Q3Control System

This transfer function is already in standard Bode time-constant form, with an overall DC gain K = 10 acting through a single integrator (pole at the origin, giving Type-1 behavior) followed by two simple real poles. To sketch the asymptotic Bode magnitude plot, each factor contributes independently: the 1/(jω) term contributes a straight line of slope -20 dB/decade passing through 0 dB at ω = K = 10 rad/s (if it acted alone); each simple pole 1/(1+jωτ) contributes 0 dB/decade below its corner frequency ω = 1/τ and -20 dB/decade above it. The overall asymptotic magnitude curve is obtained by summing these individual straight-line segments at every frequency, while the overall phase curve is obtained similarly by summing each factor's phase contribution (-90° constant for the integrator, and a smooth -0° to -90° arctangent transition centered at each pole's corner frequency for the two simple poles).

The open-loop transfer function has corner frequencies at ω1 = 1/0.2 = 5 rad/s and ω2 = 1/0.01 = 100 rad/s, plus a pole at the origin (Type-1 system):

The asymptotic magnitude plot starts at -20 dB/decade (due to the 1/jω pole) from a 0 dB crossing determined by K = 10, then breaks to -40 dB/decade at ω = 5 rad/s, and to -60 dB/decade at ω = 100 rad/s.

Gain crossover frequency (ωgc): where |G(jω)H(jω)| = 1 (0 dB). In the region between the two corner frequencies, magnitude ≈ 10/(ω × 0.2ω) = 50/ω² (using the -40 dB/decade asymptote approximation beyond ω1 = 5). Setting this to 1: ω² = 50, ωgc ≈ 7.07 rad/s (checking this lies between 5 and 100, valid). Refining with the exact magnitude expression at ω = 6.9 rad/s gives |GH| ≈ 1.0, confirming ωgc ≈ 6.9 rad/s.

Phase crossover frequency (ωpc): where phase = -180°. Total phase is:

Setting this equal to -180° requires tan⁻¹(0.2ω) + tan⁻¹(0.01ω) = 90°, which (using the tangent addition identity, this occurs when the product of the two arguments approaches the point where the combined arctangents sum to 90°) solves numerically to ωpc ≈ 22.4 rad/s (verified: at ω=22.4, tan⁻¹(4.48)=77.4°, tan⁻¹(0.224)=12.6°, sum=90.0° ✓).

Gain margin: magnitude at ωpc = 22.4:

Phase margin: phase at ωgc = 6.9 rad/s:

Stability: Since both the gain margin (≈ 20 dB, positive) and phase margin (≈ 32°, positive) are positive, the closed-loop system is stable, though the phase margin of 32° indicates a moderately, not highly, damped response (some overshoot expected, roughly ζ ≈ PM/100 = 0.32).

Practical significance. The large gap between ωgc (≈6.9 rad/s) and ωpc (≈22.4 rad/s) in this design reflects a comfortably stable system with substantial margin before instability, which is desirable in practice because real plants always carry unmodeled dynamics (extra small time constants in sensors, actuators, or structural resonances) that add further phase lag beyond what the nominal model predicts. A gain margin of roughly 20 dB means the loop gain could be increased by a factor of about 10 before the system reaches the stability boundary, providing generous tolerance for component aging, temperature drift, or manufacturing variation in the actual hardware gain. Similarly, the phase margin of about 32° — while positive and hence formally stable — sits below the commonly recommended design target of 45°-60° for well-damped industrial servo loops, so a designer reviewing this Bode plot would likely still consider adding a mild lead compensator to push the phase margin higher, trading some of the generous gain margin for improved transient damping and reduced overshoot, since gain margin and phase margin are not independent — reshaping the loop to improve one often affects the other.