Skip to main content
SpringerLink
Log in

Finite-time \(\varvec{H_{\infty }}\) control for linear systems with semi-Markovian switching

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Finite-time \(H_{\infty }\) control problem is a fascinating and hot issue in the field of control science. This paper presents a novel framework for finite-time \(H_{\infty }\) stabilization of semi-Markovian switching system. By employing Lyapunov–Krasovskii functional and matrix inequality techniques, together with properties of semi-Markovian process, sufficient conditions are proposed to guarantee finite-time boundedness, \(H_{\infty }\) finite-time boundedness and finite-time \(H_{\infty }\) state feedback stabilization for semi-Markovian switching system. At the same time, a state feedback controller is provided to ensure that the proposed closed-loop system is finite-time \(H_{\infty }\) stabilization. Finally, a numerical example and simulations are given to show the correctness and effectiveness of the proposed results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Assawinchaichote, W., Nguang, S.K., Shi, P.: Robust \(H_{\infty }\) fuzzy filter design for uncertain nonlinear singularly perturbed systems with Markovian jumps: an LMI approach. Inf. Sci. 177(7), 1699–1714 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Athans, M.: Command and control theory: a challenge to control science. IEEE Trans. Autom. Control 32(4), 286–293 (1987)

    Article  Google Scholar 

  3. Shen, L.J., Buscher, U.: Solving the serial batching problem in job shop manufacturing systems. Eur. J. Oper. Res. 221(1), 14–26 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chan, A., Englehart, K., Hudgins, B., Lovely, D.F.: Hidden markov model classification of myoelec-tric signals in speech. IEEE Eng. Med. Biol. Mag. 21(5), 143–146 (2002)

    Article  Google Scholar 

  5. Mao, X.R., Yuan, C.G.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)

    Book  MATH  Google Scholar 

  6. Zhang, Y.: Stability of discrete-time Markovian jump delay systems with delayed impulses and partly unknown transition probabilities. Nonlinear Dyn. 75, 101–111 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. De Souza, C.E., Trofino, A., Barbosa, K.A.: Mode-independent \(H_{\infty }\) filters for Markovian jump linear systems. IEEE Trans. Autom. Control 51(11), 1837–1841 (2006)

    Article  Google Scholar 

  8. Hu, L., Shi, P., Frank, P.M.: Robust sampled-data control for Markovian jump linear systems. Automatica 42(11), 2025–2030 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Karan, M., Shi, P., Kaya, Y.: Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems. Automatica 42(12), 2159–2168 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Karimi, H.R.: Robust delay-dependent \(H_{\infty }\) control of uncertain Markovian jump systems with mixed neutral, discrete and distributed time-delays. IEEE Trans. Circuits Syst. I 58(8), 1910–1923 (2011)

    Article  MathSciNet  Google Scholar 

  11. Li, H.P., Shi, Y.: Robust \(H_{\infty }\) filtering for nonlinear stochastic systems with uncertainties and Markov delays. Automatica 48(1), 1159–1166 (2012)

    Article  Google Scholar 

  12. Mahmoud, M.S.: Interconnected jumping time-delay systems: mode-dependent decentralized stability and stabilization. Int. J. Robust Nonlinear Control 22(7), 808–826 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, H.B., Zhu, C.X., Hu, P., Zhang, Y.: Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays. Nonlinear Dyn. 69, 1023–1039 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, H.B., Zhu, C.X., Hu, P., Zhang, Y.: Erratum to: Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays. Nonlinear Dyn. doi:10.1007/s11071-012-0324-3

  15. Shi, Y., Yu, B.: Robust mixed \(H_{2}/H_{\infty }\) control of networked control systems with random time delays in both forward and backward communication links. Automatica 47(4), 754–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhao, J.J., Shen, H., Li, B., Wang, J.: Finite-time \(H_{\infty }\) control for a class of Markovian jump delayed systems with input saturation. Nonlinear Dyn. 73, 1099–1110 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhu, Q., Yu, X., Song, A.G., Fei, S.M., Cao, Z.Q., Yang, Y.Q.: On sliding mode control of single input Markovian jump systems. Automatic 50(11), 2897–2904 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Johnson, B.: Design and Analysis of Fault-Tolerant Digital Systems. Addison-Wesley, Reading, MA (1989)

    Google Scholar 

  19. Limnios, N., Ouhbi, B., Sadek, A.: Empirical estimator of stationary distribution for semi-Markov processes. Commun. Stat. Theory Methods 34(4), 987–995 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schwartz, C.: Control of semi-Markov jump linear systems with application to the bunch-train cavity interaction. Ph.D. Thesis, Northwestern University. (2003)

  21. Hou, Z., Luo, J., Shi, P., Nguang, S.K.: Stochastic stability of Itô differential equations with semi-Markovian jump parameters. IEEE Trans. Autom. Control 51(8), 1383–1387 (2006)

    Article  MathSciNet  Google Scholar 

  22. Huang, J., Shi, Y.: Stochastic stability of semi-Markov jump linear systems: an LMI approach. In: 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, USA, pp. 4668–4673 (2011)

  23. Huang, J., Shi, Y.: \(H_{\infty }\) state-feedback control for semi-Markov jump linear systems with time-varying delays. ASME J. Dyn. Syst. Meas. Control, 135(4), Aritcle 041012 (2013)

  24. Huang, J., Shi, Y.: Stochastic stability and robust stabilization of semi-Markov jump linear systems. Int. J. Robust Nonlinear Control 23(18), 2028–2043 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Huang, J., Shi, Y., Zhang, X.: Active fault tolerant control systems by the semi-Markov model approach. Int. J. Adapt. Control Signal Process. 28(9), 765–858 (2014)

    Article  MathSciNet  Google Scholar 

  26. Wang, J., Shen, H.: Passivity-based fault-tolerant synchronization control of chaotic neural networks against actuator faults using the semi-Markov jump model approach. Neurocomputing 143, 51–56 (2014)

    Article  Google Scholar 

  27. Liu, X., Yu, X., Ma, G.Q., Xi, H.: On sliding mode control for networked control systems with semi-Markovian switching and random sensor delays. Inf. Sci. 337, 44–58 (2016)

    Article  Google Scholar 

  28. Li, F.B., Wu, L.G., Shi, P.: Stochastic stability of semi-Markovian jump systems with mode-dependent delays. Int. J. Robust Nonlinear Control 24(18), 3317–3330 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, F.B., Shi, P., Wu, L.G., Basin, M.V., Lim, C.C.: Quantized control design for cognitive radio networks modeled as nonlinear semi-Markovian jump systems. IEEE Trans. Ind. Electron. 62(4), 2330–2340 (2015)

    Article  Google Scholar 

  30. Yin, J., Khoo, S., Man, Z., Yu, X.: Finite-time stability and instability of stochastic nonlinear systems. Automatica 47, 2671–2677 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, L., Xiao, F.: Finite-time consensus problems for networks of dynamic agents. IEEE Trans. Autom. Control 55(4), 950–955 (2010)

    Article  MathSciNet  Google Scholar 

  33. Amato, F., Ariola, M., Dorato, P.: Finite-time stabilzation via dynamic output feedback. Automatica 42, 337–342 (2006)

  34. Zhang, W., An, X.Y.: Finite-time control of linear stochastic systems. Int. J. Innov. Comput. Inf. Control 4(3), 689–696 (2008)

    Google Scholar 

  35. Li, H.Y., Zhou, Q., Chen, B., Liu, H.H.: Parameter-dependent robust stability for uncertain Markovian jump systems with time delay. J. Frankl. Inst. 348(4), 738–748 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, H., Wang, J.M., Shi, Y.: Robust \(H_{\infty }\) sliding-mode control for Markovian jump systems subject to intermittent observations and partially known transition probabilities. Syst. Control Lett. 62(12), 1114–1124 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang, Y., Zhu, H., Zhong, S.: Robust non-fragile \(H_{\infty }\) control for a class of switched neutral systems. In: The 2nd IEEE Conference on Industrial Electronics and Applications, Harbin, China, pp. 1003–1008 (2007)

  38. Zhang, Y.S., Xu, S.Y., Zhang, J.H.: Delay-dependent robust \(H_{\infty }\) control for uncertain fuzzy Markovian jump systems. Int. J. Control Autom. Syst. 7(4), 520–529 (2009)

    Article  Google Scholar 

  39. Liu, X., Ma, G.Q., Jiang, X.F., Xi, H.: \(H_{\infty }\) stochastic synchronization for master-slave semi-Markovian switching system via sliding mode control. Complexity (2015). doi:10.1002/cplx.21702

  40. He, S.P., Liu, F.: Stochastic finite-time stabilization for uncertain jump systems via state feedback. J. Dyn. Syst. Meas. Control 132(3), 4, Article ID 034504 (2010)

  41. Xia, Y., Shi, P., Liu, G., Rees, D.: Robust mixed \(H_{\infty }/H_{2}\) state-feedback control for continuous-time descriptor systems with parameter uncertainties. Circuits Syst. Signal Process. 24(4), 431–443 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Meng, Q.Y., Shen, Y.J.: Finite-time \(H_{\infty }\) control for linear continuous system with norm-bounded disturbance. Commun. Nonlinear Sci. Numer. Simul. 14, 1043–1049 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. He, S.P., Liu, F.: Robust finite-time \(H_{\infty \infty }\) control of stochastic jump systems. Int. J. Control Autom. Syst. 8(9), 1336–1341 (2010)

    Article  Google Scholar 

  44. Luan, X.L., Liu, F., Shi, P.: Neural-network-based finite-time \(H_{\infty }\) control for extended Markov jump nonlinear systems. Int. J. Adapt. Control Signal Process. 24(7), 554–567 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Amato, F., Ariola, M., Dorato, P.: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001)

    Article  MATH  Google Scholar 

  46. Amato, F., Ariola, M.: Finite-time control of discrete-time linear systems. IEEE Trans. Autom. Control 50(5), 724–729 (2005)

    Article  MathSciNet  Google Scholar 

  47. Svishchuk, A.: Random Evolutions and Their Applications: New Trends. Springer, Berlin (2000)

    Book  Google Scholar 

  48. Xiong, J., Lam, J.: Robust \(H_{2}\) control of Markovian jump systems with uncertain switching probabilities. Int. J. Syst. Sci. 40(3), 255–265 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. Gao, S.G., Dong, H.R., Ning, B., Sun, X.: Neural adaptive control for uncertain MIMO systems with constrained input via intercepted adaptation and single learning parameter approach. Nonlinear Dyn. 82(3), 1109–1126 (2015)

    Article  MathSciNet  Google Scholar 

  50. Gao, S.G., Dong, H.R., Lyu, S., Ning, B.: Truncated adaptation design for decentralised neural dynamic surface control of interconnected nonlinear systems under input saturation. Int. J. Control (2016). doi:10.1080/00207179.2015.1135507

  51. Siraj, M., Alshebeili, S.: Performance enhancement in multi hop cognitive radio wireless mesh networks. Int. J. Innov. Comput. Inf. Control 9(10), 3929–3939 (2013)

    Google Scholar 

  52. Shah, G.A., Gungor, V.C., Akan, O.B.: A cross-layer QoS-aware communication framwork in cognitive radio sensor networks for smart grid applications. IEEE Trans. Ind. Inform. 9(3), 1477–1485 (2013)

    Article  Google Scholar 

  53. Ma, X., Djouadi, S.M., Li, H.: State estimation over a semi-Markov model based cognitive radio system. IEEE Trans. Wirel. Commun. 22(7), 2391–2401 (2012)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Auto, School of Information Science and Technology, University of Science and Technology of China, Hefei, 230027, Anhui, China

    Xinghua Liu & Hongsheng Xi

  2. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore

    Xinghua Liu

  3. Platform Technologies Research Institute, RMIT University, Melbourne, VIC, 3001, Australia

    Xinghuo Yu

  4. School of Information Science and Engineering, Central South University, Changsha, 410083, Hunan, China

    Xiaojun Zhou

Authors
  1. Xinghua Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinghuo Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaojun Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongsheng Xi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinghua Liu.

Additional information

This work was supported in part by the National Key Scientific Research Project (61233003), China Postdoctoral Science Foundation (2015M580549) and the Fundamental Research Funds for the Central Universities.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Yu, X., Zhou, X. et al. Finite-time \(\varvec{H_{\infty }}\) control for linear systems with semi-Markovian switching. Nonlinear Dyn 85, 2297–2308 (2016). https://doi.org/10.1007/s11071-016-2829-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2829-7

Keywords

Search

Navigation