufu Normal University, Rizhao, 276826, China

    Longjiang Zhao, Wenhai Qi & Lihua Zhang

  2. College of Science, Harbin Institute of Technology, Weihai, 264200, China

    Yonggui Kao

  3. School of Information Science and Engineering, Northeastern University, Shenyang, 110819, China

    Xianwen Gao

Authors
  1. Longjiang Zhao
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  2. Wenhai Qi
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  3. Lihua Zhang
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  4. Yonggui Kao
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  5. Xianwen Gao
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Correspondence to Wenhai Qi.

Additional information

Recommended by Associate Editor Hongyi Li under the direction of Editor PooGyeon Park. This work was supported in part by National Natural Science Foundation of China (61703231 and 61773235), Natural Science Foundation of Shandong (ZR2017QF001, ZR2017PF001, and ZR2017MF063), Chinese Postdoctoral Science Foundation (2017M612235 and 2018T110670), Taishan Scholar Project (TSQN20161033), and Excellent Experiment Project of Qufu Normal University (jp201728).

Longjiang Zhao was born in Rizhao, Shandong Province, P. R. China, in 1983. In 2011, he received his Ph.D. degree from Beijing Institute of Technology. He works in Qufu Normal University. His research work focus on complex industry process and intelligent control, etc.

Wenhai Qi was born in Taian, Shandong Province, P. R. China, in 1986. He received his B.S. degree in automation from Qufu Normal University in 2008 and M.S. degree from Qufu Normal University in 2013. In 2016, he received his Ph.D. degree in control theory and control engineering from Northeastern University. He works in Qufu Normal University. His research work focus on Markovian jump systems, positive systems, etc.

Lihua Zhang was born in Jinnig, Shandong Province, P. R. China, in 1972. He received his B.S. degree in automation from Shandong University in 1994 and his M.S. degree from Qufu Normal University in 2005. He works in Qufu Normal University. His research work focus on complex industry process and intelligent control, Markovian jump systems, etc.

Yonggui Kao received his B.E. degree from Beijing Jiaotong University in 1996. He received his M.E. and Ph.D. degrees from Ocean University of China in 2005 and 2008, respectively. He now is a Professor at Department of Mathematics, Harbin Institute of Technology (Weihai). His research interest covers stochastic systems, impulsive systems, singular systems, Markovian jumping systems, artificial intelligence, neural networks, stability theory and sliding mode control.

Xianwen Gao received his B.S. degree from Shenyang University of Chemical Technology in 1978 and an M.S. degree from Northeastern University in 1993. In 1998, he received a Ph.D. degree in control theory and control engineering from Northeastern University. He is currently a professor in Northeastern University. His main research interests are modeling of complex industry process and intelligent control, stochastic jump systems, etc.

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Zhao, L., Qi, W., Zhang, L. et al. Stochastic Stability, ℒ1-gain and Control Synthesis for Positive Semi-Markov Jump Systems. Int. J. Control Autom. Syst. 16, 2055–2062 (2018). https://doi.org/10.1007/s12555-017-0628-7

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