Skip to main content

Preliminary Results for Analyzing Systems with Time Delay

  • Chapter
PID Controllers for Time-Delay Systems

Part of the book series: Control Engineering ((CONTRENGIN))

Abstract

In this chapter we present an important generalization of the Hermite-Biehler Theorem due to Pontryagin for characteristic equations or systems that contain time-delay terms. Some other results for analyzing systems with time delay are also introduced and will be the basis for the work presented in the following chapters. Proofs of all results are omitted and the reader is referred to the extensive literature on the subject.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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.

5.7 Notes and References

  1. Bellman, R., and Cooke, K. L. Differential-Difference Equations, Academic Press, London, UK, 1963.

    MATH  Google Scholar 

  2. Bhattacharyya, S. P., Chapellat, H., and Keel, L. H. Robust Control: The Parametric Approach, Prentice-Hall, Upper Saddle River, NJ, 1995.

    MATH  Google Scholar 

  3. Choksy, N. H. “Time-Lag Systems—A Bibliography,” IEEE Transactions on Automatic Control, Vol. 5, pp. 66–70, 1960.

    Google Scholar 

  4. Gu, K., Kharitonov, V. L., and Chen, J. Stability of Time-Delay Systems, Birkhäuser, Boston, 2003.

    MATH  Google Scholar 

  5. Ho, K. W., and Chan, W. K. “Stability of Third-Order Control Systems with Time Lag Using Pontryagin’s Method for Zero Determination,” Proceedings of IEE, Vol. 116, No. 12, pp. 2063–2068, 1969.

    Google Scholar 

  6. Karmarkar, J. S., and Siljak, D. D. “Stability Analysis of Systems with Time Delay,” Proceedings of IEE, Vol. 117, No. 7, pp. 1421–1424, 1970.

    Google Scholar 

  7. Marshall, J. E., Gorecki, H., Korytowski, A., and Walton, K. Time-Delay Systems: Stability and Performance Criteria with Applications, Ellis Horwood, New York, 1992.

    MATH  Google Scholar 

  8. Marshall, J. E. Control of Time Delay Systems, Peter Peregrinus, London, UK, 1979.

    MATH  Google Scholar 

  9. Pontryagin, L. S. “On the Zeros of Some Elementary Transcendental Function,” American Mathematical Society Translation, Vol. 2, pp. 95–110, 1955.

    MathSciNet  Google Scholar 

  10. Silva, G. J., Datta, A., and Bhattacharyya, S. P. “Controller Design Via Padé Approximation Can Lead to Instability,” Proceedings of the 40th IEEE Conference on Decision and Control, pp. 4733–4737, December 2001.

    Google Scholar 

  11. Thompson, W. M., Vacroux, A. G., and Hoffman, C. H. “Application of Pontryagin’s Time Lag Stability Criterion to Force-Reflecting Servomechanisms,” 9th Joint Automatic Control Conference, pp. 432–443, 1968.

    Google Scholar 

  12. Walton, K., and Marshall, J. E. “Direct Method for TDS Stability Analysis,” Proceedings of IEE, Part D, Vol. 134, pp. 101–107, 1987.

    MATH  Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Boston

About this chapter

Cite this chapter

(2005). Preliminary Results for Analyzing Systems with Time Delay. In: PID Controllers for Time-Delay Systems. Control Engineering. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4423-7_5

Download citation

  • DOI: https://doi.org/10.1007/0-8176-4423-7_5

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-4266-2

  • Online ISBN: 978-0-8176-4423-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics