Page 79 - Mechatronics with Experiments

CLOSED LOOP CONTROL 65
Let us consider the output of the system shown in Figure 2.14b.
D(s)G(s) G(s) D(s)G(s)
y(s) = r(s) + w(s) − v(s)
1 + D(s)G(s) 1 + D(s)G(s) 1 + D(s)G(s)
Ideally we would like y(s) = r(s) and no output be caused as a result of w(s), variations in
G(s), and v(s). Let us consider the effects of disturbance, dynamic process variations, and
sensor noise.
1. Disturbance effect: w(s)
G(s)
y (s) = w(s)
w
1 + D(s)G(s)
The response due to the disturbance, y (s), is desired to be small or zero if possible. If
w
we can make D(s)G(s) ≫ G(s) and D(s)G(s) ≫ 1 for the frequency range where w(s)is
significant, then the response due to disturbances in that frequency range would be small.
If the response due to disturbances is small or zero, the control system is said to have good
disturbance rejection or to be insensitive to the disturbances. Another way of stating this
result is that it is desirable to have a controller with large gain.
2. Variation of process dynamics: G(s) = G (s) +ΔG(s). Consider only the com-
0
mand signal and process dynamic variations, and let us analyze the effect of the variations
in the process dynamics on the response.
D(s)(G (s) +ΔG(s))
0
y (s) = r(s)
r
1 + D(s)(G (s) +ΔG(s))
0
D(s)G (s) D(s)ΔG(s)
0
= r(s) + r(s)
1 + D(s)(G (s) +ΔG(s)) 1 + D(s)(G (s) +ΔG(s))
0 0
The second term is the main contribution of process dynamic variations to the output. In
order to make the effect of this on the system response small, the following condition must
hold,
D(s)G(s) ≫ D(s), and D(s)G(s) ≫ 1
If the response due to changes in the process dynamics is small, the control system is called
insensitive to the variations in process dynamics which is a desired property.
So far disturbance rejection capability requires,
D(s)G(s) ≫ G(s), and D(s)G(s) ≫ 1

and insensitivity to process dynamic variations requires,
D(s)G(s) ≫ D(s), and D(s)G(s) ≫ 1
Therefore, the conditions {D(s)G(s) ≫ G(s), D(s)G(s) ≫ D(s), and D(s)G(s) ≫ 1} mean
that the loop gain must be well balanced between the controller and the process in order to
have good disturbance rejection and be insensitive to process dynamics variations.
3. Sensor noise effect: sensor noise is a problem for closed loop control systems
only. The open loop control does not need feedback sensors and therefore it does not have
a sensor noise problem. Let us consider the response, y (s), of a closed loop system due to
v
sensor noise, v(s),
D(s)G(s
y (s) =− v(s)
v
1 + D(s)G(s)
In order to make y (s) small, D(s)G(s) ≪ 1 must be, which is contradictory to the loop
v
gain requirements for the previous two properties: disturbance rejection and insensitivity to