Page 175 - Servo Motors and Industrial Control Theory -

Appendix A 171

L
P
Float

h

Q

Assume that the inlet pressure is constant at 50 (bar) and the distance between
the pivot and float is 50 cm. The fluid must be controlled at h = 1 m. The cross-
2
section area of the tank is 0.25 m . The output flow rate is nonlinear function of
the fluid level. The fluid discharged at h = 1 m is 3 lit/min. The input flow rate
at fully open position (x = 2 cm) is 9 lit perminute. Draw the block diagram as
you derive the differential equation.
Assume that the command fluid level is h and the actual fluid level is h . Re-
o
i
member that although you drive the linearized equation for small variation of
variables, but the mathematical block diagram assumes no small variation and
may be used for large variation of variables. If you notice from the block dia-
gram, a negative flow rate is also possible even though in practice it has no
meaning. The only problem will be that the result does not indicate a real sys-
tem. The model is only valid from small deviation from the operating points.
After drawing the block diagram, substitute numerical values and find the
closed loop transfer function for this two inputs and single output system.
Determine the time constant and steady state gain and sketch the step input
response characteristic by finding two or three important points. Discuss the
steady error for step and ramp inputs and determine it using final value theo-
rem. Discuss what parameters reduce the steady state and increase the speed
of response. If your derivation is correct, you have a first order lag transfer
function. Also sketch the frequency response of this simple control system.
Determine the frequency band width. A frequency bandwidth is defined as the
frequency at which the amplitude ratio becomes 0.7 or—3 db.
33. Using the Routh-Hurwitz stability criteria, determine which one of the charac-
teristic equations given below represents a stable and which one represents an
unstable system. If a system is unstable how many roots are in the right hand
side of the s-plane.
s + 15 s⋅ 2 + 66 s 80 : 0.0⋅+ =
3
s + 19 s⋅ 3 + 78 s⋅ 2 − 280 s 1600 : 0.0⋅− =
4
s + 17 s⋅ 3 + 87 s⋅ 2 + 95 s 200 : 0.0⋅− =
4
s + 13 s⋅ 2 + 92 s 260 : 0.0⋅+ =
3
s − 6 s⋅ 3 + 268 s⋅ 2 + 6040 s 24000 : 0.0⋅+ =
4