Q4Control System

The Bode plot represents the open-loop frequency response G(jω)H(jω) as two separate plots versus log ω: the magnitude in decibels (20 log10|G(jω)H(jω)|) and the phase angle in degrees. Its power lies in the fact that each individual pole or zero contributes a simple, easily sketched asymptotic straight-line segment to the magnitude plot (0 dB/decade away from its corner frequency, ±20 dB/decade beyond it, for a simple real pole/zero) and a smooth arctangent-shaped transition to the phase plot; the overall magnitude and phase are then obtained simply by adding the individual contributions of every pole and zero. Gain margin and phase margin, read directly off this composite plot at the phase crossover and gain crossover frequencies respectively, give a fast, graphical assessment of relative stability without needing to solve the closed-loop characteristic equation.

Rewrite in standard Bode (time-constant) form: G(s)H(s) = 2(s+0.25)/[s²(s+1)(s+0.5)] = [2×0.25(1+s/0.25)] / [s²×1×0.5×(1+s)(1+s/0.5)] = [0.5(1+4s)] / [0.5s²(1+s)(1+2s)] = (1+4s) / [s²(1+s)(1+2s)].

So the effective gain K = 1 (0 dB at ω = 1 rad/s on the double-pole-at-origin asymptote), with corner frequencies at ω = 1/4 = 0.25 rad/s (zero, +20 dB/decade break), ω = 1 rad/s (pole, -20 dB/decade break) and ω = 1/2 = 0.5 rad/s (pole, -20 dB/decade break).

Asymptotic magnitude slope sequence (increasing ω): starts at -40 dB/decade (from the s² term) below ω = 0.25; from ω = 0.25 to 0.5, slope becomes -40+20 = -20 dB/decade (zero added); from ω = 0.5 to 1, slope becomes -20-20 = -40 dB/decade (pole at 0.5 added); above ω = 1, slope becomes -40-20 = -60 dB/decade (pole at 1 added).

Asymptotic phase slope sequence: the phase plot similarly builds up from four independent contributions: a constant -180° from the double pole at the origin (present at all frequencies), a smoothly rising +0° to +90° arctangent centered at the zero's corner frequency ω=0.25, and two independently falling 0° to -90° arctangent transitions centered at the pole corner frequencies ω=0.5 and ω=1. Because the zero's corner frequency (0.25 rad/s) is lower than both pole corner frequencies, its phase-lead contribution is already well underway (past its midpoint, contributing close to its full +90°) by the time the two poles begin contributing their phase lag around ω=0.5-1, giving the composite phase curve a local 'hump' — a partial, temporary recovery toward less negative phase — before the two poles' lag drags the total phase back down past -180° at higher frequencies. This shape (an early phase-lead bump from a low-frequency zero, followed by renewed lag from higher-frequency poles) is a classic signature of marginally-compensated Type-2 systems and is exactly why the exact phase crossover must be solved numerically rather than read off a simple asymptotic sketch.

Gain crossover frequency: the initial -40 dB/decade asymptote (for ω < 0.25) passes through 0 dB at ω = 1 (since K=1, the s² term alone crosses 0dB where ω²=1, i.e. ω=1, but this is above the first corner at 0.25 so we must check magnitude using the actual corrected asymptote). Using the exact magnitude at trial frequencies: at ω=0.66: |G|=√(1+(4×0.66)²) / [0.66²×√(1+0.66²)×√(1+(2×0.66)²)] = √(1+6.97)/[0.4356×1.198×1.640] = 2.82/0.856=3.30 (>1, i.e. >0dB, so crossover is higher). At ω=1.0: √(1+16)/[1×1.414×2.236]=4.123/3.162=1.304 (still >1). At ω=1.1: √(1+19.36)/[1.21×1.487×2.417]=4.512/4.348=1.038(~1). So gain crossover ωgc ≈ 1.1 rad/s (close to the s²-term nominal crossing, consistent with the zero at 0.25 not shifting it much since it's far below).

Phase crossover frequency: total phase = -180° (from s²) + tan⁻¹(4ω) - tan⁻¹(ω) - tan⁻¹(2ω). Setting this to -180° requires tan⁻¹(4ω) - tan⁻¹(ω) - tan⁻¹(2ω) = 0. Testing ω=0.5: tan⁻¹(2)=63.4°, tan⁻¹(0.5)=26.6°, tan⁻¹(1)=45°; 63.4-26.6-45=-8.2° (close to zero, slightly negative). Testing ω=0.6: tan⁻¹(2.4)=67.4°,tan⁻¹(0.6)=31.0°,tan⁻¹(1.2)=50.2°;67.4-31-50.2=-13.8°(worse). Testing ω=0.4: tan⁻¹(1.6)=58.0°,tan⁻¹(0.4)=21.8°,tan⁻¹(0.8)=38.7°;58-21.8-38.7=-2.5°(closer to zero). So phase crossover ωpc ≈ 0.42 rad/s (interpolating where the expression crosses zero, i.e. total phase = -180°).

Gain margin: magnitude in dB at ωpc ≈ 0.42: |G| = √(1+(4×0.42)²)/[0.42²×√(1+0.42²)×√(1+(2×0.42)²)] = √(1+2.82)/[0.1764×1.084×1.302] = 1.955/0.249 = 7.85, i.e. 20log10(7.85) ≈ 17.9 dB (positive magnitude at phase crossover means GM = -17.9 dB, i.e. negative gain margin).

Phase margin: phase at ωgc ≈ 1.1: -180° + tan⁻¹(4.4) - tan⁻¹(1.1) - tan⁻¹(2.2) = -180°+77.2°-47.7°-65.6°=-216.1°. PM = 180° + (-216.1°) = -36.1° (negative).

Stability: Both the gain margin (≈ -17.9 dB) and phase margin (≈ -36°) are negative, which conclusively indicates that the closed-loop system is unstable. This is consistent with the system being Type-2 (two poles at the origin contribute -180° phase immediately), which inherently struggles to maintain adequate phase margin unless the zero (here at ω=0.25) is positioned aggressively enough to add sufficient phase lead before the gain crosses 0 dB; in this case the poles at ω=0.5 and ω=1 add further lag that pushes the phase past -180° before crossover, confirming instability.

Design remedy. This result is instructive because it shows a Type-2 system's characteristic weakness: with two integrators, the -180° phase contribution is present at every frequency from the outset, leaving only a narrow margin (contributed by the single zero at ω=0.25 partially cancelling the lag of the two poles at ω=0.5 and ω=1) before the phase drops below -180°. Since the calculated phase margin is already negative at the current gain, simply reducing the gain K will not by itself rescue stability here in the way it often does for Type-0/Type-1 systems, because the phase crossover frequency ωpc (≈0.42 rad/s) is already below the gain crossover frequency ωgc (≈1.1 rad/s) — i.e., the magnitude is still above 0 dB at the frequency where phase has already reached -180°. The standard remedy is to add a lead compensator whose maximum phase boost is placed near ωgc to pull the phase back above -180° before the magnitude falls to 0 dB, effectively increasing the phase margin to a positive, adequate value (typically 30°-60°) without needing to reduce the loop gain (and hence without sacrificing the excellent steady-state accuracy that this Type-2 configuration otherwise provides for both step and ramp inputs).

Cross-check via Nyquist reasoning. The conclusion of instability can be independently sanity-checked using the encirclement interpretation of the Nyquist criterion: with P = 0 open-loop right-half-plane poles (both poles at the origin lie on, not inside, the imaginary axis, and are handled by the standard indentation), stability requires zero clockwise encirclements of the (-1, j0) point. A negative gain margin at the phase-crossover frequency (magnitude greater than 0 dB, i.e. greater than unity, exactly where the phase first reaches -180°) means the open-loop locus crosses the negative real axis to the left of the critical point (-1,j0) before its magnitude has decayed below unity — geometrically, this is precisely the condition under which the Nyquist plot does encircle (-1, j0), giving N > 0 and hence Z = N + P > 0 closed-loop right-half-plane poles, consistent with the instability already established from the Bode-domain margins.