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Sliding Mode Observers for Time-Dependent Set-Valued Lur’e Systems Subject to Uncertainties

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Abstract

Designing observers for dynamical systems plays an important role in the modern control theory due to the lack of full information in measured outputs. The current paper proposes a sliding mode observer for a general class of Lur’e systems subject to uncertainties where feedbacks involve time-dependent set-valued mappings. To the best of our knowledge, sliding mode observers for set-valued Lur’e systems, even for the simple static case, have not been considered in the literature. Exponential convergence of the observer state and finite-time convergence of the output estimation error are guaranteed without using any linear transformations. In addition, our design can also deduce \(H^\infty \) observers.

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References

  1. Acary, V., Bonnefon, O., Brogliato, B.: Nonsmooth Modeling and Simulation for Switched Circuits, Lecture Notes in Electrical Engineering. Springer, Dordrecht (2011)

    Book  Google Scholar 

  2. Adly, S., Hantoute, A., Le, B.K.: Nonsmooth Lur’e dynamical systems in Hilbert spaces. Set Valued Var. Anal. 24(1), 13–35 (2016)

    Article  MathSciNet  Google Scholar 

  3. Adly, S., Hantoute, A., Le, B.K.: Maximal monotonicity and cyclic-monotonicity arising in nonsmooth Lur’e dynamical systems. J. Math. Anal. Appl. 448(1), 691–706 (2017)

    Article  MathSciNet  Google Scholar 

  4. Arcak, M., Kokotovic, P.: Nonlinear observers: a circle criterion design and robustness analysis. Automatica 37, 1923–1930 (2001)

    Article  MathSciNet  Google Scholar 

  5. Alwi, H., Chen, L., Edwards, C.: Reconstruction of simultaneous actuator and sensor faults for the RECONFIGURE benchmark using a sliding mode observer. In: Boje, E., Xia, X. (eds.) IFAC Proceedings, vol. 47(3), pp. 3497–3502 (2014)

  6. Bartolini, G., Fridman, L., Pisano, A., Usai, E.: Modern Sliding Mode Control Theory: New Perspectives and Applications, Lecture Notes in Control and Information Sciences, vol. 375. Springer, Berlin (2008)

    Book  Google Scholar 

  7. Brogliato, B., Tanwani, A.: Dynamical systems coupled with monotone set-valued operators: formalisms, applications, well-posedness, and stability. SIAM Rev. 62(1), 3–129 (2020)

    Article  MathSciNet  Google Scholar 

  8. Brogliato, B.: Absolute stability and the Lagrange–Dirichlet theorem with monotone multivalued mappings. Syst. Control Lett. 51(5), 343–353 (2004)

    Article  MathSciNet  Google Scholar 

  9. Brogliato, B., Goeleven, D.: Well-posedness, stability and invariance results for a class of multivalued Lur’e dynamical systems. Nonlinear Anal. Theory Methods Appl. 74, 195–212 (2011)

    Article  Google Scholar 

  10. Brogliato, B., Lozano, R., Maschke, B., Egeland, O.: Dissipative Systems Analysis and Control, 3rd edn. Springer, Cham (2020)

    Book  Google Scholar 

  11. Brogliato, B., Heemels, W.P.M.H.: Observer design for Lur’e systems with multivalued mappings: a passivity approach. IEEE Trans. Autom. Control 54(8), 1996–2001 (2009)

    Article  Google Scholar 

  12. Camlibel, M.K., Schumacher, J.M.: Linear passive systems and maximal monotone mappings. Math. Program. 157(2), 397–420 (2016)

    Article  MathSciNet  Google Scholar 

  13. Emelyanov, S.V. (ed.): Variable Structure Control Systems. Nauka, Moscow (1967)

    MATH  Google Scholar 

  14. Golubev, A.E., Rishchenko, A.P., Kachev, S.B.: Separation principle for a class of nonlinear systems. In: Camacho, E.F., Basañes, L., Puente, J.A. (eds.) IFAC Proceedings, vol. 35(1), pp. 447–452 (2002)

  15. Huang, J., Han, Z., Cai, X., Liu, L.: Adaptive full-order and reduced-order observers for the Lur’e differential inclusion system. Commun. Nonlinear Sci. Numer. Simul. 16, 2869–2879 (2011)

    Article  MathSciNet  Google Scholar 

  16. Huang, J., Yu, L., Zhang, M., Zhu, F., Han, Z.: Actuator fault detection and estimation for the Lur’e differential inclusion system. Appl. Math. Model. 38, 2090–2100 (2014)

    Article  MathSciNet  Google Scholar 

  17. Huang, J., Zhang, J., Han, Z.: A note on adaptive observer for the Lur’e differential inclusion system. Nonlinear Dyn. 86, 1227–1237 (2016)

    Article  MathSciNet  Google Scholar 

  18. Huang, J., Zhang, W., Shi, M., Chen, L., Yu, L.: \(H^\infty \) Observer design for singular one-sided Lur’e differential inclusion system. J. Frankl. Inst. 354, 3305–3321 (2017)

    Article  MathSciNet  Google Scholar 

  19. Kou, S.R., Elliott, D.L., Tarn, T.J.: Exponential observers for nonlinear dynamic systems. Inf. Control 29, 204–216 (1975)

    Article  MathSciNet  Google Scholar 

  20. Itkis, U.: Control Systems of Variable Structure. Wiley, New York (1976)

    Google Scholar 

  21. Le, B.K.: On a class of Lur’e dynamical systems with state-dependent set-valued feedback. Set Valued Var. Anal. 28, 537–557 (2020)

    Article  MathSciNet  Google Scholar 

  22. Le, B.K.: Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions. Optimization 69, 1187–1217 (2020)

    Article  MathSciNet  Google Scholar 

  23. Luenberger, D.: Observing the state of a linear system. IEEE Trans. Mil. Electron. 8, 74–80 (1964)

    Article  Google Scholar 

  24. Luenberger, D.: Observers for multivariable systems. IEEE Trans. Autom. Control. 11, 190–197 (1966)

    Article  Google Scholar 

  25. Showalter, R.E.: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. American Mathematical Society, Providence, RI (1997)

    MATH  Google Scholar 

  26. Shtessel, Y., Edwards, C., Fridman, L., Levant, A.: Sliding Mode Control and Observation. Birkhauser, Basel (2013)

    Google Scholar 

  27. Slotine, J.J.E., Edrick, J.K., Misawa, E.A.: On sliding observers for nonlinear systems. In: Proceedings of the American Control Conference, 1794–1800, Seattle, USA (1986)

  28. Spurgeon, S.: Sliding mode observers—a survey. Int. J. Syst. Sci. 39(8), 751–764 (2008)

    Article  MathSciNet  Google Scholar 

  29. Tan, C., Edwards, C.: Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int. J. Robust. Nonlinear Control 13, 443–463 (2003)

    Article  MathSciNet  Google Scholar 

  30. Tanwani, A., Brogliato, B., Prieur, C.: Stability and observer design for Lur’e systems with multivalued, non-monotone, time-varying nonlinearities and state jumps. SIAM J. Control Optim. 52(6), 3639–3672 (2014)

    Article  MathSciNet  Google Scholar 

  31. Tanwani, A., Brogliato, B., Prieur, C.: Well-posedness and output regulation for implicit time-varying evolution variational inequalities. SIAM J. Control Optim. 56(2), 751–781 (2018)

    Article  MathSciNet  Google Scholar 

  32. Thau, F.E.: Observing the state of non-linear dynamic systems. Int. J. Control 17, 471–479 (1973)

    Article  Google Scholar 

  33. Utkin, V.I.: Variable structure systems with sliding modes. IEEE Trans. Autom. Control 22(2), 212–222 (1977)

    Article  MathSciNet  Google Scholar 

  34. Utkin, V.I.: Sliding Modes and Their Application in Variable Structure Systems. MIR Publishers, Moscow (1978)

    MATH  Google Scholar 

  35. Utkin, V.I.: Sliding Modes in Control Optimization. Springer, Berlin (1992)

    Book  Google Scholar 

  36. Walcott, B.L., Zak, S.H.: State observation of nonlinear uncertain dynamical systems. IEEE Trans. Autom. Control 32(2), 166–170 (1987)

    Article  MathSciNet  Google Scholar 

  37. Walcott, B.L., Zak, S.H.: Combined observer controller synthesis for uncertain dynamical systems with applications. IEEE Trans. Syst. Man Cybern. Syst. 18(1), 88–104 (1988)

    Article  Google Scholar 

  38. Xiang, J., Su, H., Chu, J.: On the design of Walcott-Zak sliding mode observer. In: Proceedings of the American Control Conference, Portland, OR, pp. 2451–2456 (2005)

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Acknowledgements

The author would like to thank the referees and the handling editor for valuable comments and suggestions. In addition, he wants to express his gratitude to University of O’Higgins, especially to Rector Dr. R. Correa and all his colleagues for warm welcome and hospitality during the time he worked in Chile, where some parts of this manuscript were done.

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Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Ba Khiet Le

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  1. Ba Khiet Le
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Correspondence to Ba Khiet Le.

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Communicated by Negash G. Medhin.

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Le, B.K. Sliding Mode Observers for Time-Dependent Set-Valued Lur’e Systems Subject to Uncertainties. J Optim Theory Appl 194, 290–305 (2022). https://doi.org/10.1007/s10957-022-02027-w

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  • DOI: https://doi.org/10.1007/s10957-022-02027-w

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