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Modeling and Control of Linear Two-time Scale Systems: Applied to Single-Link Flexible Manipulator

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Abstract

This paper deals with the problem of H ∞ control of linear two-time scale systems. The authors’ attention is focused on the robust regulation of the system based on a new modeling approach under the assumption of norm-boundedness of the fast dynamics. In the proposed approach, the fast dynamics are treated as a norm-bounded disturbance (dynamic uncertainty). In this view, the synthesis is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. It should be noted, however, that this scheme is significantly different from the conventional approaches of order reduction for linear two-time scale systems. Specifically, in the present work, explicitly or implicitly, all the dynamics of the system are taken into consideration. In other words, the portion that is treated as a perturbation is incorporated in the design by its maximum possible gain – in the L 2 sense – over different values of the inputs. One of the advantagesof this approach is that – unlike in the conventional approaches of the order reduction – the reduced-order system still keeps some information of the ‘deleted’ subsystem. Also, we consider the robust stability analysis and stability bound improvement of perturbed parameter (ɛ) in the two-time scale systems by using linear fractional transformations and structured singular values (μ) approach. In this direction, by introducing the parametric uncertainty and dynamic uncertainty in the two-time scale systems, we represent the system as a standard μ-interconnection framework by using linear fractional transformations, and derive a set of new stability conditions for the system in the frequency domain. The exact solution of ɛ-bound is characterized. It is shown that, in spite of the coupling between the dynamic uncertainties and certain dynamics, the designed H ∞ controller stabilizes the overall closed-loop system, in the presence of norm-bounded disturbances. To show the effectiveness of the approach, the modeling of the single-link flexible manipulator and control of the Tip-position of the manipulator by utilizing the mentioned method are presented in the case study.

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References

  1. Cannon, J.R., Schmitz, E.: Initial experiments on the end-point control of a flexible one-link robot. Int. J. Rob. Res. 3, 62–75 (1984)

    Article  Google Scholar 

  2. Chen, B.S., Lin, C.L.: On the stability bounds of singularly perturbed systems. IEEE Trans. Automat. Contr. 35(11), 1265–1270 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chiou, J.S., Kung, F.C., Li, T.H.S.: An infinite ɛ-bound stabilization design for a class of singularly perturbed systems. IEEE Trans. Circuits Syst. 46, 1507–1510 (1999)

    Article  MATH  Google Scholar 

  4. Doyle, J.C.: Analysis of feedback systems with structured uncertainties. IEE Proc. 129, Pt.D,

  5. Doyle, J.C., Glover, K., Khargonekar, P.P., Francis, B.A.: State-space solutions to standard H 2 and H ∞ control problems. IEEE Trans. Automat. Contr. 34, 831–847 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fraser, A.R., Daniel, R.W.: Perturbation Techniques for Flexible Manipulators. Kluwer (1991)

  7. Fridman, E.: Exact slow–fast decomposition of the nonlinear singularly perturbed optimal control problem. Syst. Control. Lett. 40, 121–131 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gajic, Z., Lelic, M.: Singular perturbation analysis of system order reduction via system balancing. Proc. of American Control Conference, pp. 2420–2424, 2000

  9. Gawronski, W.: Computation of H ∞ norm for flexible structures. In: Proc. Ameri. Contr. Conf., June 1993

  10. Green, M., Limebeer, D.J.N.: Linear Robust Control. Prentice Hall (1996)

  11. Karimi, H.R., Yazdanpanah, M.J.: Robust control for a class of uncertain state-delayed singularly perturbed systems. Asian J. Control. 7(2), 202–208 (2005)

    Google Scholar 

  12. Karimi, H.R., Yazdanpanah, M.J.: Robust stability and disturbance attenuation for a class of uncertain singularly perturbed systems. Int. J. Control. Autom. Syst. 3(3), 164–169 (2001)

    Google Scholar 

  13. Karimi, H.R., Yazdanpanah, M.J.: Robust stability analysis of singularly perturbed systems using the structured singular values approach. Int. J. Sci. Technol. (Iranica Scientia) 9(4), 425–432 (2002)

    MATH  Google Scholar 

  14. Karimi, H.R., Yazdanpanah, M.J.: Robust stabilization of singularly perturbed systems based on a new modeling approach. Sixth Int. Conf. Contr. Autom. Robot. Visi. (ICARCV), Dec. 2000, Singapore

  15. Khalil, H.K., Chen, F.C.: H ∞ control of two time scale systems. Syst. Control. Lett. 19(1), 35–42 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kokotovic, P.V., Khalil, R.E., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic, New York (1986)

    MATH  Google Scholar 

  17. Li, X.P., Chang, B.C.: Properties of H ∞ Riccati solutions. In: Bhattacharyya, S.P., Keel, L.H. (eds.) Control of Uncertain Dynamic Systems, pp. 77–94. CRC, San Antonio Texas (1991)

    Google Scholar 

  18. Pan, Z., Basar, T.: H ∞-optimal control for singularly perturbed systems. Part I. Perfect state measurements. Automatica 29(2), 401–423 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pan, Z., Basar, T.: H ∞-optimal control for singularly perturbed systems. Part I. Imperfect State Measurements. IEEE Trans. Automat. Contr. 39, 280–299 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Saksena, V.R., O’Reilly, J., Kokotovic P.V.: Singular perturbation and two-time-scale methods in control theory: Survey 1976–1983. Automatica 20(2), 273–293 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sanchez-Pena, R.S., Sznaier, M.: Robust Systems Theory and Applications. Wiley (1998)

  22. Shahruz, S.M., Behtash, S.: Balanced realization of singularly perturbed systems. Proc. 27th Conf. Decision and Control, pp. 1171–1172, 1988

  23. Shi, P., Dragan, V.: Asymptotic control of singularly perturbed systems with parametric uncertainties. IEEE Trans. Automat. Contr. 44, 1738–1742 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sicilano, B., Book, W.J.: A singular perturbation approach to control of lightweight flexible manipulators. Int. J. Rob. Res. 7(4), (Aug. 1988)

  25. Tan, W., Leung, T., Tu, Q.: H ∞ control for singularly perturbed systems. Automatica 34(2), 255–260 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tsai, M.C., Tsai, W.I., Sun, Y.Y.: Robustness analysis of singularly perturbed system. IEEE Proc. 30th Conf. Decis. Cont. Brighton, England, pp. 1075–1076, 1991

  27. Tuan, H.D., Hosoe, S.: On a state-space approach in robust control for singularly perturbed systems. Int. J. Control 66(3), 435–462 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. Vian, J.L., Sawan, M.E.: H ∞ control for singularly perturbed systems. Proc. 30th. IEEE Conf. Decision and Control, Brighten, U.K., pp. 1072–1074, 1991

  29. Wang, L.Y., Zhan, W.: Robust disturbance attenuation with stability for linear systems with norm-bounded nonlinear uncertainties. IEEE Trans. Automat. Contr. 41, 886–888 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  30. Yazdanpanah, M.J., Patel, R.V., Khorasani, K.: Robust regulation of a flexible-link manipulator based on a new modeling approach. Proc. Conf. Dec. Contr., San Diego, California, USA, December 1997

  31. Zhou, K., Khargonekar, P.P.: Robust stabilization of linear systems with norm-bounded time-varying uncertainty. Syst. Control. Lett. 10, 17–20 (1994)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Control & Intelligent Processing Center, School of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran, P. O. Box: 14395-515, Tehran, Iran

    H. R. Karimi & M. J. Yazdanpanah

  2. Department of Electrical and Computer Engineering, University of Western Ontario, London, Ontario, N6A 5B9, Canada

    R. V. Patel

  3. Department of Electrical and Computer Engineering, Concordia University Montreal, Quebec, H3G 1M8, Canada

    K. Khorasani

Authors
  1. H. R. Karimi
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  2. M. J. Yazdanpanah
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  3. R. V. Patel
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  4. K. Khorasani
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Correspondence to H. R. Karimi.

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Karimi, H.R., Yazdanpanah, M.J., Patel, R.V. et al. Modeling and Control of Linear Two-time Scale Systems: Applied to Single-Link Flexible Manipulator. J Intell Robot Syst 45, 235–265 (2006). https://doi.org/10.1007/s10846-006-9036-6

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  • DOI: https://doi.org/10.1007/s10846-006-9036-6

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