Korean Chemical Engineering Research

The Korean Institute of Chemical Engineers (한국화학공학회)
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New Approach Using the Continued Fraction Expansion for the Dead Time Approximation

Continued Fraction Expansion을 이용한 Dead Time 근사의 새로운 접근

  • Received : 2012.05.14
  • Accepted : 2012.08.07
  • Published : 2012.10.01
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Abstract

Dead times appear often in describing process dynamics and raise some difficulties in simulating process dynamics or analyzing process control systems. To relieve these difficulties, it is needed to approximate the infinite dimensional dead time by the finite dimensional transfer function and, for this, the Pade approximation method is often used. For the accurate approximation of the dead time, high order Pade approximation is needed and the high order Pade approximation is not easy to memorize and is not stable numerically. We propose a method based on the continued fraction expansion that provides the same transfer functions. The method is excellent numerically as well as systematic to be memorized easily. It can be used conveniently for the process control lecture and computations.

Dead time은 공정의 동특성을 기술할 때 매우 자주 나타나는 것으로 공정의 동특성 모사 혹은 제어 시스템 분석에 많은 어려움을 준다. 이 어려움을 줄이기 위해 무한 차원의 dead time을 유한 차원의 전달함수로의 근사가 필요한데, 여기에는 Pade 근사가 자주 사용된다. Dead time의 정밀한 근사를 위해서는 고차의 Pade 근사가 필요한데, 고차의 Pade 근사식은 외우기 쉽지 않고 수치적으로 안정적이지 못하다. 이 Pade 근사와 같은 전달함수를 주지만 수치적으로 우수한 continued fraction 전개를 이용하는 방법을 제안하고자 한다. 제안하는 방법은 수치적으로 우수할 뿐만 아니라 매우 체계적이어서 쉽게 기억할 수 있어 공정제어 강의와 계산에 편리하게 이용할 수 있을 것이다.

Keywords

References

  1. Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover, New York(1965).
  2. Brezinski, C., History of Continued Fractions and PadApproximants, Springer-Verlag, New York(1980).
  3. Byrnes, C. I. and Lindquist, A., "Stability and Instability of Partial Realizations," Systems and Control Letters, 2, 99-105(1982). https://doi.org/10.1016/S0167-6911(82)80018-2
  4. Chen, C. F. and Shieh, L. S., "Continued Fraction Inversion by the Routh's Algorithm," IEEE Trans. Circuit Theory, 16(2), 197-202(1969). https://doi.org/10.1109/TCT.1969.1082925
  5. Golub, G. H. and van Loan, C. F., Matrix Computations, Johns Hopkins University Press, Baltimore(1989).
  6. Kuo, B. C., Digital Control Systems, 2nd ed., Saunders College Publishing, Orlando(1992).
  7. Lee, J., Park, H. and Sung, S. W., "Analytic Expressions of Ultimate Gains and Ultimate Periods with Phase-Optimal Approximations of Time Delays," Canadian J. Chemical Engineering, 83, 990-995(2005). https://doi.org/10.1139/v05-111
  8. Lorentzen, L., "Pade Approximation and Continued Fractions," Applied Numerical Mathematics, 60, 1364-1370(2010). https://doi.org/10.1016/j.apnum.2010.03.016
  9. Richard, J. P., "Time-delay Systems: an Overview of Some Recent Advances and open Problems," Automatica, 39, 1667-1694(2003). https://doi.org/10.1016/S0005-1098(03)00167-5
  10. Seborg, D. E., Edgar, T. F., Mellichamp, D. A. and Doyle, III, F. J., Process Dynamics and Control, 2nd ed., Wiley, New Jersey (2010).
  11. Ju, S., Kim, S. J., Byeon, J., Chund, D., Sung, S. W. and Lee, J., "A Study on the First Order Plus Time Delay Model Identification from Noisy Step Responses," Korean Chem. Eng. Res.(HWAHAK KONGHAK), 46, 949-957(2008).
  12. Byeon, J., Kim, J. S., Sung, S. W., Ryoo, W. and Lee, J., "Third Quadrant Nyquist Point for the Relay Feedbak Autotuning of PI Controllers," Korean J. Chem. Eng., 28, 342-347(2011). https://doi.org/10.1007/s11814-010-0391-4