Consider the servo system of Example 3.4 on page 47. Recall that its transfer function is given by [latex]G_{o}(s) =\frac{1}{s(s+1)}[/latex] (13.6.20) Synthesize a minimal prototype controller with sampling period Δ = 0.1[s].

From Example 13.1 on page 354,we have that the system pulse transfer function (with Δ = 0.1[s]) is given by (13.3.5),i.e. [latex]G_{oq}(z) =K \frac{z-z^{q}_{o} }{(z-1)(z-\alpha _{o})}[/latex] (13.3.5) [latex]G_{oq}(z) =0.0048 \frac{z + 0.967}{(z-1)(z-0.905)}[/latex] (13.6.21) Thus [latex]B_{oq}(z) =0.0048 (z+0.967) and \bar{A}_{oq}(z) = Z-0.905 [/latex] (13.6.22) and, using (13.6.17) [latex]C_{q}(z) = [G_{oq}(z)]^{-1} \frac{1}{z^{n-m} -1} = \frac{\bar{A}_{oq}(z) }{B_{oq}(z) } \frac{z-1}{z^{n-m} -1}[/latex] (13.6.17) [latex]C_{q}(z) = 208.33\frac{z-0.905}{z+0.967}[/latex] (13.6.23) [latex]T_{oq}(z) =\frac{1}{z}[/latex] (13.6.24) The performance of the resultant control loop is evaluated for a unit step reference at t = 0.1[s]. The plant output is shown in Figure 13.6. Again we see that the sampled response settles in one sample period. However, Figure 13.6 also verifies the main characteristics of minimal prototype control: non zero errors in the intersample response and large control magnitudes.

Question 13.5: Synthesize a minimal prototype controller with sampling peri...

Control System Design

Consider the servo system of Example 3.4 on page 47. Recall that its transfer function is given by

$G_{o}(s) =\frac{1}{s(s+1)}$ (13.6.20)

Synthesize a minimal prototype controller with sampling period Δ = 0.1[s].

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From Example 13.1 on page 354,we have that the system pulse transfer function (with Δ = 0.1[s]) is given by (13.3.5),i.e.

$G_{oq}(z) =K \frac{z-z^{q}_{o} }{(z-1)(z-\alpha _{o})}$ (13.3.5)

$G_{oq}(z) =0.0048 \frac{z + 0.967}{(z-1)(z-0.905)}$ (13.6.21)

Thus

$B_{oq}(z) =0.0048 (z+0.967) and \bar{A}_{oq}(z) = Z-0.905$ (13.6.22)

and, using (13.6.17)

$C_{q}(z) = [G_{oq}(z)]^{-1} \frac{1}{z^{n-m} -1} = \frac{\bar{A}_{oq}(z) }{B_{oq}(z) } \frac{z-1}{z^{n-m} -1}$ (13.6.17)

$C_{q}(z) = 208.33\frac{z-0.905}{z+0.967}$ (13.6.23)

$T_{oq}(z) =\frac{1}{z}$ (13.6.24)

The performance of the resultant control loop is evaluated for a unit step reference at t = 0.1[s]. The plant output is shown in Figure 13.6.

Again we see that the sampled response settles in one sample period. However, Figure 13.6 also verifies the main characteristics of minimal prototype control: non zero errors in the intersample response and large control magnitudes.