International Journal of Fuzzy Systems

, Volume 20, Issue 3, pp 759–768 | Cite as

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

  • Yan Li
  • Xikui LiuEmail author


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.


\(\mathscr {H}_-\) index Nonlinear stochastic systems Fuzzy approach 



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).


  1. 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)CrossRefGoogle Scholar
  2. 2.
    Pattor, R.: Robustness in model-based fault diagnosis: the 1995 situation. Annu. Rev. Control 21, 103–123 (1997)CrossRefGoogle Scholar
  3. 3.
    Chen, J., Pattor, R.: Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer, Boston (1999)CrossRefGoogle Scholar
  4. 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)CrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)Google Scholar
  7. 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)CrossRefzbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, X., Zhou, K.: A time domain approach to robust fault detection of linear time-varying systems. Automatica 45(1), 94–102 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhong, M., Ding, S., Ding, E.: Optimal fault detection for linear discrete time-varying systems. Automatica 46(8), 1395–1400 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, X., Liu, H.H.T.: Characterization of \({H}_{-}\) index for linear time-varying systems. Automatica 49(3), 1449–1457 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)MathSciNetGoogle Scholar
  14. 14.
    Liu, N., Zhou, K.: Optimal robust fault detection for linear discrete time systems. J. Control Sci. Eng. 2008, 1–16 (2008)CrossRefGoogle Scholar
  15. 15.
    Li, X., Liu, H.H.T., Jiang, B.: Parametrization of optimal fault detection filters. Automatica 56(C), 70–77 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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)CrossRefzbMATHGoogle Scholar
  20. 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)MathSciNetzbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Persis, C., Isidori, A.: A geometric approach to nonlinear fault detection and isolation. IEEE Trans. Automat. Contr. 46(6), 853–864 (2006)MathSciNetzbMATHGoogle Scholar
  23. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yang, Y., Ding, S., Li, L.: Parameterization of nonlinear observer-based fault detection systems. IEEE Trans. Automat. Contr. 61(11), 3687–3692 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, W., Zhao, Y., Sheng, L.: Some remarks on stability of stochastic singular systems with state-dependent noise. Automatica 51, 273–277 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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. 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)CrossRefzbMATHGoogle Scholar
  28. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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)CrossRefzbMATHGoogle Scholar
  30. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Li, Y., Zhang, W., Liu, X.: Stability of nonlinear stochastic discrete-time systems. J. Appl. Math. 1–8, 2013 (2013)MathSciNetGoogle Scholar
  32. 32.
    Wu, Z.: Stability criteria of random nonlinear systems and their applications. IEEE Trans. Automat. Contr. 60(4), 1038–1049 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 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)MathSciNetzbMATHGoogle Scholar
  37. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 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. 39.
    Zhou, S., Zhang, W.: Discrete-time indefinite stochastic LQ control via SDP and LMI methods. J. Appl. Math. 2012(4), 1–7 (2012)MathSciNetzbMATHGoogle Scholar
  40. 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)MathSciNetGoogle Scholar
  41. 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)MathSciNetGoogle Scholar
  42. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 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)CrossRefGoogle Scholar
  44. 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)MathSciNetCrossRefGoogle Scholar
  45. 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)CrossRefGoogle Scholar
  46. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Lin, X., Zhang, R.: \({H}_\infty\) control for stochastic systems with Poisson jumps. J. Syst. Sci. Complex. 24(4), 683–700 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A LMI Approach. Wiley, New York (2001)CrossRefGoogle Scholar
  51. 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)CrossRefGoogle Scholar
  52. 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)CrossRefGoogle Scholar
  53. 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)Google Scholar
  54. 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)MathSciNetGoogle Scholar
  55. 55.
    Mao, X.: Stochastic Differential Equations and Applications. Horwood, England (1997)zbMATHGoogle Scholar
  56. 56.
    Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.College of Electrical Engineering and AutomationShandong University of Science and TechnologyQingdaoChina
  2. 2.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  3. 3.Shandong Province Key Laboratory of Wisdom Mine Information TechnologyShandong University of Science and TechnologyQingdaoChina

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