Question 8.CS.1: Antenna Control: Transient Design via Gain The main thrust o...
Antenna Control: Transient Design via Gain
The main thrust of this chapter is to demonstrate design of higher-order systems (higher than two) through gain adjustment. Specifically, we are interested in determining the value of gain required to meet transient response requirements, such as percent overshoot, settling time, and peak time. The following case study emphasizes this design procedure, using the root locus.
Given the antenna azimuth position control system shown in Appendix A2, Configuration 1, find the preamplifier gain required for 25% overshoot.
The block diagram for the system was derived in the Case Studies section in Chapter 5 and is shown in Figure 5.34(c), where G(s) = 6.63K/[s (s + 1.71) (s + 100)].
First a sketch of the root locus is made to orient the designer. The real-axis segments are between the origin and −1.71 and from −100 to infinity. The locus begins at the open-loop poles, which are all on the real axis at the origin, −1.71, and −100. The locus then moves toward the zeros at infinity by following asymptotes that, from Eqs. (8.27) and (8.28), intersect the real axis at −33.9 at angles of 60°, 180°, and −60°. A portion of the root locus is shown in Figure 8.28.
σ_{a} = \frac{\sum finite poles − \sum finite zeros }{\#finite poles − \#finite zeros} (8.27)
θ_{a} = \frac{(2k + 1)π}{\#finite poles − \#finite zeros} (8.28)
From Eq. (4.39), 25% overshoot corresponds to a damping ratio of 0.404. Now draw a radial line from the origin at an angle of cos^{-1} ζ = 113.8. The intersection of this line with the root locus locates the system’s dominant, second-order closed-loop poles. Using the root locus program discussed in Appendix H.2 at
www.wiley.com/go/Nise/ControlSystemsEngineering8e to search the radial line for 180° yields the closed-loop dominant poles as 2.063∠113.8° = − 0.833 ±j1.888. The gain value yields 6.63K = 425.7, from which K = 64.21.
ζ = \frac{− ln(\%OS/100)}{\sqrt{π² + ln²(\%OS/100)} } (4.39)
Checking our second-order assumption, the third pole must be to the left of the open-loop pole at −100 and is thus greater than five times the real part of the dominant pole pair, which is −0.833. The second-order approximation is thus valid.
The computer simulation of the closed-loop system’s step response in Figure 8.29 shows that the design requirement of 25% overshoot is met.
CHALLENGE:
You are now given a problem to test your knowledge of this chapter’s objectives. Referring to the antenna azimuth position control system shown in Appendix A2, Configuration 2, do the following:
a. Find the preamplifier gain, K, required for an 8-second settling time.
b. Repeat, using MATLAB.
