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268 Chapter 4 Feedback Control System Characteristics

4.9 CONTROL SYSTEM CHARACTERISTICS USING CONTROL DESIGN SOFTWARE

In this section, the advantages of feedback will be illustrated with two examples. In
the first example, we will introduce feedback control to a speed tachometer system
in an effort to reject disturbances. The tachometer speed control system example
can be found in Section 4.5. The reduction in system sensitivity to process variations,
adjustment of the transient response, and reduction in steady-state error will be
demonstrated using the English Channel boring machine example of Section 4.8.

EXAMPLE 4.5 Speed control system

The open-loop block diagram description of the armature-controlled DC motor
with a load torque disturbance T d($) is shown in Figure 4.7. The values for the vari-
ous parameters (taken from Figure 4.7) are given in Table 4.3. We have two inputs to
our system, V a(s) and T (l(s). Relying on the principle of superposition, which applies
to our linear system, we consider each input separately. To investigate the effects of
disturbances on the system, we let V a(s) = 0 and consider only the disturbance
Td(s). Conversely, to investigate the response of the system to a reference input, we
let T d(s) = 0 and consider only the input V a(s).
The closed-loop speed tachometer control system block diagram is shown in
Figure 4.9. The values for K a and K t are given in Table 4.3.
If our system displays good disturbance rejection, then we expect the distur-
bance T d(s) to have a small effect on the output o){s). Consider the open-loop sys-
tem in Figure 4.11 first. We can compute the transfer function from Tj(s) to (o(s) and
evaluate the output response to a unit step disturbance (that is, T (l(s) = 1/s). The
time response to a unit step disturbance is shown in Figure 4.29(a). The script shown
in Figure 4.29(b) is used to analyze the open-loop speed tachometer system.
The open-loop transfer function (from Equation (4.26)) is
<o(s) - 1
W = 27TT^ = ^

where sys_o represents the open-loop transfer function in the script. Since our desired
value of (o(t) is zero (remember that V a(s) = 0), the steady-state error is just the final
value of (o(t), which we denote by (o 0(t) to indicate open-loop. The steady-state error,
shown on the plot in Figure 4.29(a), is approximately the value of the speed when
/ = 7 seconds. We can obtain an approximate value of the steady-state error by look-
ing at the last value in the output vector y 0, which we computed in the process of gen-
erating the plot in Figure 4.29(a). The approximate steady-state value of (o (, is
<o„(oo) « (0,,(7) = -0.66 rad/s.

The plot verifies that we have reached steady state.

Table 4.3 Tachometer Control System Parameters

R a Km J b K5 Kg K t
i n lONm/A 2kgm 2 0.5 Nms 0.1 Vs 54 lVs