Page 189 - Servo Motors and Industrial Control Theory -

186 Appendix B

2. Repeat problem 1 for ω = 10 and ξ = 2 and discuss the difference between the
n
two problems.
3. A system has the following third order lag transfer function,
θ 10
o :=
θ i s + 25s + 600s 2500+
3
2
a. Using MathCAD or other mathematical software determine the roots of the
characteristic equation.
b. Write the system transfer function in state space form.
c. Determine the eigenvalues of the system matrix and show that they are the
same as roots of the characteristic equation.
d. Determine the dynamic matrix of the system and using the symbolic expan-
sion facility of the MathCAD software determine the determinant of the
dynamic matrix and show that it is the same as the characteristic equation of
the transfer function.
4. A system is governed by the following transfer function,

+
2
θ o := (s + 12s 32 )
θ s + 110s + 13500s 12500
+
3
2
i
a. Write the transfer function in state space form.
b. Determine the eigenvalues of the system and show that they are the same as
the roots of the characteristic equation.
c. The effect of the numerator on the transient and steady state behavior is very
complex but there are indications that the numerator does not affect the fre-
quency of oscillation but it only changes the overshoot and undershoot behav-
ior of the system. Discuss this point on the system state equation.
5. Consider a simple mass—spring—damper system as shown below,

F

x M

K C

This problem is extensively studied in vibration and classical feedback control
theory.