Skip to main content

Advertisement

Springer Nature Link
Log in

Bivariate Optimization for Control Design of Interconnected Uncertain Nonlinear Systems: A Fuzzy Set-Theoretic Approach

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

Abstract

A new fuzzy set-theoretic approach to the control design of interconnected uncertain nonlinear system is proposed. The uncertainty of the system is nonlinear and (possible) fast time-varying, which may be due to unknown parameter and input disturbance. The uncertainty bound can be described via a fuzzy set. In this paper, the control design consists of the control scheme design and control gain optimization. Under a state transformation, a new robust control scheme in deterministic form is proposed for the interconnected system, which is not fuzzy if-then rule-based. The optimal design of the control gain is then proposed via the fuzzy description of the uncertainty bound, which is formulated as a bivariate constrained optimization problem by minimizing a fuzzy-based performance index. We show that the globally unique solution to this optimization problem always exists. The closed-form (i.e., analytic) solution and the corresponding minimum value of the performance index are also presented. The resulting control is able to render the system performance in twofold. First, it guarantees uniform boundedness and uniform ultimate boundedness of the system regardless of the actual value of uncertainty. Second, it minimizes the fuzzy-based performance index associated with both the fuzzy system performance and the control cost.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Sun, K., Sui, S., Tong, S.: Fuzzy adaptive decentralized optimal control for strict feedback nonlinear large-scale systems. IEEE Trans. Cybern. 1(1), 1–14 (2017)

    Google Scholar 

  2. Zhang, Z., Lin, C., Chen, B.: New decentralized \(H_{\infty }\) filter design for nonlinear interconnected systems based on Takagi–Sugeno fuzzy models. IEEE Trans. Cybern. 45(12), 2914–2924 (2015)

    Article  Google Scholar 

  3. Li, Y., Tong, S., Li, T.: Adaptive fuzzy output feedback dynamic surface control of interconnected nonlinear pure-feedback systems. IEEE Trans. Cybern. 45(1), 138–149 (2015)

    Article  Google Scholar 

  4. Myers, M.H., Li, Y., Kivlehan, F., Lindner, E., Chaum, E.: A feedback control approach to organic drug infusions using electrochemical measurement. IEEE Trans. Biomed. Eng. 63(3), 506–511 (2016)

    Article  Google Scholar 

  5. Xu, J.Q., Chen, Y.H., Guo, H.: Fractional robust control design for fuzzy dynamical systems: an optimal approach. J. Intell. Fuzzy Syst. 29(2), 553–569 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alrayyah, A.Y., Mahmoud, M.S.: Decentralized reliable control of interconnected time-delay systems against sensor failures. J. Optim. Theory Appl. 147(2), 318–336 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hua, C., Li, Y., Wang, H., Guan, X.: Decentralised fault-tolerant finite-time control for a class of interconnected non-linear systems. IET Control Theory Appl. 9(16), 2331–2339 (2015)

    Article  MathSciNet  Google Scholar 

  8. Guo, H., Xu, J.Q., Chen, Y.H.: Robust control of fault tolerant permanent magnet synchronous motor for aerospace application with guaranteed fault switch process. IEEE Trans. Ind. Electron. 62(12), 7309–7321 (2015)

    Article  Google Scholar 

  9. Wang, Y., Song, Y., Gao, H., Lewis, F.L.: Distributed fault-tolerant control of virtually and physically interconnected systems with application to high-speed trains under traction/braking failures. IEEE Trans. Intell. Transp. Syst. 17(2), 535–545 (2016)

    Article  Google Scholar 

  10. Xu, J.Q., Chen, Y.H., Guo, H.: Robust levitation control for maglev systems with guaranteed bounded airgap. ISA Trans. 59(1), 205–214 (2015)

    Article  Google Scholar 

  11. Ebihara, Y., Peaucelle, D., Arzelier, D.: Analysis and synthesis of interconnected positive systems. IEEE Trans. Autom. Control 62(2), 652–667 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Long, L., Zhao, J.: A small-gain theorem for switched interconnected nonlinear systems and its applications. IEEE Trans. Autom. Control 59(4), 1082–1088 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lavaei, J.: Decentralized implementation of centralized controllers for interconnected systems. IEEE Trans. Autom. Control 57(7), 1860–1865 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wu, Y., Meng, X., Xie, L., Lu, R., Su, H., Wu, Z.: An input-based triggering approach to leader-following problems. Automatica 75(1), 221–228 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wu, Y., Lu, R., Shi, P., Su, H., Wu, Z.: Adaptive output synchronization of heterogeneous network with an uncertain leader. Automatica 76(1), 183–192 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhao, R., Chen, Y.H., Jiao, S.: Optimal design of constraint-following control for fuzzy mechanical systems. IEEE Trans. Fuzzy Syst. 24(5), 1108–1120 (2016)

    Article  Google Scholar 

  17. Choi, Y.H., Yoo, S.J.: Minimal-approximation-based decentralized backstepping control of interconnected time-delay systems. IEEE Trans. Cybern. 46(12), 3401–3413 (2016)

    Article  Google Scholar 

  18. Prasad, S., Purwar, S., Kishor, N.: H-infinity based non-linear sliding mode controller for frequency regulation in interconnected power systems with constant and time-varying delays. IET Gener. Transm. Distrib. 10(11), 2771–2784 (2016)

    Article  Google Scholar 

  19. Wu, H.: Decentralised adaptive robust control of uncertain large-scale non-linear dynamical systems with time-varying delays. IET Control Theory Appl. 6(5), 629–640 (2012)

    Article  MathSciNet  Google Scholar 

  20. Wang, H., Liu, X., Liu, K.: Robust adaptive neural tracking control for a class of stochastic nonlinear interconnected systems. IEEE Trans. Neural Netw. Learn. Syst. 27(3), 510–523 (2016)

    Article  MathSciNet  Google Scholar 

  21. Chang, W., Wang, W.: \(H_{\infty }\) fuzzy control synthesis for a large-scale system with a reduced number of LMIs. IEEE Trans. Fuzzy Syst. 23(4), 1197–1210 (2015)

    Article  MathSciNet  Google Scholar 

  22. Procyk, T.J., Mamdani, E.H.: A linguistic self-organizing process controller. Automatica 15(1), 15–30 (1979)

    Article  MATH  Google Scholar 

  23. Wang, H., Liu, X., Liu, K., Karimi, H.R.: Approximation-based adaptive fuzzy tracking control for a class of nonstrict-feedback stochastic nonlinear time-delay systems. IEEE Trans. Fuzzy Syst. 23(5), 1746–1760 (2015)

    Article  Google Scholar 

  24. Wu, Z.G., Shi, P., Su, H., Lu, R.: Dissipativity-based sampled-data fuzzy control design and its application to truck-trailer system. IEEE Trans. Fuzzy Syst. 23(5), 1669–1679 (2015)

    Article  Google Scholar 

  25. Wu, Z.G., Dong, S., Shi, P., Su, H., Huang, T., Lu, R.: Fuzzy-model-based nonfragile guaranteed cost control of nonlinear Markov jump systems. IEEE Trans. Syst. Man Cybern. Syst. 47(8), 2388–2397 (2017)

    Article  Google Scholar 

  26. Wei, Y., Qiu, J., Karimi, H.R.: Reliable output feedback control of discrete-time fuzzy affine systems with actuator faults. IEEE Trans. Circuits Syst. I Regul. Pap. 64(1), 170–181 (2017)

    Article  Google Scholar 

  27. Wei, Y., Qiu, J., Lam, H.K., Wu, L.: Approaches to T-S fuzzy-affine-model-based reliable output feedback control for nonlinear It\(\hat{o}\) stochastic systems. IEEE Trans. Fuzzy Syst. 25(3), 569–583 (2017)

    Article  Google Scholar 

  28. Chang, Y.C., Yen, H.M.: Design of a robust position feedback tracking controller for flexible-joint robots. IET Control Theory Appl. 5(2), 351–363 (2011)

    Article  MathSciNet  Google Scholar 

  29. Yan, X.G., Spurgeon, S.K., Edwards, C.: Global decentralised static output feedback sliding-mode control for interconnected time-delay systems. IET Control Theory Appl. 6(2), 192–202 (2012)

    Article  MathSciNet  Google Scholar 

  30. Liu, T.H., Pu, H.T., Lin, C.K.: Implementation of an adaptive position control system of a permanent-magnet synchronous motor and its application. IET Electr. Power Appl. 4(2), 121–130 (2010)

    Article  Google Scholar 

  31. Barmish, B.R., Leitmann, G.: On ultimate boundedness control of uncertain systems in the absence of matching condition. IEEE Trans. Autom. Control 27(1), 153–158 (1982)

    Article  MATH  Google Scholar 

  32. Huang, J., Chen, Y.H., Cheng, A.: Robust control for fuzzy dynamical systems: uniform ultimate boundedness and optimality. IEEE Trans. Fuzzy Syst. 20(6), 1022–1031 (2012)

    Article  Google Scholar 

  33. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

  34. Klir, J.G., Yuan, B.: Fuzzy Sets and Fuzzy Logic Theory and Applications. Prentice Hall, Englewood Cliffs (1995)

    MATH  Google Scholar 

  35. Chen, Y.H.: Robust control design for a class of mismatched uncertain nonlinear systems. J. Optim. Theory Appl. 90(3), 605–626 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, S., Yang, J., Chen, W.H., Chen, X.: Generalized extended state observer based control for systems with mismatched uncertainties. IEEE Trans. Ind. Electron. 59(12), 4792–4802 (2012)

    Article  Google Scholar 

  37. Khalil, H.K.: Nonlinear Systems. Prentice Hall International Inc, Upper Saddle River (2002)

    MATH  Google Scholar 

  38. Chen, Y.H.: On the deterministic performance of uncertain dynamical systems. Int. J. Control 43(1), 1557–1579 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hale, J.K.: Ordinary Differential Equations. Wiley-Interscience, New York (1969)

    MATH  Google Scholar 

  40. Stewart, J.: Multivariable Calculus. Thompson Brooks/Cole, Belmont (2008)

    Google Scholar 

  41. Bronshtein, I.N., Semendyayev, K.A.: Handbook of Mathematics. Van Nostrand Reinhold, New York (1985)

    MATH  Google Scholar 

  42. Hwang, C., Hsu, J.: Nonlinear control design for a Hammerstein model system. IEE Proc. Control Theory Appl. 142(4), 277–285 (1995)

    Article  MATH  Google Scholar 

  43. Wei, Y., Qiu, J., Lam, H.K.: A novel approach to reliable output feedback control of fuzzy-affine systems with time delays and sensor faults. IEEE Trans. Fuzzy Syst. 25(6), 1808–1823 (2017)

    Article  Google Scholar 

  44. Wei, Y., Qiu, J., Karimi, H.R.: Fuzzy-affine-model-based memory filter design of nonlinear systems with time-varying delay. IEEE Trans. Fuzzy Syst. PP(99), 1–1 (2017)

    Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 51707004) and Aeronautical Science Foundation of China (Grant No. 2016ZC51025).

Author information

Authors and Affiliations

  1. School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Jinquan Xu, Yutao Du & Hong Guo

  2. School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Ye-Hwa Chen

Authors
  1. Jinquan Xu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Yutao Du
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Ye-Hwa Chen
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Hong Guo
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Hong Guo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, J., Du, Y., Chen, YH. et al. Bivariate Optimization for Control Design of Interconnected Uncertain Nonlinear Systems: A Fuzzy Set-Theoretic Approach. Int. J. Fuzzy Syst. 20, 1715–1729 (2018). https://doi.org/10.1007/s40815-018-0472-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-018-0472-9

Keywords