Transient Response Design via Gain Adjustment

PROBLEM: For the position control system shown in Figure 11.2, find the value of preamplifier gain, K, to yield a 9.5% overshoot in the transient response for a step input. Use only frequency response methods.

Question

Transient Response Design via Gain Adjustment

PROBLEM: For the position control system shown in Figure 11.2, find the value of preamplifier gain, K, to yield a 9.5% overshoot in the transient response for a step input. Use only frequency response methods.

Accepted Answer

We will now follow the previously described gain adjustment design
procedure.

1. Choose K = 3.6 to start the magnitude plot at 0 dB at ω = 0.1 in Figure 11.3.

2. Using Eq. (4.39),

[latex]\zeta =\frac{-\ln (\%OS/100)}{\sqrt{\pi ^{2}+\ln ^{2}(\%OS /100) } }[/latex]

a 9.5% overshoot implies ζ = 0.6 for the closed-loop dominant poles. Equation (10.73)

[latex]\Phi _{M} =90- an ^{-1} \frac{\sqrt{-2\zeta ^{2}+\sqrt {1+4\zeta ^{4} } } }{2\zeta }[/latex]

[latex]= an ^{-1}\frac{2\zeta }{\sqrt{-2\zeta ^{2}+\sqrt{1+4\zeta ^{4} } } }[/latex]

yields a 59.2° phase margin for a damping ratio of 0.6.

3. Locate on the phase plot the frequency that yields a 59.2° phase margin. This frequency is found where the phase angle is the difference between -180° and 59.2°, or - 120.8°. The value of the phase-margin frequency is 14.8 rad/s.

4. At a frequency of 14.8 rad/s on the magnitude plot, the gain is found to be -44.2 dB. This magnitude has to be raised to 0 dB to yield the required phase margin. Since the log-magnitude plot was drawn for K = 3.6, a 44.2 dB increase, or [latex]K=3.6 imes 162.2=583.9[/latex], would yield the required phase margin for 9.48% overshoot.

The gain-adjusted open-loop transfer function is

[latex]G(s)=\frac{58.390}{s(s+36)(s+100)}[/latex]

Table 11.1 summarizes a computer simulation of the gain-compensated system.

Parameter	Proposed specification	Actual value
[latex]K_{ u }[/latex]	_	16.22
Phase margin	59.2°	59.2°
Phase-margin frequency	_	14.8 rad/s
Percent overshoot	9.5	10
Peak time	_	0.18 second

Students who are using MATLAB should now run ch11p1 in Appendix B. You will learn how to use MATLAB to design a gain to meet a percent
overshoot specification using Bode plots. This exercise solves Example 11.1 using MATLAB.

Question 11.1: Transient Response Design via Gain Adjustment PROBLEM: For t...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions