Page 176 - Servo Motors and Industrial Control Theory -

172 Appendix A

34. Using the Routh-Hurwitz stability criteria, determine the condition of stabil-
ity as a parameter of the system is changed. Usually the gain of the system is
changed. Other parameters of the system may also be variable. The power of
this technique is to determine the condition of stability as one or several param-
eters are variable.

3
s + 19 s⋅ 2 + 111 s 189 K:=0.0⋅+ +
4
s + 100 s⋅ 3 + 5800 s⋅ 2 + (194000 K ) s 1450000 K : 0.0+ d ⋅+ + p =
3
s + 150 s⋅ 2 + 17200 s 408000 K:=0.0⋅+ +
35. Using the mathematical software such as MathCad or any other mathematical
software that you are familiar with, determine the roots of the characteristic
equations given in problem 33 and hence confirm the accuracy of the Routh-
Hurwitz method.
36. Calculate the roots of the characteristic equation given below as the gain K
changes. Calculate a few points and sketch the root-locus. Determine which
root dominates the response for step input and hence determine the value of K
for which the damping ratio of the dominant root is 0.7. You may use MathCad
for this purpose.

s + 80 s⋅ 2 + 1700 s 10000 K: 0.0⋅+ + =
3
37. The following characteristic equation is a function of two parameters. Again
using the MathCad program calculate the roots of the characteristic equation as
two parameters change. Keep one parameter constant and vary the other param-
eter and find the resulting root locus. Change the first parameter and repeat the
process until you find a clear indication how the roots change as both param-
eters are changed. Determine the dominant root(s) in the step input response
and make sure that at least the system has a damping ratio 0.7.
s + 170 s⋅ 3 + 10600 s⋅ 2 + (490000 K ) s 13000000 K : 0.0+ d ⋅+ + p =
4

38. The following block diagram shows a control system with three first order lags
in cascade with unity feedback. The overall gain of the system is shown by K.
Using MathCad draw the Nyquist plot. First set the gain to unity and remember
that the Nyquist Plot is the frequency response of the open loop transfer func-
tion. Find the gain and phase margins. Determine the maximum value of gain
which gives a gain margin of 6 db and phase margin of 60°.

θ i + K 1 1 1 θ o
– 20s + 1 10s + 1 15s + 1