Skip to main content

Advertisement

SpringerLink
Log in

\({\mathscr{H}}_-\) Index for Nonlinear Stochastic Systems with State- and Input-Dependent Noises

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

This brief focuses on the \(\mathscr {H}_-\) index problem for nonlinear systems involving state- and input-dependent noises in finite and infinite horizon. A sufficient condition of the \(\mathscr {H}_-\) index in finite/infinite case is developed for such systems based on Hamilton–Jacobi equations/inequalities (HJEs/HJIs). Generally, one can hardly solve these HJEs/HJIs. By fuzzy approach, the characterization of \(\mathscr {H}_{-}\) index is derived via solving a linear matrix inequality (LMI). Finally, an example verifies the effect of the obtained 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. Frank, P., Ding, X.: Survey of robust residual generation and evaluation methods in observer-based fault detection systems. J. Process Control 7(6), 403–424 (1997)

    Article  Google Scholar 

  2. Pattor, R.: Robustness in model-based fault diagnosis: the 1995 situation. Annu. Rev. Control 21, 103–123 (1997)

    Article  Google Scholar 

  3. Chen, J., Pattor, R.: Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer, Boston (1999)

    Book  Google Scholar 

  4. Ding, S., Jeinsch, T., Frank, P., Ding, E.: A unified approach to the optimization of fault detection systems. Int. J. Adapt. Control Signal Process. 14(7), 725–745 (2000)

    Article  MATH  Google Scholar 

  5. Jaimoukha, I., Li, Z., Papakos, V.: A matrix factorization solution to the \({H}_{-}/{H}_{\infty }\) fault detection problem. Automatica 42(11), 1907–1912 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hou, M., Pattor, R.: An LMI approach to \({H}_{-}/{H}_{\infty }\) fault detection observers. In: Proceedings of UKACC International Conference on Control, pp. 305–310 (1996)

  7. Wang, J., Yang, G., Liu, J.: An LMI approach to \({H}_{-}\)index and mix \({H}_{-}/{H}_{\infty }\) fault detection observer design. Automatica 43, 1656–1665 (2007)

    Article  MATH  Google Scholar 

  8. Iwasaki, T., Hara, S., Yamauchi, H.: Dynamical systems design from a control perspective: finite frequency positive-realness approach. IEEE Trans. Automat. Contr. 48(8), 1337–1354 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, J., Wang, J., Yang, G.: An LMI approach to minimum sensitivity analysis with application to fault detection. Automatica 41(11), 1995–2004 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, X., Zhou, K.: A time domain approach to robust fault detection of linear time-varying systems. Automatica 45(1), 94–102 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhong, M., Ding, S., Ding, E.: Optimal fault detection for linear discrete time-varying systems. Automatica 46(8), 1395–1400 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, X., Liu, H.H.T.: Characterization of \({H}_{-}\) index for linear time-varying systems. Automatica 49(3), 1449–1457 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, X., Liu, H.H.T.: Minimum system sensitivity study of linear discrete time systems for fault detection. Math. Probl. Eng. 1–13, 2013 (2013)

    MathSciNet  Google Scholar 

  14. Liu, N., Zhou, K.: Optimal robust fault detection for linear discrete time systems. J. Control Sci. Eng. 2008, 1–16 (2008)

    Article  Google Scholar 

  15. Li, X., Liu, H.H.T., Jiang, B.: Parametrization of optimal fault detection filters. Automatica 56(C), 70–77 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Khan, A., Abid, M., Ding, S.: Fault detection filter design for discrete-time nonlinear systems-A mixed \({H}_{-}/{H}_{\infty }\) optimization. Syst. Control Lett. 67, 46–54 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Saravanakumar, R., Ali, M.: Robust \({H}_{\infty }\) control for uncertain Markovian jump systems with mixed delays. Chin. Phys. B 25(7), 108–113 (2016)

    Article  Google Scholar 

  18. Fu, J., Wang, J., Li, Z.: Leader-following control of perturbed second-order integrator systems with binary relative information. Int. J. Syst. Sci. 48(3), 485–493 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Long, L., Zhao, J.: \({H}_{\infty }\) control of switched nonlinear systems in \(p-\)normal form using multiple Lyapunov functions. IEEE Trans. Automat. Contr. 57(5), 1285–1291 (2012)

    Article  MATH  Google Scholar 

  20. Niu, B., Zhao, J.: Robust \({H}_{\infty }\) control for a class of switched nonlinear cascade systems via multiple Lyapunov functions approach. Appl. Math. Comput. 218, 6330–6339 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Niu, B., Zhao, J.: Robust \({H}_{\infty }\) control for a class of uncertain nonlinear switched systems with average dwell time. Int. J. Control 86(6), 1107–1117 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Persis, C., Isidori, A.: A geometric approach to nonlinear fault detection and isolation. IEEE Trans. Automat. Contr. 46(6), 853–864 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Demetriou, M., Armaou, A.: Dynamic online robust detection and accommodation of incipient component faults for nonlinear dissipative distributed processes. Int. J. Robust Nonlinear Control 22, 3–23 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, Y., Ding, S., Li, L.: Parameterization of nonlinear observer-based fault detection systems. IEEE Trans. Automat. Contr. 61(11), 3687–3692 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, W., Zhao, Y., Sheng, L.: Some remarks on stability of stochastic singular systems with state-dependent noise. Automatica 51, 273–277 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sun, M., Zhang, W., Li, G.: Stochastic admissibility of continuous-time singular Markov jump systems with general uncertain transition rates. J. Shandong Univ. Sci. Technol. 35(4), 86–92 (2016)

    Google Scholar 

  27. Saravanakumar, R., Ali, M., Hua, M.: \({H}_{\infty }\) state estimation of stochastic neural networks with mixed time-varying delays. Soft Comput. 20, 3475–3487 (2016)

    Article  MATH  Google Scholar 

  28. Hu, J., Wang, Z., Shen, B., Gao, H.: Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurement. Int. J. Control 86(4), 650–663 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hu, J., Wang, Z., Liu, S., Gao, H.: A variance-constrained approach to recursive state estimation for time-varying complex networks with missing measurement. Automatica 64, 155–162 (2016)

    Article  MATH  Google Scholar 

  30. Saravanakumar, R., Ali, M., Cao, J., Huang, H.: \({H}_{\infty }\) state estimation of generalised neural networks with interval time-varying delays. Int. J. Syst. Sci. 47(16), 3888–3899 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, Y., Zhang, W., Liu, X.: Stability of nonlinear stochastic discrete-time systems. J. Appl. Math. 1–8, 2013 (2013)

    MathSciNet  Google Scholar 

  32. Wu, Z.: Stability criteria of random nonlinear systems and their applications. IEEE Trans. Automat. Contr. 60(4), 1038–1049 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, W., Lin, X., Chen, B.S.: LaSalle-type theorem and its applications to infinite horizon optimal control of discrete-time nonlinear stochastic systems. IEEE Trans. Automat. Contr. 62(1), 250–261 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, Q., Zhang, W.: Properties of storage functions and applications to nonlinear stochastic \({H}_\infty\) control. J. Syst. Sci. Complex. 24(5), 850–861 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, W., Chen, B.S.: \({H}\)-representation and applications to generalized Lyapunov equations and linear stochastic systems. IEEE Trans. Automat. Contr. 57(12), 3009–3022 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu, X., Li, Y., Zhang, W.: Stochastic linear quadratic optimal control with constraint for discrete-time systems. Appl. Math. Comput. 228(2), 264–270 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Lin, X., Zhang, W.: A maximum principle for optimal control of discrete-time stochastic systems with multiplicative noise. IEEE Trans. Automat. Contr. 60(4), 1121–1126 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Jiang, D., Li, Y.: Robust \({H}_{2}/{H}_{\infty }\) control of stochastic control systems with multiplicative noise. J. Shandong Univ. Sci. Technol. 35(3), 92–98 (2016)

    Google Scholar 

  39. Zhou, S., Zhang, W.: Discrete-time indefinite stochastic LQ control via SDP and LMI methods. J. Appl. Math. 2012(4), 1–7 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Li, G., Chen, M.: Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems. Adv. Differ. Equ. 2015(14), 1–12 (2015)

    MathSciNet  Google Scholar 

  41. Gao, M., Sheng, L., Zhang, W.: Stochastic \({H}_2/{H}_\infty\) control of nonlinear systems with time-delay and state-dependent noise. Appl. Math. Comput. 266, 429–440 (2015)

    MathSciNet  Google Scholar 

  42. Shen, M., Ye, D.: Improved fuzzy control design for nonlinear Markovian jump systems with incomplete transition descriptions. Fuzzy Sets Syst. 217, 80–95 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Shen, M., Park, J.H., Ye, D.: A separated approach to control of Markov jump nonlinear systems with general transition probabilities. IEEE Trans. Cybern. 46(9), 2010–2018 (2016)

    Article  Google Scholar 

  44. Shen, M., Ye, D., Zhang, G.: Finite-time \({H}_{\infty }\) static output control of Markov jump systems with an auxiliary approach. Appli. Math. Comput. 273, 553–561 (2016)

    Article  MathSciNet  Google Scholar 

  45. Shen, M., Ye, D., Wang, Q.: Model-dependent filter design for Markov jump systems with sensor nonlinearities in finite frequency domain. Signal Process. 134, 1–8 (2017)

    Article  Google Scholar 

  46. Tan, C., Zhang, W.: On observability and detectability of continuous-time stochastic Markov jump systems. J. Syst. Sci. Complex. 28(4), 830–847 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhao, Y., Zhang, W.: Observer-based controller design for singular stochastic Markov jump systems with state dependent noise. J. Syst. Sci. Complex. 29(4), 946–958 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ni, Y., Zhang, W., Fang, H.: On the observability and detectability of linear stochastic systems with Markov jumps and multiplicative noise. J. Syst. Sci. Complex. 23(1), 102–115 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lin, X., Zhang, R.: \({H}_\infty\) control for stochastic systems with Poisson jumps. J. Syst. Sci. Complex. 24(4), 683–700 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A LMI Approach. Wiley, New York (2001)

    Book  Google Scholar 

  51. Zhang, X., Liu, X., Li, Y.: Adaptive fuzzy tracking control for nonlinear strict-feedback systems with unmodeled dynamics via backstepping technique. Neurocomputing 235, 182–191 (2017)

    Article  Google Scholar 

  52. Chen, B.S., Chang, Y.: Fuzzy state-space modeling and robust observer-based control design for nonlinear. IEEE Trans. Fuzzy Syst. 17(5), 1025–1043 (2009)

    Article  Google Scholar 

  53. Boyd, S., Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics, vol. 15. SIAM, Philadelphia, PA, USA (1994)

  54. Limebeer, D., Anderson, B., Khargonekar, P., Francis, B.: State-space solutions to standard \({H}_{2}\) and \({H}_{\infty }\) control problems. IEEE Trans. Automat. Contr. 46(6), 831–847 (1989)

    MathSciNet  Google Scholar 

  55. Mao, X.: Stochastic Differential Equations and Applications. Horwood, England (1997)

    MATH  Google Scholar 

  56. Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (61402265); the SDUST Research Fund (2015TDJH105); the Fund for Postdoctoral Application Research Project of Qingdao (01020120607).

Author information

Authors and Affiliations

  1. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, China

    Yan Li

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Xikui Liu

  3. Shandong Province Key Laboratory of Wisdom Mine Information Technology, Shandong University of Science and Technology, Qingdao, 266590, China

    Xikui Liu

Authors
  1. Yan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xikui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xikui Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Liu, X. \({\mathscr{H}}_-\) Index for Nonlinear Stochastic Systems with State- and Input-Dependent Noises. Int. J. Fuzzy Syst. 20, 759–768 (2018). https://doi.org/10.1007/s40815-017-0356-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-017-0356-4

Keywords

Search

Navigation