Time to Frequency Domain Conversion of Step Response Data A. Rajakumar and P. R. Krishnaswamy* Department of Chemical Engineering, lndian Institute of Technology, Madras 600036, India
Conversion of step response data to frequency form in t h e higher ranges of frequency, not hitherto reported, has been achieved. Theoretical analysis a n d simulation study indicate that t h e technique, although it necessitates numerical differentiation of time response data, causes significantly less error in transform amplitudes and angles. It has been found possible by this technique to identify correctly not only t h e major system lag but also lags d u e to additional subsystems. System identification, even in t h e presence of random noise, could also b e successfully carried out. T h e technique, w h e n applied to available experimental transient response data on a heat exchanger, enabled evaluation of its distributed parameter characteristics with considerably greater precision.
Process identification by direct experimentation is an area of active investigation in which many advances have been reported in recent years. A number of experimental techniques have been proposed, the conventional and proven ones used in chemical engineering practice being step testing and pulse testing, besides the classical method of sine-wave testing. Since experimental sinusoidal response testing involves laborious and time-consuming data acquisition procedure, it is not considered as a very practical approach. A more convenient way to establish process dynamics comprises converting time domain (step or pulse response) data into frequency response form. The feasibility of such a conversion of time-based data to frequency domain has been reported for step testing by Schechter and Wissler (1959), and Nyquist et al. (1963), and for pulse testing by Hougen and Walsh (1961), Hougen (1964), and Clements and Schnelle (1963). These authors and also a number of others (see, for example, Krishnaswamy and Shemilt, 1973) have applied these techniques to evaluate a variety of real and simulated chemical process systems. A cursory glance a t the frequency response results reported by them indicate that these techniques yield reliable dynamic information mostly below a dimensionless frequency UT,of 20 radians, that is, in the low-frequency region. This restriction on frequency bound is essentially due to numerical, computational, and data errors inherent in the two testing techniques. Although low-frequency results are sufficient to cover the range of w of practical interest in a number of systems, high-frequency data are, however, required to increase the reliability and accuracy of identification of certain complex systems. Typical of these are distributed parameter systems and higher order lumped parameter systems of the commonly encountered (Latour et al., 1967) form (TIS
K e - Ls l)(T,s
+
+
1)
In order to recover system dynamics at elevated frequencies it is imperative that, experimentally, the width of the input pulse or rise time of the input step (depending upon whether the forcing function used is a pulse or a step) be kept as small as possible and, computationally, errors due to data reduction procedures be kept to the minimum. To satisfy these requirements it is felt that a step forcing will be more amenable than a pulse because (i) within a short interval of time (as could possibly be dictated by the magnitude of the smallest subsystem time constant of interest) it is easier to implement a step change rather than a pulse change; (ii) even if the time 250
Ind. Eng. Chern., Process D e s . Dev., Vol. 14, No. 3, 1975
constants of interest are large enough to permit pulsing, the frequency content of the input pulse function drops off with frequency (Clements and Schnelle, 1963) restricting the upper frequency limit; and (iii) errors involved in step to frequency transformation can be significantly reduced, especially a t higher frequencies, if the procedure described here is adopted. One method of translating step response data to frequency form is based on evaluating the Fourier integral transform of the first derivative of the response function. This general idea was applied rather in a simplified manner by Guillemin (1954) and Teasdale (1955). Subsequent authors (Schechter and Wissler, 1959; Nyquist et al., 1963), however, rejected the idea of taking derivatives on the ground that error introduced by numerical differentiation might be so serious as to prevent establishing meaningful results. The main objective of this paper is to show that, contrary to this general belief, the numerical differentiation procedure as applied here to exponential type of step responses yields high-frequency results that are much more reliable and precise than the previously published results.
Theory For a linear process with an input, x ( t ) , and corresponding output, y ( t ) , the process transfer function, when all initial conditions are zero, is defined as
If the function x ( t ) is a step of magnitude A defined by
x ( t ) = 0, t
