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Finite-Time H Inverse Optimal Control of Affine Nonlinear Systems

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Abstract

This paper is focused on the problem of the finite-time H inverse optimal control for affine nonlinear systems. Based on the finite-time control Lyapunov function, we derive a sufficient condition for the existence of time-invariant, continuous, finite-time stabilizing and inverse optimal state feedback control law, and propose a universal formula for constructing the finite-time H inverse optimal control law. We investigate the relationship between the finite-time stabilization and the finite-time H inverse optimal control. Finally, some examples are given to illustrate the effectiveness of the presented results.

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References

  1. M. Angulo, L. Fridman, A. Levant, Output-feedback finite-time stabilization of disturbed LTI systems. Automatica 48, 606–611 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Z. Artstein, Stabilization with relaxed control. Nonlinear Anal. 7, 1163–1173 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Bhat, D. Bernstein, Finite-time stability of continuous autonomous system. SIAM J. Control Optim. 38(3), 751–766 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. X. Cai, Z. Han, Inverse optimal control of nonlinear systems with structural uncertainty. IET Control Theory Appl. 152(1), 79–83 (2005)

    Article  Google Scholar 

  5. S. Ding, S. Li, Q. Li, Stability analysis for a second-order continuous finite-time control system subject to a disturbance. J. Control Theory Appl. 3, 271–276 (2009)

    Article  MathSciNet  Google Scholar 

  6. R.A. Freeman, P.V. Kokotovic, Inverse optimality in robust stabilization. SIAM J. Control Optim. 34(4), 1365–1391 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Haimo, Finite-time controllers. SIAM J. Control Optim. 24, 760–770 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Hong, Finite-time stabilization and stabilizability of a class of controllable systems. Syst. Control. Lett. 46(3), 231–236 (2002)

    Article  MATH  Google Scholar 

  9. X. Huang, W. Lin, B. Yang, Global finite-time stabilization of a class of uncertain nonlinear system. Automatica 41, 881–888 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Li, C. Qian, Global finite-time stabilization by dynamic output feedback for a class of continuous nonlinear systems. IEEE Trans. Autom. Control 51(5), 879–884 (2006)

    Article  MathSciNet  Google Scholar 

  11. P. Li, Z. Zheng, Global finite-time stabilization of planar nonlinear systems with disturbance. Asian J. Control 14(3), 851–858 (2012)

    Article  MathSciNet  Google Scholar 

  12. S. Li, Y. Tian, Finite-time stability of cascaded time varying systems. Int. J. Control 80(4), 646–657 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. X. Luan, F. Liu, P. Shi, Robust Finite-Time H control for nonlinear jump systems via neural networks. Circuits Syst. Signal Process. 29(3), 481–498 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Mo, Robust stabilization for multi-input polytopic nonlinear systems. J. Syst. Sci. Complex. 24, 93–104 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Mo, Y. Jia, Z. Zheng, Finite-time disturbance attenuation of nonlinear systems. Sci. China Ser. F 52, 2163–2171 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Mo, Y. Jia, Z. Zheng et al., Finite-time disturbance attenuation of a class of uncertain nonlinear systems, in Proceedings of the 29th Chinese Control Conference, (2010), pp. 4721–4727

    Google Scholar 

  17. E. Moulay, W. Perruquetti, Finite-time stability and stabilization of a class of continuous systems. J. Math. Anal. Appl. 323(2), 1430–1443 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. N. Nakamura, Global inverse optimal control with guaranteed convergence rates of input affine nonlinear systems. IEEE Trans. Autom. Control 56(2), 358–369 (2011)

    Article  Google Scholar 

  19. S.G. Nersesov, W.M. Haddad, Q. Hui, Finite-time stabilization of nonlinear dynamical systems via control Lyapunov functions. J. Franklin Inst. 345, 819–837 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. E.P. Ryan, Singular optimal control for second order saturating systems. Int. J. Control 30(4), 549–564 (1979)

    Article  MATH  Google Scholar 

  21. E.P. Ryan, Finite-time stabilization of uncertain nonlinear planar systems. Dyn. Control 1(1), 83–89 (1991)

    Article  MATH  Google Scholar 

  22. Y. Shen, Y. Huang, Global finite-time stabilisation for a class of nonlinear systems. Int. J. Syst. Sci. 43(1), 73–78 (2012)

    Article  MathSciNet  Google Scholar 

  23. E.D. Sontag, A Lyapunov-like characterization of asymptotic controllability. SIAM J. Control Optim. 21, 462–471 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. E.D. Sontag, A ‘universal’ constructive of Artstein’s theorem on nonlinear stabilization. Syst. Control Lett. 12, 542–550 (1989)

    Article  MathSciNet  Google Scholar 

  25. A.J. Van Der Schaft, On the state space approach to nonlinear H control. Syst. Control Lett. 16, 1–8 (1991)

    Article  MATH  Google Scholar 

  26. A.J. Van Der Schaft, L 2 gain analysis of nonlinear systems and nonlinear H control. IEEE Trans. Autom. Control 37, 770–784 (1992)

    Article  MATH  Google Scholar 

  27. Z. Xiang, C. Qiao, M. Mahmoud, Finite-time analysis and H control for switched stochastic systems. J. Franklin Inst. 349, 915–927 (2012)

    Article  MathSciNet  Google Scholar 

  28. J. Yin, S. Khoo, Z. Man, X. Yu, Finite-time stability and instability of stochastic nonlinear systems. Automatica 47, 2671–2677 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. K. Zhou, J. Doyle, K. Glover, Robust and Optimal Control (Prentice Hall, New York, 1995)

    Google Scholar 

Download references

Acknowledgements

The work described in this paper was supported by the National Nature Science Foundation of China (Grant No. 11126210, 11001004), the Research Foundation for Youth Scholars of Beijing Technology and Business University (Grant No. QNJJ2011-34), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR) (IHLB 201106206) and the Beijing Municipal Natural Science Foundation (Grant No. 4122019).

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Correspondence to Lipo Mo.

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Mo, L. Finite-Time H Inverse Optimal Control of Affine Nonlinear Systems. Circuits Syst Signal Process 32, 47–60 (2013). https://doi.org/10.1007/s00034-012-9442-x

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  • DOI: https://doi.org/10.1007/s00034-012-9442-x

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