Skip to main content
SpringerLink
Log in

H∞ Asynchronous Deconvolution Fuzzy Filter Design for Nonlinear Singular Markov Jump Systems with Time-Varying Delays

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

Abstract

This thesis focuses on the asynchronous deconvolution fuzzy filtering for time-varying delays Lur’e singular Markov jump fuzzy systems via the Takagi–Sugeno fuzzy control technique. The stochastic admissibility and \(H_\infty\) performance index of Lur’e singular Markov jump fuzzy systems are obtained by establishing mode-dependent Lyapunov-Krasovskill functional. In addition, the asynchrony phenomenon between the original system modes and filter modes is described by a hidden Markov model. The parallel distribute compensation approach is used to design an asynchronous deconvolution fuzzy filter, and the desired filter parameters are achieved via solving linear matrix inequalities. Lastly, the validity and practicality of the proposed method are shown by a numerical example and a single-link robot arm.

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data availability statement

The data used to support the findings of this study are available from the corresponding author upon request.

References

  1. Feng, Z., Shi, P.: Sliding mode control of singular stochastic Markov jump systems. IEEE Trans. Autom. Control 62(8), 4266–4273 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Dai, L.: Singular Control Systems. Springer, Berlin (1989)

    MATH  Google Scholar 

  3. Wang, J., Liang, K., Huang, X., Wang, Z., Shen, H.: Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback. Appl. Math. Comput. 328, 247–262 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Liu, G., Park, J.H., Xu, S., Zhuang, G.: Robust non-fragile H∞ fault detection filter design for delayed singular Markovian jump systems with linear fractional parametric uncertainties. Nonlinear Anal. Hybrid Syst. 32, 65–78 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Zhuang, G., Xu, S., Zhang, B., Xu, H., Chu, Y.: Robust H∞ deconvolution filtering for uncertain singular Markovian jump systems with time-varying delays. Int. J. Robust Nonlinear Control. 26(12), 2564–2585 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Shu, Y., Zhu, Y.: Optimistic value based optimal control for uncertain linear singular systems and application to a dynamic input-output model. ISA Trans. 71(2), 235–251 (2017)

    Google Scholar 

  7. Park, I.S., Kwon, N.K., Park, P.: Dynamic output-feedback control for singular Markovian jump systems with partly unknown transition rates. Nonlinear Dyn. 95, 3149–3160 (2019)

    MATH  Google Scholar 

  8. Ma, Q., Xu, S., Lewis, F.L., Zhang, B., Zou, Y.: Cooperative output regulation of singular heterogeneous multiagent systems. IEEE Trans. Cybern. 46(6), 1471–1475 (2016)

    Google Scholar 

  9. Zhuang, G., Xia, J., Feng, J., Sun, W., Zhang, B.: Admissibilization for implicit jump systems with mixed retarded delays based on reciprocally convex integral inequality and Barbalat’s lemma. IEEE Trans. Syst. Man Cybern. Syst. 51(11), 6808–6818 (2021)

    Google Scholar 

  10. Sakthivel, R., Suveetha, V.T., Divya, H., Sakthivel, R.: Fault detection finite-time filter design for T-S fuzzy Markovian jump system with missing measurements. Circ. Syst. Signal Process. 40, 1607–1634 (2021)

    Google Scholar 

  11. Shen, H., Li, F., Xu, S., Sreeram, V.: Slow state variables feedback stabilization for semi-Markov jump systems with singular perturbations. IEEE Trans. Autom. Control 63(8), 2709–2714 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Qi, W., Zong, G., Karim, H.R.: Observer-based adaptive SMC for nonlinear uncertain singular semi-Markov jump systems with applications to DC motor. IEEE Trans. Circ. Syst. I 65(9), 2951–2960 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Zhu, Y., Zhang, L., Zheng, W.: Distributed \({{\cal{H} }}_{\infty }\) filtering for a class of discrete-time Markov jump Lur’e systems with redundant channels. IEEE Trans. Ind. Electron. 63(3), 1876–1885 (2016)

    Google Scholar 

  14. Zhang, J., Raïssi, T., Li, S.: Non-fragile saturation control of nonlinear positive Markov jump systems with time-varying delays. Nonlinear Dyn. 97, 1495–1513 (2019)

    MATH  Google Scholar 

  15. Li, F., Xu, S., Shen, H., Ma, Q.: Passivity-based control for hidden Markov jump systems with singular perturbations and partially unknown probabilities. IEEE Trans. Autom. Control 65(8), 3701–3706 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Allen, I.E.: Semi-Markov Processes: applications in system reliability and maintenance. Qual. Prog. 50(2), 60 (2017)

    Google Scholar 

  17. Wang, B., Zhu, Q.: Stability analysis of semi-Markov switched stochastic systems. Automatica 94, 72–80 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Wang, Y., Zhuang, G., Chen, X., Wang, Z., Chen, F.: Dynamic event-based finite-time mixed H∞ and passive asynchronous filtering for T-S fuzzy singular Markov jump systems with general transition rates. Nonlinear Anal. Hybrid Syst. 36, 100874 (2020)

    MathSciNet  MATH  Google Scholar 

  19. Tao, J., Wu, Z., Su, H., Wu, Y., Zhang, D.: Asynchronous and resilient filtering for Markovian jump neural networks subject to extended dissipativity. IEEE Trans. Cybern. 49(7), 2504–2513 (2019)

    Google Scholar 

  20. Ma, H., Wang, Y.: Full information H2 control of Borel-measurable Markov jump systems with multiplicative noises. Mathematics 10(1), 37 (2022)

    Google Scholar 

  21. Zhuang, G., Xia, J., Ma, Q., Sun, W., Wang, Y.: Event-triggered H∞ feedback control for delayed singular jump systems based on sampled observer and exponential detector. Int. J. Robust Nonlinear Control 31(15), 7298–7316 (2021)

    MathSciNet  Google Scholar 

  22. Li, F., Du, C., Yang, C., Gui, W.: Passivity-based asynchronous sliding mode control for delayed singular Markovian jump systems. IEEE Trans. Autom. Control 63(8), 2715–2721 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Wang, Y., Xia, Y., Shen, H., Zhou, P.: SMC design for robust stabilization of nonlinear Markovian jump singular systems. IEEE Trans. Autom. Control 63(1), 219–224 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Jiang, B., Kao, Y., Karimi, H.R., Gao, C.: Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates. IEEE Trans. Autom. Control 63(11), 3919–3926 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Zhuang, G., Xia, J., Feng, J., Zhang, B., Lu, J., Wang, Z.: Admissibility analysis and stabilization for neutral descriptor hybrid systems with time-varying delays. Nonlinear Anal. Hybrid Syst. 33, 311–321 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Zhuang, G., Sun, W., Su, S., Xia, J.: Asynchronous feedback control for delayed fuzzy degenerate jump systems under observer-based event-driven characteristic. IEEE Trans. Fuzzy Syst. 29(12), 3754–3768 (2021)

    Google Scholar 

  27. Ahn, C.K., Shi, P., Wu, L.: Receding horizon stabilization and disturbance attenuation for neural networks with time-varying delay. IEEE Trans. Cybern. 45(12), 2680–2692 (2015)

    Google Scholar 

  28. Niu, B., Ahn, C.K., Li, H., Liu, M.: Adaptive control for stochastic switched nonlower triangular nonlinear systems and its application to a one-link manipulator. IEEE Trans. Syst. Man Cybern. Syst. 48(10), 1701–1714 (2018)

    Google Scholar 

  29. Sakthivel, R., Suveetha, V.T., Nithya, V., Sakthivel, R.: Finite-time reliable filtering for Takagi-Sugeno fuzzy semi-Markovian jump systems. Math. Comput. Simul 185, 403–418 (2021)

    MathSciNet  MATH  Google Scholar 

  30. Gu, Z., Ahn, C.K., Yue, D., Xie, X.: Event-triggered H∞ filtering for T-S fuzzy-model-based nonlinear networked systems with multisensors against DoS attacks. IEEE Trans. Cybern. 52(6), 5311–5321 (2022)

    Google Scholar 

  31. Lee, T.H., Trinh, H.M., Park, J.H.: Stability analysis of neural networks with time-varying delay by constructing novel Lyapunov functionals. IEEE Trans. Neural Netw. Learn. Syst. 29(9), 4238–4247 (2018)

    Google Scholar 

  32. Wu, H., Cai, K.: Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control. IEEE Trans. Syst. Man Cybern. Part B 36(3), 509–519 (2006)

    Google Scholar 

  33. Kwon, N.K., Park, I.S., Park, P.: \({\cal{H} }_\infty\) state-feedback control for continuous-time Markovian jump fuzzy systems using a fuzzy weighting-dependent Lyapunov function. Nonlinear Dyn. 90, 2001–2011 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Shi, P., Zhang, Y., Chadli, M., Agarwal, R.K.: Mixed H∞ and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays. IEEE Trans. Neural Netw. Learn. Syst. 27(4), 903–909 (2016)

    MathSciNet  Google Scholar 

  35. Nie, R., He, S., Liu, F., Luan, X., Shen, H.: HMM-based asynchronous controller design of Markovian jumping Lur’e systems within a finite-time interval. IEEE Trans. Syst. Man Cybern. Syst. 51(11), 6885–6891 (2021)

    Google Scholar 

  36. Wang, R., Lin, W., Wang, W.: Stabilizability of linear quadratic state feedback for uncertain fuzzy time-delay systems. IEEE Trans. Syst. Man Cybern. Part B 34(2), 1288–1292 (2004)

    Google Scholar 

  37. Chang, J., Wang, R., Wang, W., Huang, C.: Implementation of an object-grasping robot arm using stereo vision measurement and fuzzy control. Int. J. Fuzzy Syst. 17, 193–205 (2015)

    Google Scholar 

  38. Wang, R.: Nonlinear decentralized state feedback controller for uncertain fuzzy time-delay interconnected systems. Fuzzy Sets Syst. 151(1), 191–204 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Wu, Z., Dong, S., Su, H., Li, C.: Asynchronous dissipative control for fuzzy Markov jump systems. IEEE Trans. Cybern. 48(8), 2426–2436 (2018)

    Google Scholar 

  40. Yuan, W., Sun, W., Liu, Z., Zhang, F.: Adaptive fuzzy tracking control of stochastic mechanical system with input saturation. Int. J. Fuzzy Syst. 21, 2600–2608 (2019)

    MathSciNet  Google Scholar 

  41. Li, B., Xia, J., Zhang, H., Shen, H., Wang, Z.: Event-triggered adaptive fuzzy tracking control for nonlinear systems. Int. J. Fuzzy Syst. 22, 1389–1399 (2020)

    MATH  Google Scholar 

  42. Wang, X., Park, J.H., Yang, H., Zhong, S.: An improved fuzzy event-triggered asynchronous dissipative control to T-S MJSs with nonperiodic sampled data. IEEE Trans. Fuzzy Syst. 29(10), 2926–2937 (2021)

    Google Scholar 

  43. Park, P., Ko, J.W., Jeong, C.: Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1), 235–238 (2011)

    MathSciNet  MATH  Google Scholar 

  44. Li, M., Chen, Y., Xu, L., Chen, Z.: Asynchronous control strategy for semi-Markov switched system and its application. Inf. Sci. 532, 125–138 (2020)

    MathSciNet  MATH  Google Scholar 

  45. Zhuang, G., Su, S., Xia, J., Sun, W.: HMM-based asynchronous H∞ filtering for fuzzy singular Markovian switching systems with retarded time-varying delays. IEEE Trans. Cybern. 51(3), 1189–1203 (2021)

    Google Scholar 

  46. Shen, Y., Wu, Z., Shi, P., Shu, Z., Karimi, H.R.: H∞ control of Markov jump time-delay systems under asynchronous controller and quantizer. Automatica 99, 352–360 (2019)

    MathSciNet  MATH  Google Scholar 

  47. Zhuang, G., Xia, J., Sun, W., Feng, J., Ma, Q.: Asynchronous admissibility and H∞ fault detection for delayed implicit Markovian switching systems under hidden Markovian model mechanism. Int. J. Robust Nonlinear Control 31(15), 7261–7279 (2021)

    MathSciNet  Google Scholar 

  48. Lin, Y., Zhuang, G., Xia, J., Chen, G., Zhang, H., Liu, G.: Asynchronous dynamic output feedback control for delayed fuzzy stochastic Markov jump systems based on HMM strategy. Int. J. Fuzzy Syst. (2022). https://doi.org/10.1007/s40815-022-01273-4

    Article  Google Scholar 

  49. Suveetha, V.T., Sakthivel, R., Nithya, V., Sakthivel, R.: Finite-time fault detection filter design for T-S fuzzy Markovian jump systems with distributed delays and incomplete measurements. Circ. Syst. Signal Process. 41, 28–56 (2022)

    MATH  Google Scholar 

  50. Zhang, H., Hong, Q., Yan, H., Yang, F., Guo, G.: Event-based distributed H∞ filtering networks of 2-DOF quarter-car suspension systems. IEEE Trans. Ind. Inf. 13(1), 312–321 (2017)

    Google Scholar 

  51. Gu, Z., Shi, P., Yue, D., Ding, Z.: Decentralized adaptive event-triggered H∞ filtering for a class of networked nonlinear interconnected systems. IEEE Trans. Cybern. 49(5), 1570–1579 (2019)

    Google Scholar 

  52. Zhu, Y., Zhong, Z., Zheng, W., Zhou, D.: HMM-based \({mathcal H }_{\infty }\) filtering for discrete-time Markov jump LPV systems over unreliable communication channels. IEEE Trans. Syst. Man Cybern. Syst. 48(12), 2035–2046 (2018)

    Google Scholar 

  53. Liu, H., Gao, L., Wang, Z., Liu, Z.: Asynchronous \(L_2-L_\infty\) filtering of discrete-time impulsive switched systems with admissible edge-dependent average dwell time switching signal. Int. J. Syst. Sci. 52(8), 1564–1585 (2021)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Editors and the Referees for the important comments and valuable suggestions for improving the paper. This work was supported by National Natural Science Foundation of China under Grants 62173174, 61773191, 62173183, 61973148; Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions under Grant 2019KJI010; Natural Science Foundation of Shandong Province for Key Projects under Grant ZR2020KA010; Graduate education high-quality curriculum construction project for Shandong Province under Grant SDYKC20185.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Liaocheng University, Liaocheng, 252000, People’s Republic of China

    Yuexia Yin, Guangming Zhuang, Jianwei Xia & Wei Sun

  2. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Qian Ma

Authors
  1. Yuexia Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guangming Zhuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianwei Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qian Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Wei Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangming Zhuang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix

Appendix

Conditions of Theorem 2

$$\begin{aligned}&\hat{\varXi }_{iir\eta 11}=\left[ \begin{array}{cccc} \varXi _{iir\eta 11}&\,\varXi _{iir\eta 12}&\,\varXi _{iir\eta 13}&\,\varXi _{iir\eta 14}\\ *&\,\varXi _{iir\eta 22}&\,\varXi _{iir\eta 23}&\,\varXi _{iir\eta 24}\\ *&\,*&\,\varXi _{iir\eta 33}&\,\varXi _{iir\eta 34}\\ *&\,*&\,*&\,\varXi _{iir\eta 44} \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 12}=\left[ \begin{array}{cccc} \varXi _{iir\eta 15}&\,\varXi _{iir\eta 16}&\,\varXi _{iir\eta 17}\\ \varXi _{iir\eta 25}&\,\varXi _{iir\eta 26}&\,\varXi _{iir\eta 27}\\ \varXi _{iir\eta 35}&\,0&\,\varXi _{iir\eta 37}\\ \varXi _{iir\eta 45}&\,0&\,\varXi _{iir\eta 47} \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 13}=\left[ \begin{array}{cccc} \varXi _{iir\eta 18}&\,\varXi _{iir\eta 19}&\,\varXi _{iir\eta 110}\\ \varXi _{iir\eta 28}&\,\varXi _{iir\eta 29}&\,\varXi _{iir\eta 210}\\ \varXi _{iir\eta 38}&\,\varXi _{iir\eta 39}&\,0\\ \varXi _{iir\eta 48}&\,\varXi _{iir\eta 49}&\,0 \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 22}=\left[ \begin{array}{cccc} \varXi _{iir\eta 55}&\,\varXi _{iir\eta 56}&\,0\\ *&\,\varXi _{iir\eta 66}&\,0\\ *&\,*&\,-2{\breve{\varLambda }}\\ \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 33}=\left[ \begin{array}{cccc} -\chi I&\,0&\,\varXi _{iir\eta 810}\\ *&\,-\chi I&\,\varXi _{iir\eta 910}\\ *&\,*&\,-I\\ \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 33}=\left[ \begin{array}{cccc} -2\chi I&\,0&\,\varXi _{ijr\eta 810}\\ *&\,-2\chi I&\,\varXi _{ijr\eta 910}\\ *&\,*&\,-2I\\ \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 12}=\left[ \begin{array}{cccc} \tilde{\varXi }_{ijr\eta 15}&\,\tilde{\varXi }_{ijr\eta 16}&\,\tilde{\varXi }_{ijr\eta 17}\\ \tilde{\varXi }_{ijr\eta 25}&\,\tilde{\varXi }_{ijr\eta 26}&\,\tilde{\varXi }_{ijr\eta 27}\\ \tilde{\varXi }_{ijr\eta 35}&\,0&\,\tilde{\varXi }_{ijr\eta 37}\\ \tilde{\varXi }_{ijr\eta 45}&\,0&\,\tilde{\varXi }_{ijr\eta 47} \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 13}=\left[ \begin{array}{cccc} \tilde{\varXi }_{ijr\eta 18}&\,\tilde{\varXi }_{ijr\eta 19}&\,\tilde{\varXi }_{ijr\eta 110}\\ \tilde{\varXi }_{ijr\eta 28}&\,\tilde{\varXi }_{ijr\eta 29}&\,\tilde{\varXi }_{ijr\eta 210}\\ \tilde{\varXi }_{ijr\eta 38}&\,\tilde{\varXi }_{ijr\eta 39}&\,0\\ \tilde{\varXi }_{ijr\eta 48}&\,\tilde{\varXi }_{ijr\eta 49}&\,0 \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 22}=\left[ \begin{array}{cccc} \tilde{\varXi }_{ijr\eta 55}&\,\tilde{\varXi }_{ijr\eta 56}&\,0\\ *&\,\tilde{\varXi }_{ijr\eta 66}&\,0\\ *&\,*&\,-4{\breve{\varLambda }}\\ \end{array}\right] ,\\&\check{\varXi }_{ijr\eta 11}=\left[ \begin{array}{cccc} \tilde{\varXi }_{ijr\eta 11}&\,\tilde{\varXi }_{ijr\eta 12}&\,\tilde{\varXi }_{ijr\eta 13}&\,\tilde{\varXi }_{ijr\eta 14}\\ *&\,\tilde{\varXi }_{ijr\eta 22}&\,\tilde{\varXi }_{ijr\eta 23}&\,\tilde{\varXi }_{ijr\eta 24}\\ *&\,*&\,\tilde{\varXi }_{ijr\eta 33}&\,\tilde{\varXi }_{ijr\eta 34}\\ *&\,*&\,*&\,\varXi _{ijr\eta 44} \end{array}\right] ,\\&\hat{\varXi }_{iir\eta 23}=\left[ \begin{array}{cccc} 0&\,0&\,\varXi _{iir\eta 510}\\ *&\,0&\,0\\ *&\,*&\,0\\ \end{array}\right] , \check{\varXi }_{iir\eta 23}=\left[ \begin{array}{cccc} 0&\,0&\,\tilde{\varXi }_{ijr\eta 510}\\ *&\,0&\,0\\ *&\,*&\,0\\ \end{array}\right] , \\&\hat{\varTheta }_{iir\eta 11}=sym(Y_{r1}A_{ir}+{\mathcal {B}}_{fi\eta }C_{ir})+\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s1}E\\ \,&-E^{T}Z_{11}E, \varXi _{iir\eta 26}=E^{T}_{f}Z_{22}E_{f},\\&\hat{\varTheta }_{iir\eta 12}={\mathcal {A}}_{fi\eta }+A^{T}_{ir}Y^{T}_{r2}+C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta }+\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s2}E_{f}\\ \,&-E^{T}Z_{12}E_{f}, \hat{\varTheta }_{iir\eta 24}=E^{T}_{f}P^{T}_{r3}+U_{r4}R^{T}_{2}-Y+{\mathcal {A}}^{T}_{fi\eta },\\&\hat{\varTheta }_{iir\eta 13}=E^{T}P^{T}_{r1}+U_{r1}R^{T}_{1}-Y_{r1}+A^{T}_{ir}Y^{T}_{r3}+C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta }, \\&\hat{\varTheta }_{iir\eta 14}=E^{T}P_{r2}+U_{r2}R^{T}_{2}-Y+A^{T}_{ir}Y^{T}_{r4}+C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta },\\&\hat{\varTheta }_{iir\eta 22}=sym({\mathcal {A}}_{fi\eta })+\sum \limits _{s=1}^N\lambda _{rs}E^{T}_{f}P_{s3}E_{f}-E_{f}^{T}Z_{22}E_{f},\\&\hat{\varTheta }_{iir\eta 23}=E^{T}_{f}P^{T}_{r2}+U_{r3}R^{T}_{1}-Y_{r2}+{\mathcal {A}}^{T}_{fi\eta }, \\&\check{\varTheta }_{ijr\eta 11}=sym(Y_{r1}A_{ir}+Y_{r1}A_{jr}+{\mathcal {B}}_{fj\eta }C_{ir}+{\mathcal {B}}_{fi\eta }C_{jr})+\\ \,&2\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s1}E-2E^{T}Z_{11}E,\\&\check{\varTheta }_{ijr\eta 12}={\mathcal {A}}_{fj\eta }+{\mathcal {A}}_{fi\eta }+A^{T}_{ir}Y^{T}_{r2}+C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }\\ \,&+A^{T}_{jr}Y^{T}_{r2}+C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta }+2\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s2}E_{f}-2E^{T}Z_{12}E_{f},\\&\check{\varTheta }_{ijr\eta 13}=2E^{T}P^{T}_{r1}+2U_{r1}R^{T}_{1}-2Y_{r1}+A^{T}_{ir}Y^{T}_{r3}\\ \,&+A^{T}_{jr}Y^{T}_{r3}+C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta },\\&\check{\varTheta }_{ijr\eta 14}=2E^{T}P_{r2}+2U_{r2}R^{T}_{2}-2Y+A^{T}_{ir}Y^{T}_{r4}+A^{T}_{jr}Y^{T}_{r4}\\ \,&+C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta },\\&\check{\varTheta }_{ijr\eta 22}=sym({\mathcal {A}}_{fj\eta }+{\mathcal {A}}_{fi\eta })+2\sum \limits _{s=1}^N\lambda _{rs}E^{T}_{f}P_{s3}E_{f}\\ \,&-2E^{T}_{f}Z_{22}E_{f},\\&\check{\varTheta }_{ijr\eta 23}=2E^{T}_{f}P^{T}_{r2}+2U_{r3}R^{T}_{1}-2Y_{r2}+{\mathcal {A}}^{T}_{fj\eta }+{\mathcal {A}}^{T}_{fi\eta },\\&\check{\varTheta }_{ijr\eta 24}=2E^{T}_{f}P^{T}_{r3}+2U_{r4}R^{T}_{2}-2Y+{\mathcal {A}}^{T}_{fj\eta }+{\mathcal {A}}^{T}_{fi\eta },\\&\varXi _{iir\eta 11}=sym(Y_{r1}A_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{ir})\\ \,&+\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s1}E+Q_{11}-E^{T}Z_{11}E,\\&\varXi _{iir\eta 12}=A^{T}_{ir}Y^{T}_{r2}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta }+\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s2}E_{f}\\ \,&+Q_{12}-E^{T}Z_{12}E_{f}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fi\eta },\\&\varXi _{iir\eta 13}=U_{r1}R^{T}_{1}+E^{T}P_{r1}-Y_{r1}+A^{T}_{ir}Y^{T}_{r3}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta },\\&\varXi _{iir\eta 14}=U_{r2}R^{T}_{2}+E^{T}P_{r2}+A^{T}_{ir}Y^{T}_{r4}-Y+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fi\eta },\\&\varXi _{iir\eta 15}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{dir}+E^{T}Z_{11}E+Y_{r1}A_{dir},\\&\varXi _{iir\eta 66}=-(1-\mu )Q_{22}-E^{T}_{f}Z_{22}E_{f},\\&\varXi _{iir\eta 16}=E^{T}Z_{12}E_{f}, \varXi _{iir\eta 18}=Y_{r1}B_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{ir},\\&\varXi _{iir\eta 17}=Y_{r1}D_{ir}+L^{T}O,\varXi _{iir\eta 27}=Y_{r2}D_{ir}, \\&\varXi _{iir\eta 19}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{ir}+Y_{r1}H_{ir},\\&\varXi _{iir\eta 22}=sym(\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fi\eta })+\sum \limits _{s=1}^N\lambda _{rs}E^{T}_{f}P_{s3}E_{f}+Q_{22}\\ \,&-E^{T}_{f}Z_{22}E_{f}, \varXi _{iir\eta 44}=-Y^{T}-Y+\tau ^{2}Z_{22},\\&\varXi _{iir\eta 23}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fi\eta }+E^{T}_{f}P^{T}_{r2}+U_{r3}R^{T}_{1}-Y_{r2},\\&\varXi _{iir\eta 24}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fi\eta }+E^{T}_{f}P^{T}_{r3}+U_{r4}R^{T}_{2}-Y, \\&\varXi _{iir\eta 25}=Y_{r2}A_{dir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{dir}+E^{T}_{f}Z^{T}_{12}E,\\ \\&\varXi _{iir\eta 28}=Y_{r2}B_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{ir}, \\&\varXi _{iir\eta 29}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{ir}+Y_{r2}H_{ir},\\&\varXi _{iir\eta 33}=-sym(Y_{r3})+\tau ^{2}Z_{11},\\&{\varXi _{iir\eta 34}=-Y^{T}_{r4}-Y+\tau ^{2}Z_{12},}\\&\varXi _{iir\eta 35}=Y_{r3}A_{dir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{dir}, \\&\varXi _{iir\eta 38}=Y_{r3}B_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{ir},\\&\varXi _{iir\eta 37}=Y_{r3}D_{ir}, \varXi _{iir\eta 39}=Y_{r3}H_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{ir}, \\&\varXi _{iir\eta 45}=Y_{r4}A_{dir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{dir}, \varXi _{iir\eta 47}=Y_{r4}D_{ir},\\&\varXi _{iir\eta 48}=Y_{r4}B_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{ir}, \\&\varXi _{iir\eta 49}=Y_{r4}H_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{ir},\\&\varXi _{iir\eta 55}=-(1-\mu )Q_{11}-E^{T}Z_{11}E,\\&\varXi _{iir\eta 56}=-(1-\mu )Q_{12}-E^{T}Z_{12}E_{f},\\&\varXi _{iir\eta 110}=[\sqrt{\pi _{r1}}C^{T}_{ir}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{ir}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{ir}D^{T}_{fi\kappa }],\\&\varXi _{iir\eta 210}=[\sqrt{\pi _{r1}}C^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{fi\kappa }~],\\&\varXi _{iir\eta 510}=[\sqrt{\pi _{r1}}C^{T}_{dir}D^{T}_{fi1}~\sqrt{\pi _{r2}}C^{T}_{dir}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{dir}D^{T}_{fi\kappa }],\\&\varXi _{iir\eta 810}=[\sqrt{\pi _{r1}}F^{T}_{ir}D^{T}_{fi1}-I\quad \sqrt{\pi _{r2}}F^{T}_{ir}D^{T}_{fi2}-I \\&\cdots \sqrt{\pi _{r\kappa }}F^{T}_{ir}D^{T}_{fi\kappa }-I],\\&\varXi _{iir\eta 910}=[\sqrt{\pi _{r1}}G^{T}_{ir}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}G^{T}_{ir}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}G^{T}_{ir}D^{T}_{fi\kappa }],\\&\tilde{\varXi }_{ijr\eta 11}=2\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s1}E+2Q_{11}-2E^{T}Z_{11}E+sym\\ \,&{(Y_{r1}A_{ir}+Y_{r1}A_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{ir}),}\\&{\tilde{\varXi }_{ijr\eta 12}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fi\eta }+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fj\eta }}+A^{T}_{ir}Y^{T}_{r2}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+A^{T}_{jr}Y^{T}_{r2} +\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta }\\ \,&+2\sum \limits _{s=1}^N\lambda _{rs}E^{T}P_{s2}E_{f}+2Q_{12}-2E^{T}Z_{12}E_{f},\\&\tilde{\varXi }_{ijr\eta 13}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+2E^{T}P_{r1}+2U_{r1}R^{T}_{1}\\ \,&-2Y_{r1}+A^{T}_{ir}Y^{T}_{r3}+A^{T}_{jr}Y^{T}_{r3}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta },\\&\tilde{\varXi }_{ijr\eta 14}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{ir}{\mathcal {B}}^{T}_{fj\eta }+2E^{T}P_{r2}-2Y\\ \,&+A^{T}_{ir}Y^{T}_{r4}+A^{T}_{jr}Y^{T}_{r4}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }C^{T}_{jr}{\mathcal {B}}^{T}_{fi\eta }+2U_{r2}R^{T}_{2},\\ \\&\tilde{\varXi }_{ijr\eta 15}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{djr}+Y_{r1}A_{dir}+Y_{r1}A_{djr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{dir}+2E^{T}Z_{11}E,\\&\tilde{\varXi }_{ijr\eta 16}=2E^{T}Z_{12}E_{f}, \tilde{\varXi }_{ijr\eta 17}=2Y_{r1}D_{ir}+2L^{T}O,\\&\tilde{\varXi }_{ijr\eta 34}=-2Y^{T}_{r4}-2Y+2\tau ^{2}Z_{12},\\&\tilde{\varXi }_{ijr\eta 18}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{jr}+Y_{r1}B_{ir}+Y_{r1}B_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }F_{ir}, {\tilde{\varXi }_{ijr\eta 19}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{jr}+Y_{r1}H_{ir}}\\ \,&{+Y_{r1}H_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }G_{ir},}\\&\tilde{\varXi }_{ijr\eta 22}=2Q_{22}-2E^{T}_{f}Z_{22}E_{f}+sym(\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fi\eta }\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}_{fj\eta })+2\sum \limits _{s=1}^N\lambda _{rs}E^{T}_{f}P_{s3}E_{f},\\&\tilde{\varXi }_{ijr\eta 23}=2E^{T}_{f}P^{T}_{r2}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fi\eta }+2U_{r3}R^{T}_{1}-2Y_{r2}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fj\eta },\\&\tilde{\varXi }_{ijr\eta 24}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fi\eta }+2E^{T}_{f}P^{T}_{r3}-2Y+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {A}}^{T}_{fj\eta }\\ \,&+2U_{r4}R^{T}_{2}, \tilde{\varXi }_{ijr\eta 26}=2E^{T}_{f}Z_{12}E_{f},\\&\tilde{\varXi }_{ijr\eta 25}=Y_{r2}A_{dir}+Y_{r2}A_{djr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{djr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{dir}+2E^{T}_{f}Z^{T}_{12}E, \\&\tilde{\varXi }_{ijr\eta 27}=2Y_{r2}D_{ir}, \tilde{\varXi }_{ijr\eta 28}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{jr}+Y_{r2}B_{ir}\\ \,&+Y_{r2}B_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }F_{ir},\\&\tilde{\varXi }_{ijr\eta 29}=Y_{r2}H_{ir}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{jr}+Y_{r2}H_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }G_{ir},\\&\tilde{\varXi }_{ijr\eta 33}=-2Y^{T}_{r3}-2Y_{r3}+2\tau ^{2}Z_{11},\\&\tilde{\varXi }_{ijr\eta 44}=-2Y^{T}-2Y+2\tau ^{2}Z_{22},\\&\tilde{\varXi }_{ijr\eta 35}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{djr}+Y_{r3}A_{dir}+Y_{r3}A_{djr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{dir}, \\&\tilde{\varXi }_{ijr\eta 45}=Y_{r4}A_{dir}+Y_{r4}A_{djr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }C_{djr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }C_{dir}, \tilde{\varXi }_{ijr\eta 37}=2Y_{r3}D_{ir},\\&\tilde{\varXi }_{ijr\eta 39}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{jr}+Y_{r3}H_{ir}+Y_{r3}H_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }G_{ir}, \tilde{\varXi }_{ijr\eta 48}=Y_{r4}B_{ir}+Y_{r4}B_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{jr}{+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }F_{ir},}\\&\tilde{\varXi }_{ijr\eta 49}=Y_{r4}H_{ir}+Y_{r4}H_{jr}+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }G_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }G_{ir}, \tilde{\varXi }_{ijr\eta 55}=-2(1-\mu )Q_{11}-2E^{T}Z_{11}E,\\&\tilde{\varXi }_{ijr\eta 38}=\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fi\eta }F_{jr}+Y_{r3}B_{ir}+Y_{r3}B_{jr}\\ \,&+\sum \limits _{\eta =1}^\kappa \pi _{r\eta }{\mathcal {B}}_{fj\eta }F_{ir}, \tilde{\varXi }_{ijr\eta 47}=2Y_{r4}D_{ir},\\&\tilde{\varXi }_{ijr\eta 56}=-2(1-\mu )Q_{12}-2E^{T}Z_{12}E_{f},\\&\tilde{\varXi }_{ijr\eta 66}=-2(1-\mu )Q_{22}-2E^{T}_{f}Z_{22}E_{f},\\&\tilde{\varXi }_{ijr\eta 110}=[\sqrt{\pi _{r1}}C^{T}_{ir}D^{T}_{fj1}\quad \sqrt{\pi _{r2}}C^{T}_{ir}D^{T}_{fj2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{ir}D^{T}_{fj\kappa }~]\\ \,&+[~\sqrt{\pi _{r1}}C^{T}_{jr}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{jr}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{jr}D^{T}_{fi\kappa }],\\&\tilde{\varXi }_{ijr\eta 210}=[\sqrt{\pi _{r1}}C^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{fi2}~\cdots ~\sqrt{\pi _{r\kappa }}C^{T}_{fi\kappa }~]\\ \,&+[~\sqrt{\pi _{r1}}C^{T}_{fj1}\quad \sqrt{\pi _{r2}}C^{T}_{fj2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{fj\kappa }],\\&\tilde{\varXi }_{ijr\eta 510}=[\sqrt{\pi _{r1}}C^{T}_{dir}D^{T}_{fj1}~\sqrt{\pi _{r2}}C^{T}_{dir}D^{T}_{fj2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{dir}D^{T}_{fj\kappa }]\\ \,&+[\sqrt{\pi _{r1}}C^{T}_{djr}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}C^{T}_{djr}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}C^{T}_{djr}D^{T}_{fi\kappa }],\\&\tilde{\varXi }_{ijr\eta 810}=[\sqrt{\pi _{r1}}F^{T}_{ir}D^{T}_{fj1}-I~\sqrt{\pi _{r2}}F^{T}_{ir}D^{T}_{fj2}-I\cdots \\ \,&\sqrt{\pi _{r\kappa }}F^{T}_{ir}D^{T}_{fj\kappa }-I] +[\sqrt{\pi _{r1}}F^{T}_{jr}D^{T}_{fi1}-I~\sqrt{\pi _{r2}}F^{T}_{jr}D^{T}_{fi2}-I\\ \,&\cdots \sqrt{\pi _{r\kappa }}F^{T}_{jr}D^{T}_{fi\kappa }-I],\\&\tilde{\varXi }_{ijr\eta 910}=[\sqrt{\pi _{r1}}G^{T}_{ir}D^{T}_{fj1}\quad \sqrt{\pi _{r2}}G^{T}_{ir}D^{T}_{fj2}\cdots \sqrt{\pi _{r\kappa }}G^{T}_{ir}D^{T}_{fj\kappa }]\\ \,&+[\sqrt{\pi _{r1}}G^{T}_{jr}D^{T}_{fi1}\quad \sqrt{\pi _{r2}}G^{T}_{jr}D^{T}_{fi2}\cdots \sqrt{\pi _{r\kappa }}G^{T}_{jr}D^{T}_{fi\kappa }]. \end{aligned}$$

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yin, Y., Zhuang, G., Xia, J. et al. H∞ Asynchronous Deconvolution Fuzzy Filter Design for Nonlinear Singular Markov Jump Systems with Time-Varying Delays. Int. J. Fuzzy Syst. 25, 763–779 (2023). https://doi.org/10.1007/s40815-022-01400-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-022-01400-1

Keywords

Search

Navigation