Page 38 - Dynamics and Control of Nuclear Reactors

3.8 Stability 29

Input x(t) + Plant y(t) Output
X(s) – G(s) Y(s)

Feedback
H(s)
FIG. 3.3
Transfer functions in a feedback configuration. Note that, in some cases, the feedback effect
may be positive.

3.7 Frequency response function
The frequency response of a linear system is defined as the sinusoidal deviations in a
system’s output resulting from a sinusoidal deviation in the system’s input. There is
an initial transient following start of a sinusoidal input followed by a sinusoidal evo-
lution of the output. The frequency response is characterized by the ratio of the out-
put’s amplitude to the input’s amplitude and the phase shift between the two
sinusoids. Appendix E addresses frequency response theory.
As shown in Appendix E, the frequency response may be calculated by substitut-
p ffiffiffiffiffiffiffi
ing s¼jω in the transfer function (where j¼ 1 and ω is the frequency in rad/s).
Performing the complex manipulation provides the real and imaginary parts of
the solution.

3.8 Stability

Stability is an issue for any dynamic system. Stability analysis methods are well-
developed for linear systems and are computationally simple. Before the advent
of modern digital computers and simulation software, formal stability analysis
methods were easier to perform than transient analysis (which would also reveal sta-
bility problems). Stability analysis methods (those other than time-domain simula-
tions) currently find little use in reactor analysis and are not addressed here. The
exception is in analyzing coupled thermal-hydraulic/neutronic instabilities in BWRs
(see Chapter 13).
Linear stability is a universal concept. That is, a stable linear system demonstrates
bounded outputs for bounded inputs. Linear stability analysis also provides a tool for
assessing the suitability of a candidate control system.
One purpose of a controller is to cancel out the effect of an input disturbance. To
emphasize this fact, the block diagram usually shows the feedback subtracted from
the input. In a system with feedback due to processes within the system, it is more