Question 10.18: 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 10.12 using the Bode plot technique.
The Bode plot for the open-loop transfer function K G(s), where
G(s)=\frac{3.778}{s^{2}+16.883 s} \text { and } K=1
is shown in Figure 10.73.
Note that the requirements are given as overshoot M_{\mathrm{p}}<10 \% and rise time t_{\mathrm{r}}<0.15 \mathrm{~s}. These conditions correspond to \zeta>0.59 and \omega_{\mathrm{n}}>12.33 \mathrm{rad} / \mathrm{s}. It can be shown that the relationship between the damping ratio and \mathrm{PM} is
\mathrm{PM} \approx 100 \zeta_{\prime}, (10.63)
which, for the current example, yields the requirement \mathrm{PM}>59^{\circ}. In addition, the closed-loop natural frequency \omega_{n} is related to the closed-loop bandwidth, which is somewhat greater than the frequency when the Bode magnitude plot of K G(s) crosses -3 \mathrm{~dB}. Denote this crossover frequency as \omega_{c}, and we have
\omega_{\mathrm{C}} \leq \omega_{\mathrm{BW}} \leq 2 \omega_{\mathrm{C}} . (10.64)
The higher the crossover frequency, the higher the bandwidth and the natural frequency.
As shown in Figure 10.73, PM =89.2^{\circ}, which meets the requirement. However, the crossover frequency \omega_{c} is only approximately 0.3 \mathrm{rad} / \mathrm{s}, which is too small. We must adjust the value of the proportional control gain K to meet both requirements. Because the current \mathrm{PM} is way above the requirement, let us decrease it and pick P M=60^{\circ}. Based on the definition of P M, this implies that the frequency at which the magnitude plot crosses 0 \mathrm{~dB} should be -120^{\circ}. It is observed from Figure 10.73 that the frequency corresponding to -120^{\circ} is 9.7 \mathrm{rad} / \mathrm{s}, at which the magnitude is -34 \mathrm{~dB}. To make the magnitude 0 \mathrm{~dB}, the magnitude plot should slide upward by 34 \mathrm{~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 10.15 using the root locus design method. The Bode plot of the open-loop transfer function K G(s) with the new value of K is shown in Figure 10.74 . The PM is 60.1^{\circ} and the crossover frequency \omega_{c} is 12.6 \mathrm{rad} / \mathrm{s}. The Bode plot with K=1 is also shown in Figure 10.74. Comparing the two magnitude plots, we find the magnitude plot corresponding to K=50 to be 34 \mathrm{~dB} above the one corresponding to K=1, as designed.
