9-12] SERVO SYSTEM TRANSFER FUNCTION RELATIONSHIP 503 



Laplace integral is converted to a set of impulses in order that the integral 

 may be evaluated without numerical methods. Two operations are 

 involved. The time function, or a derivative of it, is first approximated 

 with a series of broken lines. A broken-line approximation of the first 

 derivative is equivalent to fitting the time function itself with a series of 

 parabolic segments. Broken-line approximation of the acceleration curve 

 is equivalent to fitting the original time function with a series of cubic 

 segments. When the original function is based upon other than an analytic 

 function, the errors resulting from graphical differentiations tend to 

 mitigate against any increased accuracy resulting from fitting with seg- 

 ments of higher-order curves, so that a point of diminishing return is 

 reached. Once the broken-line approximation has been made, the second 

 step in the process is to differentiate the broken-line approximation twice 

 to obtain a set of impulses. The impulses are an approximation representing 

 the information within the time-function interval and are based upon the 

 tacit assumption of a smooth function at the start and end of the interval. 

 That is, immediately before and immediately after the starting and ending 

 points, the position, velocity, acceleration, and all higher order derivatives 

 of motion have the same value. When other initial and final conditions are 

 to be imposed, additional impulses (doublets, etc.) appear at these two 

 points. 



Using the Transformed Input Information. A log-magnitude 

 log-frequency plot is prepared from the mathematical expression equivalent 

 to the transformation to the frequency domain of the set of impulses. 

 Straight-line segments are drawn (having integral values of slope) tan- 

 gent to the maximum values on the log-log plot to form the asymptotic 

 equivalent. The servo-system excitation function as represented by the 

 asymptotic segments is then compared with the spectral density of the 

 extraneous noise. This comparison may lead to careful shaping of a 

 regulation frequency response in the case of an AGC design or to merely 

 emphasizing the necessity for minimizing the servo-system transmission 

 bandwidth. The excitation function is then studied in relationship to 

 possible servo open-loop transfer characteristics with a view to providing, 

 in general, gain variation with frequency matching the excitation function. 

 Closed-loop transient response requirements preclude steeper slopes 

 through unity gain than perhaps — 1| ( — 2 results in instability). 



Choice of Gain Level. Graham's^^ design procedure makes use of a 

 steady-state error series having the general form 



^' + %^ + 5^+ - (9-4) 



24R. E. Graham, "Linear Servo Theory," Be// System Tech. J. 25, 616-651 (1946) 



