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National Advisory Committee for Aeronautics, Report - Frequency Response of Linear Systems from Transient Data

Five Methods are presented that use general correlatiue time-
response input and output data for a linear system to determine
the frequency-response function of that. system. These methods
gice an exact description of any linear system for which, such
transient data are available.

Examples are shown of application of a method to both- an
underdamped and a critically damped exact second-order system,
and to an exact first-order system with and without dead time.
Experimental data for a turbine-propeller engine showing the
response of engine speed to change in propeller-blade angle are
presented and analyzed.

A basic problem confronting the control designer is that
of determining the behavior of the controlled system under
varying conditions of operation. This question becomes
particularly acute when various system parameters, for
example, engine shaft torque or turbine-inlet temperature,
must be closely controlled in order to prevent system damage
or even failure. The problem then becomes one of matching
transient behavior of the control to that of the uncontrolled
system in order to attain a desired response of the controlled
system.

General methods do not exist for the solution of the
equations of motion of nonlinear systems. As a result, the
analysis of such systems may be impracticably difficult, if,
indeed, a solution can be found at all. Because the behavior
of many nonlinear systems may be satisfactorily approxi-
mated by_ the assumption of system linearity and because
general mathematical methods and techniques for the analy-
sis of linear systems are readily available and (in comparison
with present nonlinear methods) relatively simple to apply,
the assumption is generally made that the system being
studied is linear. The methods of this report are based on
such an assumption.

For inputs such as step or impulse functions, the inherent
system characteristics might be obtained by fitting an
equation to the output function and from it deriving the
differential equation of the system. If the input and output
functions were of any general form, fitting differential
equations to the data might- still be possible, but for systems
of inherently high order the accuracy of such a procedure
would be low.