Page 67 - Biosystems Engineering

48 Chapter Two

1. Steady state is the time after which the transient behavior of a
system in response to a step input has died out and the sys-
tem behavior no longer changes in time.
2. Settling time T is the time required for the output signal to
s
reach and remain within a given error band (e.g., 2 percent)
around the steady-state value.
3. Peak response y is the maximal value of the step response,
max
reached at the peak time T .
p
4. Rise time T is the time required for the output signal to change
r
from a predefined low value to a predefined higher value,
typically 10 and 90 percent, respectively, of the step height.
2.3.2 Magnitude and Phase of a Transfer Function
The description of a system in the frequency domain is given in terms of
the response to a sinusoidal input signal after all initial transients have
died out. Provided the system is linear, this steady-state output is a sinu-
soid of the same frequency (i.e., the forcing frequency) as the input, but
with a shift of phase and a change of amplitude. The ratio of the ampli-
tude of the output sine wave to the amplitude of the input sine wave is
usually referred to as the magnitude (or sometimes as the magnitude
ratio, amplitude ratio, or gain); the shift of phase of the output sine wave
relative to the input is simply called the phase. Magnitude and phase are
dependent both on the system transfer function and on the forcing fre-
quency but not, for a linear system, on the amplitude (because magni-
tude represents an amplitude ratio). Variation of the magnitude and the
phase with the frequency, traditionally known as the frequency response
or the harmonic response information, can be found from an arbitrary
transfer function G(s) by replacing the Laplace variable s by jω. This varia-
tion can be presented graphically by a Bode diagram or Bode plot.
Note that the Laplace variable s is actually a complex number a + jω
with a real part a and a complex part ω in radians per second. Replac-
ing s in G(s) with jω indicates that G(s) is evaluated on an imaginary
axis for Re(s) = 0.
We will show that the magnitude and phase (i.e., the ratio of the
amplitude of the steady-state output to the amplitude of the input
sine wave and the phase shift between the output and input sinu-
soids) are given by the modulus and argument of G(jω), the transfer
function where the Laplace variable s has been replaced by jω.
For a linear system, the transfer function G(s) is the ratio of two
polynomials in s, a polynomial of degree m in the numerator, and a
polynomial of degree n in the denominator. Each polynomial can be
given in factorized form:

−
−
−
(
Ns() Ks z )( s z ) ... ( s z )
Gs() = = 1 2 m (2.37)
−
−
Ds() ( sp )(ssp− ) ... ( sp )
1 2 n