Skip to main content
SpringerLink
Log in

Tracking performance with finite input energy

  • Conference paper
  • First Online:
Perspectives in robust control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 268))

Abstract

This paper studies an optimal control problem which is to minimize jointly the error in tracking a step reference and the energy of the plant input. We derive an analytical expression for the best attainable performance. It is found that this performance depends not only on the plant nonminimum phase zeros-a fact known previously-but also on the plant gain in the entire frequency range. The result thus reveals and quantifies another source of fundamental performance limitations beyond those already known, which are nonexistent when only conventional performance objectives such as tracking and regulation are addressed individually. It shows, among other observations, that the bandwidth as well as minimum phase zeros of the plant may all impose constraints on the achievable performance.

This research was supported in part by the NSF/USA under Grant ECS-9623228, and in part by The Grant-in-Aid for COE Research Project of Super Mechano-Systems by The Ministry of Education, Science, Sport and Culture, Japan.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Boyd and C.A. Desoer, “Subharmonic functions and performance bounds in linear time-invariant feedback systems,” IMA J. Math. Contr. and Info., vol. 2, pp. 153–170, 1985.

    Article  Google Scholar 

  2. J. Chen, “Sensitivity integral relations and design tradeoffs in linear multivariable feedback systems,” IEEE Trans. Auto. Contr., vol. AC-40, no. 10, pp. 1700–1716, Oct. 1995.

    Article  Google Scholar 

  3. J. Chen, “Multivariable gain-phase and sensitivity integral relations and design tradeoffs,” IEEE Trans. Auto. Contr., vol. 43, no. 3, March 1998, pp. 373–385.

    Article  MATH  Google Scholar 

  4. J. Chen, “Logarithmic integrals, interpolation bounds, and performance limitations in MIMO systems,” IEEE Trans. on Automatic Control, vol. 45, no. 6, June 2000, pp. 1098–1115.

    Article  MATH  Google Scholar 

  5. J. Chen, Z. Ren, S. Hara, and L. Qiu, “Optimal tracking performance: preview control and exponential signals,” Proc. 39th IEEE Conf. Decision Contr., Sydney, Australia, Dec. 2000.

    Google Scholar 

  6. J. Chen, O. Toker, and L. Qiu, “Limitations on maximal tracking accuracy,” IEEE Trans. on Automatic Control, vol. 45, no. 2, pp. 326–331, Feb. 2000.

    Article  MATH  MathSciNet  Google Scholar 

  7. B.A. Francis, A Course in H ∞ Control Theory, Lecture Notes in Control and Information Sciences, Berlin: Springer-Verlag, 1987.

    Google Scholar 

  8. J.S. Freudenberg and D.P. Looze, “Right half plane zeros and poles and design tradeoffs in feedback systems,” IEEE Trans. Auto. Contr., vol. AC-30, no. 6, June 1985, pp. 555–565

    Article  MathSciNet  Google Scholar 

  9. S. Hara and N. Naito, “Control performance limitation for electro-magnetically levitated mechanical systems,” Proc. 3rd MOVIC, Zurich, 1998, pp. 147–150.

    Google Scholar 

  10. S. Hara and H.K. Sung, “Constraints on sensitivity characteristics in linear multivariable discrete-time control systems,” Linear Algebra and its Applications, vol. 122/123/124, pp. 889–919, 1989.

    Article  MathSciNet  Google Scholar 

  11. T. Iwasaki, S. Hara, and Y. Yamauchi, “Structure/control design integration with finite frequency positive real property,” Proc. 2000 Amer. Contr. Conf., Chicago, IL, June 2000, pp. 549–553.

    Google Scholar 

  12. P.P. Khargonekar and A. Tannenbaum, “Non-Euclidean metrics and the robust stabilization of systems with parameter uncertainty,” IEEE Trans. Auto. Contr., vol. AC-30, no. 10, pp. 1005–1013, Oct. 1985.

    Article  MathSciNet  Google Scholar 

  13. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, New York, NY: Wiley-Interscience, 1972.

    MATH  Google Scholar 

  14. N. Levinson and R.M. Redheffer, Complex Variables, Baltimore: Holden-Day, 1970.

    MATH  Google Scholar 

  15. R.H. Middleton, “Trade-offs in linear control system design,” Automatica, vol. 27, no. 2, pp. 281–292, Feb. 1991.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Morari and E. Zafiriou, Robust Process Control, Englewood Cliffs, NJ: Prentice Hall, 1989.

    Google Scholar 

  17. L. Qiu and J. Chen, “Time domain performance limitations of feedback control”, in Mathematical Theory of Networks and Systems, eds. A. Beghi, L. Finesso, and G. Picci, Il Poligrafo, 1998, pp. 369–372.

    Google Scholar 

  18. L. Qiu and E.J. Davison, “Performance limitations of non-minimum phase systems in the servomechanism problem,” Automatica, vol. 29, no. 2, pp. 337–349, Feb. 1993.

    Article  MATH  MathSciNet  Google Scholar 

  19. M.M. Seron, J.H. Braslavsky, and G.C. Goodwin, Fundamental Limitations in Filtering and Control, London: Springer-Verlag, 1997.

    MATH  Google Scholar 

  20. M.M. Seron, J.H. Braslavsky, P.V. Kokotovic, and D.Q. Mayne, “Feedback limitations in nonlinear systems: from Bode integrals to cheap control,” IEEE Trans. Auto. Contr., vol. 44, no. 4, April 1999, pp. 829–833.

    Article  MATH  MathSciNet  Google Scholar 

  21. G. Zames and B.A. Francis, “Feedback, minimax sensitivity, and optimal robustness,” IEEE Trans. Auto. Contr., vol. AC-28, no. 5, pp. 585–600, May 1985.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, University of California, 92521, Riverside, CA

    Jie Chen

  2. Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Ookayama, Meguro-ku, 152-8552, Tokyo, Japan

    Shinji Hara

Authors
  1. Jie Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shinji Hara
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

S.O. Reza Moheimani BSc, MengSc, PhD

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag London Limited

About this paper

Cite this paper

Chen, J., Hara, S. (2001). Tracking performance with finite input energy. In: Moheimani, S.R. (eds) Perspectives in robust control. Lecture Notes in Control and Information Sciences, vol 268. Springer, London. https://doi.org/10.1007/BFb0110613

Download citation

  • DOI: https://doi.org/10.1007/BFb0110613

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-452-9

  • Online ISBN: 978-1-84628-576-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation