Skip to main content
SpringerLink
Log in

Cheap Control Fundamental Limitations: The Linear Time-Varying Case

  • Chapter
Model Identification and Adaptive Control

  • 381 Accesses

Abstract

It is well-known that for linear time-invariant systems, the cheap control quadratic tracking error for step reference trackings has a value proportional to the reciprocal of the open loop non-minimum-phase zeros. We extend the result to linear finite dimensional time-varying operators. It is shown a particular cost function that is closely related to the cheap control quadratic tracking error cost has a value proportional to the reciprocal of the time-varying analogue of non- minimum-phase zeroes.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hara, S., B. D. O. Anderson, and H. Fujioka. Relating mathcalH2 and H∞ norm bounds for hybrid systems. Proc 33rd IEEE Conference on Decision and Control, pages 724–729, 1994.

    Google Scholar 

  2. Ananthakrishna, P., S. K. Mitra, and B. D. O. Anderson. Digital lattice and ladder block structures. Proc. 1980 Asilomar Conf. on Circuits, Systems and Computers, pages 234–238, 1980.

    Google Scholar 

  3. Pulford, G. W., R. A. Kennedy, and B. D. O. Anderson. A neural network structure for emulating decision feedback equalizers. Proc Australian Conf. on Neural Networks, pages 223–226, 1991.

    Google Scholar 

  4. Pulford, G. W., R. A. Kennedy, and B. D. O. Anderson. An upper bound on the performance of a novel feedforward perceptron equalizer. IEEE Info. Theory, 39:1923–1929, 1993.

    Article  MATH  Google Scholar 

  5. Yamamoto, Y., Madievski A. G., and B. D. O Anderson. Approximation of frequency response for sampled-data control systems. Automatica, 35:729–734, April 1999.

    MathSciNet  MATH  Google Scholar 

  6. Benvenuti, L., L. Farina, B. D. O. Anderson, and F. De Bruyne. Minimal discrete-time positive realizations of transfer functions with positive real poles. Proceedings of MTNS 98, pages 81–84, July 1998.

    Google Scholar 

  7. Ikeda, T., T.-K. Nam, T. Mita, X. Xin, and B. D. O. Anderson. Variable con-straint control of nonholomic free flying robots— mechanical design and convergence. Technical report, Tokyo Institute of Technology, 2000.

    Google Scholar 

  8. Richardson, D. W., B. D. O. Anderson, and N. K. Bose. Matrix-fraction de-scription from frequency samples. Linear Algebra and its Applications 18 pages 251–268, 1993.

    Article  MathSciNet  Google Scholar 

  9. Tsai, M. C., D.-W. Gu, I. Postlethwaite, and B. D. O. Anderson. A pseudo singular value decomposition and inner functions in super optimal control. Int J Control, 51:1119–1131, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  10. Hara, S., B. D. O. Anderson, and H. Fujioka. Relating H2 and H∞-norm bounds for sampled-data systems. IEEE Transactions on Automatic Control, 42(6):858–863, June 1997.

    Article  MathSciNet  MATH  Google Scholar 

  11. De Bruyne, F., B. D. O. Anderson, and I.D. Landau. Recursive iterative feedback tuning. Proc 38th IEEE Conference on Decision and Control 1999.

    Google Scholar 

  12. Antoulas, A. C. and B. D. O. Anderson. On the choice of inputs in identification for robust control. Automatica, 35:1009–1031, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  13. Yamamoto, Y., A. G. Madievski, and B. D. O Anderson. Computation and convergence of frequency response via fast sampling for sampled-data control systems. Proc 36th IEEE Conference on Decision and Control 3:2157–2162, 1997.

    Article  Google Scholar 

  14. Byrnes, C. I. and B. D. O. Anderson. On the stabilizability of multivariable systems by minimum order compensation. Proc 22nd IEEE Conference on Decision and Control, pages 1358–1362, 1983.

    Google Scholar 

  15. Rao, T. N., B. D. Anderson, W. New, and R. W. Newcomb. Integrated variable capacitors for large capacitance variation. Proc of the IEEE, 54(11): 1570, November 1966.

    Article  Google Scholar 

  16. loan Dore Landau , B.D.O. Anderson, and F. De Bruyne. Closed-loop output- error identification algorithms for nonlinear plants. Proc. 38th, IEEE Conf. on Decision and Control, July 1999.

    Google Scholar 

  17. Antoulas, A. C. and B. D. O. Anderson. On optimal inputs for identification: A summary. Proceedings 33rd IEEE Conf. on Dec. and Cont. pages 2886–2887, 1994.

    Google Scholar 

  18. Newcomb, R. W., and B. D. O. Anderson. The description of coloured noise in dynamical systems. Int. J. of Control, 3(3):221–227, 1966.

    MathSciNet  MATH  Google Scholar 

  19. B. D. Anderson and J. B. Moore. Time-varying version of the lemma of lya- punov. Electronic Letters, 3(7), July 1967.

    Google Scholar 

  20. B. D. Anderson and J. B. Moore. New results in linear system stability. SI AM J. Control, 7(3):398–414, August 1969.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. D. O. Anderson. From Youla-Kucera to identification, adaptive and robust control. Automatica, 34(12):1485–1505, July 1998.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. D. O. Anderson, T. S. Brinsmead, and F. De Bruyne. The Vinnicombe metric for nonlinear operators. In E. Behr, editor, Applied and Computational Control, Signals, and Circuits, 2nd Ed. Birkhauser, 2000. to appear.

    Google Scholar 

  23. B. D. O. Anderson and J. B. Moore. Linear Optimal Control Prentice-Hall International, Englewood Cliffs, NJ, 1971.

    Google Scholar 

  24. K. J. Astrom. Limitations on control system performance. Proc, European Control Conf, July 1997.

    Google Scholar 

  25. J. H. Braslavsky, M. M. Seron, D. Q. Mayne, and P. Kokotovic. Limiting performance of optimal linear filters. Automatica, 35(2): 189–199, February 1999.

    Article  MathSciNet  MATH  Google Scholar 

  26. T. S. Brinsmead and G. C. Goodwin. Cost of cheap decoupled control. Proc. Conf. Decision and Control, December 1998.

    Google Scholar 

  27. J. Chen. Logarithmic integrals, interpolation bounds, and performance limitations in MIMO systems. IEEE Trans. Auto. Control, 2000. to appear.

    Google Scholar 

  28. J. Chen, L. Qiu, and O. Toker. Limitations on maximal tracking accuracy. IEEE Trans. Auto. Control, 45(2):326–331, February 2000.

    Article  MathSciNet  MATH  Google Scholar 

  29. W. Coppel. Dichotomies in Stability Theory, volume 629 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1978.

    Google Scholar 

  30. W. Coppel . Dichotomies and lyapunov functions. Journal of Differential Equations, 52(l):58–65, March 1984.

    Article  MathSciNet  MATH  Google Scholar 

  31. G. E. Dullerud and S. Lall. A new approach for analysis and synthesis of time- varying systems. IEEE Trans. Auto Control, 44(8):1486–1497, August 1999.

    Article  MathSciNet  MATH  Google Scholar 

  32. B. A. Francis. The optimal linear quadratic time-invariant regulator with cheap control. IEEE Trans. Auto. Control, 24(4):616–621, August 1979.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. S. Freudenberg and D. P. Looze. Frequency Domain Properties of Scalar and Multivariate Feedback Systems. Springer-Verlag, Berlin, 1988.

    Book  Google Scholar 

  34. T. T. Georgiou . Differential stability and robust control of nonlinear systems. Math. Control Signals Systems, pages 289–306, 1993.

    Google Scholar 

  35. K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their L∞ error bounds. Int. J. Control, 39(6): 1115–1193, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  36. W. Hahn. Theory and Application of Lyapunov’s Direct Method. Prentice-Hall, Englewood Cliffs, NJ, 1963.

    Google Scholar 

  37. B. R. Holt and M. Morari. Design of resilient processing plants—VL The effect of right-half-plane zeros on dynamic resilience. Chemical Engineering, 40(1):59–74, 1985.

    Article  Google Scholar 

  38. R. E. Kalman . Contributions to the theory of optimal control. Boletin de la Sociedad Matematica Mexicana, pages 102–119, 1960.

    Google Scholar 

  39. H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley- Interscience, NY, 1972.

    MATH  Google Scholar 

  40. H. Kwakernaak and R. Sivan. The maximally achievable accuracy of linear optimal regulators and linear optimal filters. IEEE Trans. Auto. Control, AC- 17(l):79–86, February 1972.

    Article  MathSciNet  Google Scholar 

  41. R. H. Middleton. Tradeoffs in linear control system design. Automatica, 27(2):281–292, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  42. R. R. Mohler. Nonlinear Systems: Volume 1, Dynamics and Control Prentice Hall, NJ, 1991.

    Google Scholar 

  43. M. Morari and E. Zafirou. Robust Process Control Prentice Hall, NJ, 1989.

    Google Scholar 

  44. R. T. O’Brien, Jr. and P. A. Iglesias. Closed loop cancellations in H∞ of time-varying systems. International J. of Control, 71(1): 163–189, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  45. R. T. O’Brien, Jr. and P. A. Iglesias. Pole/zero cancellations for time-varying systems. International J. of Control, 71(1): 103–130, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  46. S. D. O’Young and B. A. Francis. Sensitivity tradeoffs for multivariable plants. IEEE Trans. Auto. Control, 30(7):625–632, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  47. M. A. Peters and P. A. Iglesias. Minimum entropy control for discrete-time time-varying systems. Automatica 33(4):591–605, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  48. L. Qiu and E. J. Davidson. Performance limitations of non-minimum phase systems in the servomechanism problem. Automatica 29(2):337–349, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  49. R. Ravi, A. M. Pascoal, and P. P. Khargonekar. Normalized coprime factoriza¬tions for linear time-varying systems. Systems and Control Letters, 18:455–465, 1992.

    Article  MathSciNet  Google Scholar 

  50. Z. Ren, L. Qiu, and J. Chen. Performance limitations in estimation. Proc. 38th IEEE Conf. on Decision and Control, pages 3204–3209, December 1999.

    Google Scholar 

  51. M. Seron, J. Braslavsky, P. Kokotovic, and D. Mayne. Feedback limitations in nonlinear systems: From Bode integrals to cheap control. IEEE Trans. Auto. Control, 44(4):829–833, April 1999.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. M. Seron, J. H. Braslavsky, and G. C. Goodwin. Fundamental Limitations in Filtering and Control. Springer, 1997.

    Google Scholar 

  53. U. Shaked. The effect of system zeros on the achievable H∞ estimation level. IEEE Trans. Auto. Control, 42(10):1469–1471, October 1997.

    Article  MathSciNet  MATH  Google Scholar 

  54. L. M. Silverman and B. D. O. Anderson. Controllability, observability and stability of linear systems. SIAM J. control, 6(1):121–130, 1968.

    Article  MathSciNet  MATH  Google Scholar 

  55. M. Vidyasagar. Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, Massachusetts, 1985.

    MATH  Google Scholar 

  56. G. Zames and B. A. Francis. Feedback, minimax sensitivity and optimal ro-bustness. IEEE Trans. Auto. Control, AC-28(5):585–601 May 1983.

    Article  MathSciNet  Google Scholar 

  57. Z. Zhang and J. S. Freudenberg. Loop transfer recovery for nonmmimum phase plants. IEEE Trans. Auto. Control, 35(5):547–553, May 1990.

    Article  MathSciNet  MATH  Google Scholar 

  58. K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control Prentice Hall, NJ, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RSISE, Australian National University, Canberra, ACT 0200, Australia

    Thomas S. Brinsmead

Authors
  1. Thomas S. Brinsmead
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, The University of Newcastle, Callaghan, 2308, NSW, Australia

    Graham Goodwin BSc, BE PhD, FIEEE, Hon FIE Aust., FTS, FAA

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag London Limited

About this chapter

Cite this chapter

Brinsmead, T.S. (2001). Cheap Control Fundamental Limitations: The Linear Time-Varying Case. In: Goodwin, G. (eds) Model Identification and Adaptive Control. Springer, London. https://doi.org/10.1007/978-1-4471-0711-8_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-0711-8_12

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-1185-6

  • Online ISBN: 978-1-4471-0711-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation