Q8Control System

(a) Bode Magnitude Plot and Gain Crossover Frequency

Given: G(s) = 10/[s(1+5s)(1+0.25s)], already in standard Bode time-constant form, with gain K=10, and corner (break) frequencies at ω = 1/5 = 0.2 rad/s (from the 1+5s term) and ω = 1/0.25 = 4 rad/s (from the 1+0.25s term).

Magnitude plot construction: below ω=0.2 rad/s, the slope is -20 dB/decade (due to the single pole at the origin alone); between ω=0.2 and ω=4 rad/s, the slope steepens to -40 dB/decade (with the 1+5s pole now also contributing); above ω=4 rad/s, the slope further steepens to -60 dB/decade (with the 1+0.25s pole additionally contributing).

Gain crossover frequency: solving |G(jωgc)|=1 numerically:

This falls within the -40 dB/decade slope region of the plot (between the two corner frequencies 0.2 and 4 rad/s), consistent with the fact that at this frequency, the ω term and the 1+5s term are both contributing significantly to the overall roll-off, while the 1+0.25s term (with its higher corner frequency of 4 rad/s) has not yet begun contributing significant additional attenuation.

(b) Definitions of Bode Plot Terms

(i) Phase crossover frequency (ωpc): the specific frequency at which the open-loop phase angle response φ(ω) equals exactly -180°; this is the frequency at which the open-loop frequency-response phasor G(jω) points exactly in the negative-real-axis direction on a polar plot.

(ii) Gain crossover frequency (ωgc): the specific frequency at which the open-loop magnitude response |G(jω)| equals exactly 1 (equivalently, 0 dB); this is the frequency at which the open-loop frequency-response phasor G(jω) crosses the unit circle on a polar plot.

(iii) Phase margin (PM): the additional amount of phase lag that can be introduced into the system at the gain crossover frequency before the closed-loop system becomes unstable, calculated as PM = 180° + φ(ωgc) (where φ(ωgc) is the actual open-loop phase, a negative number, at the gain crossover frequency); a positive phase margin indicates a stable system, with larger positive values indicating greater relative stability margin.

(iv) Gain margin (GM): the factor (usually expressed in dB) by which the open-loop gain can be increased at the phase crossover frequency before the closed-loop system becomes unstable, calculated as GM = -20log₁₀|G(jωpc)| (in dB); a positive gain margin (dB) indicates a stable system, meaning the actual open-loop gain at the phase-crossover frequency is currently below 0 dB (unity), leaving room for additional gain increase before instability, while a negative gain margin indicates the system is already unstable at the nominal gain setting.

Computing the Actual Phase Crossover Frequency, Gain Margin and Phase Margin for This System

Having defined the four quantities in part (b), it is instructive to actually evaluate all four for the specific system given in part (a), G(s) = 10/[s(1+5s)(1+0.25s)], completing the numerical picture alongside the already-found gain crossover frequency ωgc≈1.368 rad/s. The open-loop phase is:

Phase crossover frequency: setting φ(ωpc)=-180° requires tan⁻¹(5ωpc)+tan⁻¹(0.25ωpc)=90°. Using the identity that this sum equals 90° exactly when the product of the two arguments equals 1 (i.e., 5ωpc×0.25ωpc=1, since tan⁻¹A+tan⁻¹B=90° iff AB=1 for positive A,B), gives 1.25ωpc²=1, so ωpc²=0.8, and:

Gain margin: evaluating the magnitude at ωpc=0.894 rad/s:

Converting to dB: 20log₁₀(2.51)≈7.99 dB, so:

Phase margin: evaluating the phase at the previously-found gain crossover frequency ωgc≈1.368 rad/s: φ(ωgc) = -90°-tan⁻¹(5×1.368)-tan⁻¹(0.25×1.368) = -90°-tan⁻¹(6.84)-tan⁻¹(0.342) = -90°-81.68°-18.87° ≈ -190.55°, so:

Conclusion: since both GM (-7.99 dB) and PM (-10.55°) are negative, this system, like the one in the companion (non-OR) part of this same question, is closed-loop unstable at the nominal gain K=10. This can be independently cross-checked via Routh-Hurwitz on the closed-loop characteristic equation s(1+5s)(1+0.25s)+10=0, i.e., 1.25s³+5.25s²+s+10=0: the Routh array (s³ row: 1.25, 1; s² row: 5.25, 10; s¹ row: (5.25×1-1.25×10)/5.25 = (5.25-12.5)/5.25 = -1.381; s⁰ row: 10) shows a sign change (5.25 to -1.381), confirming at least one right-half-plane closed-loop pole and independently verifying the instability found from the Bode-based margins.

Critical Gain for Stability and Design Implication

As with the companion problem in this same unit, it is useful to determine the actual maximum gain Kmax for which this pole configuration would remain stable. From the general Routh array in terms of K for 1.25s³+5.25s²+s+K=0: the s¹-row condition requires (5.25×1-1.25K)/5.25>0, i.e., 5.25>1.25K, giving K<4.2.

Since the given gain K=10 substantially exceeds Kmax=4.2 (by a factor of about 2.38), this confirms quantitatively that the system operates well beyond its stability boundary — consistent with the sizeable negative margins found above (a gain ratio of 10/4.2≈2.38 corresponds to 20log₁₀(2.38)≈7.53 dB, reasonably close to the GM magnitude of 7.99 dB found from the Bode analysis, with the small discrepancy attributable to the fact that gain margin is evaluated at the phase-crossover frequency of the actual K=10 system rather than as a simple linear gain ratio at DC). This shows that, exactly as in the companion problem, the corrective action for stabilization here is straightforward gain reduction (bringing K below 4.2, with an appropriate additional margin left for robustness) rather than requiring a full lead/lag compensator redesign, since the underlying pole locations (at s=0, s=-0.2, s=-4) are themselves all in the left-half plane and would yield a stable, well-behaved closed-loop system at a sufficiently reduced gain.

Comparison with the Companion (Non-OR) Problem in This Unit

It is instructive to directly compare this system, G(s)=10/[s(1+5s)(1+0.25s)], with the companion problem's system, G(s)=200/[s(s+1)(s+10)], since both are Type-1, third-order plants with broadly similar pole spacing (one pole at the origin plus two widely-separated real poles) yet differ substantially in their corner-frequency locations: this system's break frequencies (0.2 rad/s and 4 rad/s) are both considerably lower than the companion problem's break frequencies (1 rad/s and 10 rad/s), meaning the -40 dB/decade and -60 dB/decade roll-off regions begin at much lower frequencies here. This is precisely why, despite this system's nominal gain (K=10) being twenty times smaller than the companion problem's (K=200), both systems still end up unstable — the lower corner frequencies here mean the phase has already dropped substantially by the time even a comparatively modest gain crossover is reached, so a comparatively small nominal gain is still sufficient to push the gain crossover frequency past the phase crossover frequency, producing negative margins in both cases. This comparison reinforces the general principle that closed-loop stability depends jointly on both the loop gain and the pole/corner-frequency locations together, not on the gain value in isolation — two systems with very different nominal gains can both end up unstable, or both stable, depending on how their respective corner frequencies are arranged relative to the resulting crossover frequencies.