Second-Order Dominant-Poles Models — s-Domain, Step, and Frequency Response: 17-Phys-B5 Systems and Control, Undated paper, Q4

Question 4: Second-Order Dominant-Poles Models — s-Domain, Step, and Frequency Response (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

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The source page's own printed denominator $(s+5.67)(s^2+1.33s+20.46)$ (page 8) does not algebraically match the cubic $s^3+17s^2+38s+96$ derived independently in Question 3, although BOTH share the identical numerator $2(s+42)$ — an internal rounding/illustration inconsistency in the source exam. This question is answered using the printed $G_{cl}(s)$ AS GIVEN, since Figures Q4.1–Q4.2 (the actual data these sub-parts read from) are generated from exactly this formula — its response matches every feature of the printed plots (peak $\approx1.07$ at $t\approx0.85$ s settling near $0.72$–$0.74$; resonant peak $\approx2.0$ near $\omega\approx4$ rad/s), so no figure-reading uncertainty is introduced.

Given. $G_{cl}(s)=\dfrac{2(s+42)}{(s+5.67)(s^2+1.33s+20.46)}$.

Find. Three second-order dominant-pole models built from three different characterizations of the same system, and a comparison.

Approach. Extract the exact poles for $G_{m1}$; simulate the exact step response and read percent-overshoot/peak-time for $G_{m2}$; simulate the exact frequency response and read the resonant peak/frequency for $G_{m3}$; compare all three against each other and against the full third-order response.

Part 1) — why dominant-poles, and $G_{m1}(s)$. The denominator factors exactly as given: a real pole at $s=-5.67$ and a complex pair from $s^2+1.33s+20.46=0$, i.e. $s=-0.665\pm j4.4744$ ($\omega_n=\sqrt{20.46}=4.5233$, $\zeta=1.33/(2\times4.5233)=0.1470$). The real pole's magnitude ($5.67$) is $8.5\times$ the complex pair's real part ($0.665$) — well past the standard $5\times$-separation rule — so the complex pair dominates the transient and a 2nd-order model built purely from these poles is justified. Matching the exact DC gain $G_{cl}(0)=2(42)/[5.67\times20.46]=0.7241$: $$\boxed{G_{m1}(s)=\frac{0.7241\times20.46}{s^2+1.33s+20.46}=\frac{14.82}{s^2+1.33s+20.46}}.$$
Part 2) — from the step response, $G_{m2}(s)$. Simulating $G_{cl}(s)$'s exact step response reproduces Figure Q4.1: peak $y_{max}=1.0678$ at $t_p=0.8430$ s, settling to $y_{ss}=0.7241$ (matching the plot's dashed reference band). $$PO=\frac{1.0678-0.7241}{0.7241}\times100\%=47.45\%.$$ $$\zeta=\frac{-\ln(PO/100)}{\sqrt{\pi^2+\ln^2(PO/100)}}=0.2309,\qquad \omega_n=\frac{\pi}{t_p\sqrt{1-\zeta^2}}=3.830\ \text{rad/s}.$$ $$\boxed{G_{m2}(s)=\frac{0.7241\times3.830^2}{s^2+2(0.2309)(3.830)s+3.830^2} =\frac{10.62}{s^2+1.769s+14.67}}.$$
Part 3) — from $|G_{cl}(j\omega)|$, $G_{m3}(s)$. The exact frequency response reproduces Figure Q4.2: DC magnitude $0.7243$, resonant peak $M_r=1.9769$ at $\omega_r=4.3866$ rad/s. Solving the standard pair $M_r=\dfrac{1}{2\zeta\sqrt{1-\zeta^2}}$, $\omega_r=\omega_n\sqrt{1-2\zeta^2}$ simultaneously: $$\zeta=0.2621,\qquad \omega_n=4.723\ \text{rad/s}.$$ $$\boxed{G_{m3}(s)=\frac{0.7241\times4.723^2}{s^2+2(0.2621)(4.723)s+4.723^2} =\frac{16.15}{s^2+2.476s+22.31}}.$$

Part 4) — comparison. Predicted transient specs from each model:

Model	$\zeta$	$\omega_n$ (rad/s)	Predicted PO	Predicted $T_s$(2%)
$G_{m1}$ (exact poles)	0.147	4.523	62.7%	6.02 s
$G_{m2}$ (step data)	0.231	3.830	47.5% (exact, by construction)	4.52 s
$G_{m3}$ (freq. data)	0.262	4.723	42.6%	3.23 s

$G_{m1}$, built from the exact dominant poles alone, OVER-predicts overshoot (62.7% vs. the true 47.5%) because it ignores the damping contribution of the third, non-dominant real pole at $-5.67$: even at an 8.5:1 separation the real pole is not entirely negligible for a precise PO estimate, only for the general shape/settling behaviour. $G_{m2}$ is exact for PO/peak-time by construction (it was fit to reproduce them) but its $\zeta,\omega_n$ still differ from the true pole location. $G_{m3}$, read from the resonant peak, is the closest of the three to the exact pole pair in $\omega_n$ but under-estimates $\zeta$. $G_{m2}$ is the most practically useful model (it reproduces the actual time-domain transient exactly, which is usually the design-relevant quantity), while $G_{m1}$ is the most theoretically correct (built from exact poles) but the least accurate for overshoot specifically.

Final results — Question 4
Item	Result
$G_{m1}(s)$ (from poles)	$14.82/(s^2+1.33s+20.46)$
$G_{m2}(s)$ (from step response)	$10.62/(s^2+1.769s+14.67)$
$G_{m3}(s)$ (from $\|G_{cl}(j\omega)\|$)	$16.15/(s^2+2.476s+22.31)$
Most accurate for overshoot	$G_{m2}$ (fit directly to the true transient)

Topic: Dominant-pole 2nd-order modelling from three independent characterizations.

Key relations: $PO=100e^{-\zeta\pi/\sqrt{1-\zeta^2}}$; $t_p=\pi/(\omega_n\sqrt{1-\zeta^2})$; $M_r=1/(2\zeta\sqrt{1-\zeta^2})$, $\omega_r=\omega_n\sqrt{1-2\zeta^2}$ (valid for $\zeta\lt0.707$); dominant-pole validity rule of thumb: non-dominant real pole at least $5\times$ farther from the imaginary axis.

Why this works: each characterization (poles, step response, frequency response) is a different experimental "view" of the same underlying system, so comparing the three $(\zeta,\omega_n)$ pairs is a genuine cross-validation exercise, not just three ways of computing the same number.

Common pitfall: assuming the dominant-pole model built from the EXACT poles is automatically the most accurate; a non-dominant pole/zero not far enough away still measurably shifts the observed transient even when the theoretical separation ratio "passes" the rule of thumb.

Source: Nise, Control Systems Engineering, Ch. 4–8 (second-order approximations, dominant poles, frequency-response correlation to time domain).

Question 4 of 8: Second-Order Dominant-Poles Models — s-Domain, Step, and Frequency Response

Question 4: Second-Order Dominant-Poles Models — s-Domain, Step, and Frequency Response (20 marks)