Smith's Second-Order Method in Process Dynamics
Consider an overdamped model with a step response represented by , where and are the time constant and the damping coefficient, respectively. It is possible to use the step-response data in order to obtain approximate values for and . Such a method was developed by Smith [1, 2] and uses the times at which the normalized response reaches 20% and 60% (at times and . Indeed, the ratio gives (red curve) and (blue curve). It is found that the approximate values of the time constant and the damping coefficient ( and ) match the exact values, which are selected by the user, perfectly. A separate plot (select "output") shows the exact response of the second-order process to a unit step, as well as the values of and , identified by the blue and red dots.
 C. L. Smith, Digital Computer Process Control, Scranton: Intext Educational Publishers, 1972.
 D. E. Seborg, T. F. Edgar, D. A. Mellichamp, and F. J. Doyle III, Process Dynamics and Control, 3rd ed., New York: Wiley, 2011.