ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
SIDNEY LEES Massachusetts lnstifute o f Technology, Cambridge, Mass.
JOEL 0. HOUGEN Rensselaer P olyfechnic Institute, Troy,
N. Y.
ETHODS for testing dynamic systems are sometimes classified either as transient response or as frequency response procedures. Although the classification is often considered t o establish two independent techniques, it is recognized that the information yielded by one method is related to the other. The pulse method employs the transient response to a known input. By suitable manipulation in accordance with Fourier transformation theory, the frequency response of the system is computed from the transient data. The motivation for the pulse method is inherent in the control point concept of system performance. Process systems, like many other dynamic systems, have properties that depend on the operating conditions. It is convenient, in acquiring an understanding of t.he system performance to associate the system performance with linear differential equation forms having constant coefficients for each setting of the operating condit,ions. The association is valid when the manner by which the system restores itself after a slight perturbation from its equilibrium condition or control point may be calcula,ted from a linear differential equation. The control point concept is an extension of conventional linear theory in that the complete performance of the system is associated v i t h an equation form. The equation cocfficients are represented by a field of numbers uniquely related to the field representing the operating conditions t,hat establish the control point. For each set of operating conditions, the differential equation is linear with a particular set of constant coefficients. The pulse method is a technique for determining values for the equstion coefficients for a specified set of operating conditions exactly as prescribed by the basic concept. I n order to be applicable, the system should satisfy the folloving assumptions: 1. The system performance may be associated with a linear integro-differential equation with const,ant coefficients for small disturbances from the control point. 2 . The system performance parameters change relatively slowly for changes in the operating conditions. 3. The uncertainties associated with t,he performance of the subsystems and components may be neglected. For example, the motion of valves must be essentially free and xvithout appreciable stiction (static friction). Pulse methods for testing systems have been employed since 1948 (1, 6, 7 ) to test aircraft while in flight. The technique has been employed to test military fire control systems ( d ) , but no reports have appeared in the literature. IIougen (5) demonstrated the applicability to heat exchangers in 1953. H e s h o m d that the results of conventional frequency response methods and those obtained from the pulse method are comparable Tithin experimental error. The theory of the pulse method including suitable approximation formulas of the Fourier transformation lor obtaining the frequency response is given in chapter 25 of “Instrument Engineering” (3). 1C64
The first n-orlr in the field assumed that rectangular-shaped pulses are suitable inputs for the pulse method. EIowever, in an unpublished thesis, Coppedge and Wolf (2) have shown that the discontinuity of the input characterized by the instantaneous rise of the rectangular pulse produces saturation in a system capable of this condition. Coppedge and Wolf havc also shown that pulses must be carefully chosen with respect to shape, duration, and strength. Saturation effects may be avoided by using
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