선형시불변 단일입출력 시스템의 영점과 계단응답 특성

Zeros and Step Response αlaracteristics in LTI SISO Systems

  • 이상용 (인하대학교 전기공학과)
  • 발행 : 2009.08.01


This paper deals with the relationship between zeros and step response of the second and third order LTI(Linear Time Invariant) SISO(Single-Input and Single-Output) systems. As well known, if a system has a single unstable zero, it shows the step response with undershoot and, on the other hand, a stable zero slower than the dominant pole causes the system to have the step response with overshoot. Generally, in the case of a system with two unstable real zeros, it is known to have B type undershoot[7]. But there are many complex cases of the step response extrema corresponding to zeros location in third order systems. This paper investigates the whole cases depending on DC gains of the additive equivalence systems and they are to be classified by the region of zeros which are related to the shape of the step response. Moreover, monotone nondecreasing conditions are proposed in the case of complex conjugate zeros as well as the case of two stable zeros.


  1. B. M. Kwon, M. E. Lee, and O. K. Kwon, "On nonovershooting or Monotone nondecreasing step response of second-order systems," Transaction on Control, Automation, and Systems Engineering, vol. 4, no. 4, pp. 283-288, Dec. 2002
  2. B. Kouvaritakis and A. G. J. Macfarlane, "Geometric approach to analysis and synthesis of system zeros: Part l. Square systems," International Journal of Control, vol. 23, no. 2, pp. 149-166, 1976
  3. B. C. Kuo, Automatic Control Systems 7th Ed., Prentice Hall, 1995
  4. G. C. Goodwin, S. F. Graebe, and M. E. Salgado, Control System Design, Prentice Hall, 2001
  5. M. Vidyasagar, "On undershooting and nonminimum phase zeros," IEEE Transactions on Automatic Control, vol. AC-31, no. 5, p. 440, May 1986
  6. K. Iinoya and R. J. Altpeter, "Inverse response in process control," Industrial and Engineering Chemistry, vol. 54, no. 7, pp. 39-43, July 1962
  7. T. Mita and H. Yoshida, "Undershooting phenomenon and its control in linear multivariable servomechanisms," IEEE Transactions on Automatic Control, vol. AC-26, no. 2, pp. 402-407, Apr. 1981
  8. M. El-khoury, O. D. Crisalle, and R. Longchatnp, "Influence of zero locations on the number of step-response extrema," Automatica, vol. 29, no. 6, pp. 1571-1574, 1993