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Robust \(H_\infty \) Repetitive Control for a Class of Linear Stochastic Switched Systems with Time Delay

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Abstract

This paper investigates the problem of robust \(H_\infty \) repetitive control for a class of linear stochastic switched systems with time delay. Based on the lifting technique, a continuous-discrete stochastic 2D (two-dimensional) delayed model is firstly proposed to describe the control and learning actions of the repetitive control system. Then a sufficient condition for the asymptotical stability with \(H_\infty \) performance of the 2D model is derived by choosing an appropriate common Lyapunov functional. The feedback controller gains are then obtained by solving a set of linear matrix inequalities. One example is given to illustrate the effectiveness of the proposed method.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. M.S. Branicky, V.S. Borkar, S.K. Mitter, A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Autom. Control 43(1), 31–45 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Chen, Z. Xiang, Robust \(H_\infty \) control of switched stochastic systems with time delays under asynchronous switching. Adv. Differ. Equ. 2013(86), 1–15 (2013)

    MathSciNet  Google Scholar 

  4. G. Chen, Z. Xiang, M.S. Mahmoud, Stability and \(H_\infty \) performance analysis of switched stochastic neutral systems. Circuits Syst. Signal Process. 32(1), 387–400 (2013)

    Article  MathSciNet  Google Scholar 

  5. W.H. Chen, W.X. Zheng, Delay-dependent stochastic stability and \(H_\infty \)-control of uncertain neutral stochastic systems with time delay. IEEE Trans. Autom. Control 54(7), 1660–1667 (2009)

    Article  MathSciNet  Google Scholar 

  6. J. Daafouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans. Autom. Control 47(11), 1883–1887 (2002)

    Article  MathSciNet  Google Scholar 

  7. L. Fainshil, M. Margaliot, P. Chigansky, On the stability of positive linear switched systems under arbitrary switching laws. IEEE Trans. Autom. Control 54(4), 897–899 (2009)

    Article  MathSciNet  Google Scholar 

  8. M.M. Fateh, H.A. Tehrani, S.M. Karbassi, Repetitive control of electrically driven robot manipulators. Int. J. Syst. Sci. 44(4), 775–785 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. W. Feng, J. Tian, P. Zhao, Stability analysis of switched stochastic systems. Automatica 47(1), 148–157 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Goebel, R.G. Sanfelice, A.R. Teel, Hybrid dynamical systems. IEEE Contr. Syst. Mag. 29(2), 28–93 (2009)

    Article  MathSciNet  Google Scholar 

  11. X. Gu, S. Zhu, D. Wu, Two different kinds of time delays in a stochastic system. Eur. Phys. J. 42(3), 461–466 (2007)

    Google Scholar 

  12. J.P. Hespanha, Uniform stability of switched linear systems: Extensions of LaSalle’s invariance principle. IEEE Trans. Autom. Control 49(4), 470–482 (2004)

    Article  MathSciNet  Google Scholar 

  13. C. Hu, B. Yao, Z. Chen, Adaptive robust repetitive control of an industrial biaxial precision gantry for contouring tasks. IEEE Trans. Control Syst. Technol. 19(6), 1559–1568 (2011)

    Article  Google Scholar 

  14. T. Inoue, M. Nakano, S. Iwai, High accuracy control of a proton synchrotron magnet power supply, in Proceedings of the 8th World Congress of IFAC, Oxford, (1981), pp. 3137–3142

  15. J.W. Lee, P.P. Khargonekar, Optimal output regulation for discrete-time switched and markovian jump linear systems. SIAM J. Control Optim. 47(1), 40–72 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. J.R. Lemon, P.C. Ackermann, Application of self-excited machine-tool chatter theory: contribution to machine-tool chatter research-4. J. Manuf. Sci. Eng. 87(4), 471–479 (1965)

    Google Scholar 

  17. Z. Li, Y. Soh, C. Wen, Switched and Impulsive Systems: Analysis, Design and Applications (Springer, Berlin, 2005)

    Google Scholar 

  18. W. Li, E. Todorov, Iterative linearization methods for approximately optimal control and estimation of non-linear stochastic system. Int. J. Control 80(9), 1439–1453 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. D. Liberzon, Switching in Systems and Control (Springer, Boston, 2003)

    MATH  Google Scholar 

  20. H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 308–322 (2009)

    Article  MathSciNet  Google Scholar 

  21. R.W. Longman, On the theory and design of linear repetitive control systems. Eur. J. Control 16(5), 447–496 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Margaliot, J.P. Hespanha, Root-mean-square gains of switched linear systems: a variational approach. Automatica 44(9), 2398–2402 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. B. Niu, J. Zhao, Barrier Lyapunov functions for the output tracking control of constrained nonlinear switched systems. Syst. Control Lett. 62(10), 963–971 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Qiu, G. Feng, H. Gao, Fuzzy-model-based piecewise \(H_\infty \) static output feedback controller design for networked nonlinear systems. IEEE Trans. Fuzzy Syst. 18(5), 919–934 (2010)

    Article  Google Scholar 

  25. J. Qiu, G. Feng, H. Gao, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements. IEEE Trans. Fuzzy Syst. 20(6), 1046–1062 (2012)

    Article  Google Scholar 

  26. J. Qiu, G. Feng, J. Yang, A new design of delay-dependent robust \(H_\infty \) filtering for discrete-time T-S fuzzy systems with time-varying delay. IEEE Trans. Fuzzy Syst. 17(5), 1044–1058 (2009)

    Article  Google Scholar 

  27. N. Sadegh, Synthesis and stability analysis of repetitive controllers, in American Control Conference, Boston, (1991), pp. 2634–2639.

  28. J. She, L. Zhou, M. Wu, J. Zhang, Y. He, Design of a modified repetitive control system based on a continuous-discrete 2D model. Automatica 48(5), 844–850 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems. SIAM Rev. 49(4), 545–592 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Steinbuch, Repetitive control for systems with uncertain period-time. Automatica 38(12), 2103–2109 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  31. W. Sun, H. Gao, B. Yao, Adaptive robust vibration control of full-car active suspensions with electrohydraulic actuators. IEEE Trans. Control Syst. Technol. 21(6), 2417–2422 (2013)

    Article  Google Scholar 

  32. W. Sun, Y. Zhao, J. Li, L. Zhang, H. Gao, Active suspension control with frequency band constraints and actuator input delay. IEEE Trans. Ind. Electron. 59(1), 530–537 (2012)

    Article  Google Scholar 

  33. M. Wu, L. Zhou, J. She, Y. He, Design of robust output-feedback repetitive controller for class of linear systems with uncertainties. Sci. China Inform. Sci. 53(5), 1006–1015 (2010)

    Article  MathSciNet  Google Scholar 

  34. Y. Yamamoto, A function space approach to sampled data control systems and tracking problems. IEEE Trans. Autom. Control 39(4), 703–713 (1994)

    Article  MATH  Google Scholar 

  35. G. Zhai, B. Hu, K. Yasuda, Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. Int. J. Syst. Sci. 32(8), 1055–1061 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  36. W. Zhang, B.S. Chen, C.S. Tseng, Robust \(H_\infty \) filtering for nonlinear stochastic systems. IEEE Trans. Signal Process. 53(2), 589–598 (2005)

    Article  MathSciNet  Google Scholar 

  37. B. Zhang, D. Wang, K. Zhou, Linear phase lead compensation repetitive control of a CVCF PWM inverter. IEEE Trans. Ind. Electron. 55(4), 1595–1602 (2008)

    Article  Google Scholar 

  38. P. Zhao, W. Feng, Y. Kang, Stochastic input-to-state stability of switched stochastic nonlinear systems. Automatica 48(10), 2569–2576 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. X. Zhao, L. Zhang, P. Shi, Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2012)

    Article  MathSciNet  Google Scholar 

  40. X. Zhao, L. Zhang, P. Shi, Stability of switched positive linear systems with average dwell time switching. Automatica 48(6), 1132–1137 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  41. L. Zhou, J. She, M. Wu, Design of a discrete-time output-feedback based repetitive-control system. Int. J. Autom. Comput. 10(4), 343–349 (2013)

    Article  Google Scholar 

  42. L. Zhou, J. She, M. Wu, Y. He, LMI-based design method of robust modified repetitive-control systems, in Proceedings of 2010 8th IEEE International Conference on Control and Automation, Xiamen, (2010), pp. 440–445.

  43. L. Zhou, M. Wu, J. She, Design of observer-based robust repetitive-control system. IEEE Trans. Autom. Control 56(6), 1452–1457 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 61273120, the Postgraduate Innovation Project of Jiangsu Province (Grant No. CXZZ13_0208), and the Jiangsu Scientific and Technological Support Plan (Grant No. BE2012175).

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Correspondence to Zhengrong Xiang.

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Shao, Z., Huang, S. & Xiang, Z. Robust \(H_\infty \) Repetitive Control for a Class of Linear Stochastic Switched Systems with Time Delay. Circuits Syst Signal Process 34, 1363–1377 (2015). https://doi.org/10.1007/s00034-014-9905-3

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