Page 52 - Servo Motors and Industrial Control Theory -

44 2 Feedback Control Theory Continued

where θ , θ are the output and the required position respectively. The stability of the
o
i
system can quickly be checked by Routh–Hurwitz method. The array can be built
up as shown below,
s 3 0.5 0 0
s 2 1 K 0
s −0.5K 0 0
s 0 K 0 0

There are two sign changes in the first column and it shows that the system is un-
stable for all values of K. It also shows that there are two roots in the right hand side
of the complex plane. This method is an easy method for checking the stability of
control systems but it does not show how oscillatory the system is if it is stable. The
MathCAD or other control software can be used to calculate the roots of character-
istic equation. In MathCAD program, a vector ( v) is defined in which the first ele-
ment should be the constant and with the coefficients of the characteristic equation.
Then the statement polyroots ( V) gives the roots. In this method, the gain can be
changed to see how roots move in the complex plane. The root locus can be plotted
if required. Otherwise the gain is varied so that all roots lie between the + 45 and
the − 45° lines and they are located as far as the origin possible so that the system
responds quickly with minimum amount of oscillation. If the system is unstable as
the case is in this example compensation network must be used to stabilize the sys-
tem with acceptable transient response. For this example, the MathCAD program is
used to calculate all roots for various values of K. For example, for K = 1, the vector
v is defined as

1.0
0.0
: v =
1.0
0.5
− 2.359
polyroots(v) = 0.18 − 0.903i
0.18 + 0.903i

It can be seen that there is a real root with negative value and there are two complex
values with positive real part indicating that the above characteristic with the value
of gain defined is unstable. In fact, the above process can be repeated to show that
for all values of the gain the system is unstable. The table below shows the roots for
some values of gain.

K = 0 s = –2.0 s = 0.0
2,3
1
K = 1 s = –2.4 s = 0.3 ± 1.2i
2,3
1
K = 5 s = –3.0 s = 0.5 ± 1.8i
2,3
1