Page 87 - Biosystems Engineering

68 Chapter Two

where the return ratio L(s) is defined as

Ls() = Gs C s H s() (2.82)
()
()
The modulus of L(s) [i.e., | (Ljω )|] is called the loop gain. Besides track-
ing performance, the influence of disturbance signals w(t) and mea-
surement noise v(t) on the output of the controlled system are also
very important:

Ss() = Ys() = 1 = 1 (2.83)
()
()
Ws() 1 + G s C s Hs() 1 + L s()
()
Ys() − Gs C s() − Gs()CCs()
Ts() = = = (2.84)
()
Vs() 1 + Gs C s H s() 1 + Ls()
()
Finally, the transfer from the reference signal R(s) into an actuator sig-
nal U(s) applied to the system is given by the following expression:

Us() = Gs() = Cs() = Cs() = Cs S s() (2.85)
c
()
+
()
Rs() Gs() 1 + Gs C s H s() 1 + Ls()
()
2.5.2 Stability
The simplified Nyquist stability criterion states that if an open-loop
system is stable then the system with a closed loop is also stable pro-
vided that the magnitude of the loop gain | (Ljω )| does not exceed 1
)]
when its phase shift arg[ (Ljω is –180°. Physically, the condition of
instability can be visualized as follows. If the reference input to a
closed-loop system is a sine wave, then the signal returning to the
error detector will have a different amplitude and phase (which are
known for each frequency of the sine wave from the Bode diagrams).
If the phase lag is 180°, then the returning signal, when inverted and
added to the reference input, will reinforce the signal. If the ampli-
tude of the returning signal is less than that of the input signal at this
phase shift [| (Ljω< 1
)| ] a steady-state condition will be reached, but if
the amplitude is greater than 1, then the amplitude will build up con-
tinuously until the system is saturated. Even if the input signal is
removed, the system will continue to oscillate.
Models of physical systems are only approximations. Some dynami-
cal phenomena are not known or are neglected. These phenomena can
cause extra gain variations and extra phase shifts. For example, in elec-
trohydraulic control systems, where the dynamics of the hydraulic group
as pump, hoses, and valves are not taken into account, these dynamic
phenomena can cause extra phase shifts and gain variations at the input
u(t) of the feedback system that are unknown by the controller because
they are not taken into account during controller design.