Page 275 - Dynamics and Control of Nuclear Reactors

APPENDIX D Laplace transforms and transfer functions 277

Simplifying

δYsðÞ GsðÞ
¼ (D.25)
δXsðÞ 1 GsðÞHsðÞ
The feedback is often subtracted from instead of added to the input (negative feed-
back system). In this case, the closed-loop transfer function is given by

δYsðÞ GsðÞ
¼ (D.26)
δXsðÞ 1+ GsðÞHsðÞ
Since industrial controllers usually provide a signal that is subtracted from the input,
the form of Eq. (D.26) is applicable in that case.

D.7 The convolution integral
If the input (x) and output (y) variables are related by a linear time-invariant system,
and if G(s) is the transfer function relating x(t) and y(t), then
YsðÞ ¼ GsðÞXsðÞ (D.27)
If the input is a unit impulse function, then X(s)¼1, and Y(s)¼G(s). Thus the
impulse response function, g(t), of this system is simply the inverse Laplace trans-
form of G(s). That is,
y I tðÞ gtðÞ ¼ L 1 ½ Gsðފ (D.28)
Thus, once the impulse response of a linear system is known, the response to any
other input is determined from inverting Eq. (D.27). This inversion of the product
of two Laplace transforms is called the convolution integral and is given by
ð ∞
ytðÞ ¼ L 1 ½ GsðÞXsðފ ¼ gt τÞx τðÞ dτ (D.29)
ð
0
This is a fundamental property of a linear system. By using the property that for a
physically realizable system, the impulse response is zero for time t<0, integral
(D.29) may be rewritten by changing the upper limit to the current time, t.
ð t
ð
ytðÞ ¼ gt τÞx τðÞ dτ (D.30)
0
Eq. (D.30) is the form of the convolution integral used in numerical calculations.
The convolution integral is directly applicable to multivariate state variable
equations.