Consider the control system discussed in the last two long paragraphs of Box 22.2. It has GH

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Consider the control system discussed in the last two long paragraphs of Box 22.2. It has G̃H̃ = −κ(1+ iz)[iz(1− iz)]−1, with z = ωτ a dimensionless frequency and τ some time constant.

(a) Show that there are no poles of D = 1+ G̃H̃ in the upper half of the complex frequency plane (z plane).

(b) Construct the Nyquist diagram for various feedback strengths κ. Show that for κ > 1 the curve encircles the origin twice (diagram a in the third figure in Box 22.2), so the control system is unstable, while for κ

(c) Show that the phase margin and gain margin, defined in diagram c in the third figure in Box 22.2, approach zero as κ increases toward the instability point, κ = 1.

(d) Compute explicitly the zeros of D = 1 + G̃H̃, and plot their trajectories in the complex frequency plane as κ increases from zero through one to ∞. Verify that two zeros enter the upper half of the frequency plane as κ increases through one, and they remain in the upper half-plane for all κ > 1, as is guaranteed by the Nyquist diagrams.


Box 22.2

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