Question 10.20: Proportional Control Design Using a Bode Plot Design a propo...
Proportional Control Design Using a Bode Plot
Design a proportional controller for the cart system in Example 10.13 using the Bode plot technique.
The Bode plot for the open-loop transfer function KG(s), where
G(s) = \frac{3.778}{s^2+16.883s} and K=1
is shown in Figure 10.76.
Note that the requirements are given as overshoot M_p < 10\% and rise time t_r < 0.15 \text{s}. These conditions correspond to ζ > 0.59 and ω_n > 12.33 \text{rad/s}. It can be shown that the relationship between the damping ratio and \text{PM} is
\text{PM} ≈ 100ζ (10.63)
which, for the current example, yields the requirement \text{PM} > 59°. In addition, the closedloop natural frequency ω_n is related to the closed-loop bandwidth, which is somewhat greater than the frequency when the Bode magnitude plot of KG(s) crosses −3 \text{dB}. Letting ω_c denote this crossover frequency, we have
ω_c ≤ ω_{BW} ≤ 2ω_c (10.64)
As the crossover frequency increases, so do the bandwidth and the natural frequency. Figure 10.76 shows that \text{PM} = 89.2°, which meets the requirement. However, the crossover frequency ω_c is only approximately 0.3 \text{rad/s}, which is too slow. We must adjust the value of the proportional control gain K to meet both requirements. Because the current \text{PM} is well above the requirement, we decrease the \text{PM} and pick \text{PM} = 60°. Based on the definition of \text{PM}, this implies that the frequency at which the magnitude plot crosses 0 \text{dB} should be −120°. It is observed from Figure 10.76 that the frequency corresponding to −120° is 9.7 \text{rad/s}, where the magnitude is −34 \text{dB}. To make the magnitude 0 \text{dB}, the magnitude plot should slide upward by 34 \text{dB}. This is the effect of multiplying a constant term of
10^{34/20} = 50
which is the value of the proportional control gain K.
Let us set K to be 50, which is also what was found in Example 10.16 using the root locus design method. The Bode plot of the open-loop transfer function KG(s) with the new value of K is shown in Figure 10.77. The \text{PM} is 60.1° and the crossover frequency ω_c is 12.6 \text{rad/s}. The Bode plot with K = 1 is also shown in Figure 10.77. Comparing the two magnitude plots, we find that the magnitude corresponding to K = 50 is 34 \text{dB} above the one corresponding to K = 1, as designed.
