Page 54 - Servo Motors and Industrial Control Theory -

46 2 Feedback Control Theory Continued

Similar to the previous example, the MathCAD software is used to find the roots of
the characteristic equation for various values of k and k . The result is summarized
d
below;
K = 1
d
K = 0.2 s =− 1.9 s 2,3 =− 0.5 ± 0.5i
1
K = 1 s =− 1.55 s 2,3 =− 0.23 1.1i
±
1
K = 5 s =− 1.1 s 2,3 =− 0.4 ± 3i
1
K = 2
d
K = 1 s =− 0.64 s =− 0.4 ± 3i
1 2,3
K = 2 s =− 0.56 s =− 0.7 ± 2.6i
1 2,3
K = 5
d
K = 1 s =− 0.2 s 2,3 =− 0.9 3i
±
1
K = 5 s =− 0.2 s 2,3 =− 0.9 ± 7i
1
The value of K has been changed first so that the zero of the open loop transfer
d
function remains unchanged as K is changed. It is clear that the poles of the open
loop transfer function move towards the zero and infinity as both parameters are
changed. The response usually dominated by the roots that are nearer to the imagi-
nary axis. For small values of K , the complex roots dominate the response and the
d
system becomes oscillatory.
For large value of K , the real root becomes dominant and the response becomes
d
very slow. When K = 2, K = 2 the real part of the real root and imaginary root are
d
very close together and these values are suitable for the system. As will be shown
later, the gain of the system must be selected as large as possible so that the steady
state error becomes smaller. In this example, there are three roots and two param-
eters to change; so it is not possible to locate the three roots as it is wished on the
s-plane. In the next chapter, it will be shown that with state variable feedback it is
possible to locate the roots anywhere in the s-plane. The only limitation will be the
noise and saturation limit present in practical transducers and amplifiers.

2.8 Steady State Error

It is desirable to find the steady state error for some standard input without solving
the differential equation. The final value theorem can be used to achieve this. It
states that if the Laplace Transform of the function f( t) is F( s), then the final value
theorem states that

c Lim(f(t)) : Lim(sF(s))=
t →∞ → 0
s