Second Order Dominant Poles Model, Step Response Specifications (20 marks): 22-Elec-A2 Systems and Control, Undated paper, Q4

Question 4: Second Order Dominant Poles Model, Step Response Specifications (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.

Given. The closed-loop transfer function obtained in Question 3,

Quantity	Symbol	Value
Closed-loop TF	$G_{cl}(s)$	$\dfrac{2(s+42)}{(s+5.67)(s^2+1.33s+20.46)}$
Real pole	$p_3$	$-5.67$
Complex pair	$p_{1,2}$	$-0.665\pm j4.474$
Finite zero	$z_1$	$-42$
Step-response reads (Fig. Q4.1)	—	$y_{ss}\approx0.725$, peak $\approx1.07$, period $\approx1.40$ s
Frequency-response reads (Fig. Q4.2)	—	DC $\approx0.725$, $M_r\approx2.0$ at $\omega_r\approx4.4$ rad/s

Find. Three second-order dominant-pole models — $G_{m1}$ from the pole locations, $G_{m2}$ from the step plot, $G_{m3}$ from the magnitude plot — and a comparison of the three.

Figure S4.3 - Pole-zero map of Gcl(s). The lightly damped pair sits 8.5x closer to the jw axis than the real pole at -5.67; the zero at -42 is remote. Note the -42 zero is plotted at the axis edge.

Approach. Confirm that one complex pair dominates, extract $(\zeta,\omega_n,K_{dc})$ three independent ways, and compare. Each model has the standard form $G_m(s)=K_{dc}\,\omega_n^2/(s^2+2\zeta\omega_n s+\omega_n^2)$.

Justify the dominant-pole reduction (part 1). The quadratic factor gives $\omega_n=\sqrt{20.46}=4.523$ rad/s and $\zeta=1.33/(2\times4.523)=0.1470$, so the pair sits at $-0.665\pm j4.474$ with real part $\sigma=\zeta\omega_n=0.665$. The third pole is at $-5.67$, i.e.$$\frac{|p_3|}{\sigma}=\frac{5.67}{0.665}=8.53,$$well beyond the usual factor-of-five guideline, so its transient decays roughly $8.5$ times faster than the oscillation envelope. The only finite zero, $s=-42$, is nearly ten times further out still and barely shapes the response. The lightly damped pair therefore dominates.
Model 1 — from the exact pole locations. The DC gain must be preserved: $K_{dc}=G_{cl}(0)=\dfrac{2(42)}{5.67\times20.46}=\dfrac{84}{116.01}=0.7241$. Keeping the dominant quadratic and matching DC gain,$$\boxed{G_{m1}(s)=\frac{0.7241\times20.46}{s^2+1.33s+20.46}=\frac{14.81}{s^2+1.33s+20.46}}$$with $\zeta=0.147$, $\omega_n=4.523$ rad/s.
Model 2 — from the step response (part 2). Reading Figure Q4.1: the response settles at $y_{ss}=0.725$ (so $K_{dc}=0.725$) and peaks at $1.07$, giving$$PO=\frac{1.07-0.725}{0.725}\times100=47.6\%.$$Inverting $PO=100e^{-\zeta\pi/\sqrt{1-\zeta^2}}$,$$\zeta=\frac{-\ln(PO/100)}{\sqrt{\pi^2+\ln^2(PO/100)}}=0.2300.$$The peak-to-peak spacing of the oscillation is about $1.40$ s, so $\omega_d=2\pi/1.40=4.488$ rad/s and $\omega_n=\omega_d/\sqrt{1-\zeta^2}=4.612$ rad/s. Hence$$\boxed{G_{m2}(s)=\frac{15.42}{s^2+2.122s+21.27}}$$
Model 3 — from the closed-loop magnitude plot (part 3). Figure Q4.2 gives a DC value of $0.725$ and a resonant peak $M_r=2.0$ at $\omega_r=4.4$ rad/s. Using the chart relation $M_r/K_{dc}=1/(2\zeta\sqrt{1-\zeta^2})$ with $M_r/K_{dc}=2.0/0.725=2.759$ and taking the lightly damped root,$$\zeta=0.1844,\qquad \omega_n=\frac{\omega_r}{\sqrt{1-2\zeta^2}}=\frac{4.4}{0.9654}=4.558\ \text{rad/s}.$$Therefore$$\boxed{G_{m3}(s)=\frac{15.06}{s^2+1.681s+20.78}}$$
Compare the three (part 4). All agree closely on $\omega_n$ ($4.52$–$4.61$ rad/s, a spread under 2%) and on $K_{dc}$, but disagree on damping: $\zeta=0.147$, $0.230$ and $0.184$ respectively.

Model	Source	$K_{dc}$	$\zeta$	$\omega_n$ (rad/s)	Transfer function
$G_{m1}$	exact poles	0.7241	0.1470	4.523	$\dfrac{14.81}{s^2+1.33s+20.46}$
$G_{m2}$	step plot	0.725	0.2300	4.612	$\dfrac{15.42}{s^2+2.122s+21.27}$
$G_{m3}$	magnitude plot	0.725	0.1844	4.558	$\dfrac{15.06}{s^2+1.681s+20.78}$

Which model is most accurate? $G_{m1}$ is, because it is derived from the exact denominator of $G_{cl}(s)$ rather than from a chart reading. The other two are systematically biased upward in damping, and for a physical reason worth stating: the neglected third pole at $-5.67$ slows the initial rise, so the first overshoot of the true third-order response reaches only $1.068$ rather than the $1.18$ a pure $\zeta=0.147$ second-order system would produce. Any method that infers $\zeta$ from the observed peak (Model 2) or from the observed resonant peak (Model 3) therefore reports a damping ratio that is too large. Model 2 is furthest off because peak height is the quantity most contaminated by the third pole; Model 3 sits between the two because a resonant peak is a narrow-band measurement and is less affected. The consistent message is that the three models describe the same lightly damped $\approx4.5$ rad/s oscillation, and the differences quantify how much the “non-dominant” pole still matters.

[Figure not reproduced: Figure S4.1 - Redrawn Figure Q4.1: unit step response of Gcl(s). Peak 1.068 at t = 0.843 s, steady state 0.7241, period 1.402 s. See the official exam paper.]

[Figure not reproduced: Figure S4.2 - Redrawn Figure Q4.2: closed-loop magnitude |Gcl(jw)|. DC gain 0.724, resonant peak Mr = 1.977 at wr = 4.386 rad/s. See the official exam paper.]

Topic: Second-order dominant-pole modelling in the s-, time- and frequency domains.

Key relations: $G_m=K_{dc}\dfrac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$; $PO=100e^{-\zeta\pi/\sqrt{1-\zeta^2}}$; $\omega_d=\omega_n\sqrt{1-\zeta^2}$; $\dfrac{M_r}{K_{dc}}=\dfrac{1}{2\zeta\sqrt{1-\zeta^2}}$; $\omega_r=\omega_n\sqrt{1-2\zeta^2}$.

Why this works: When one complex pair lies far closer to the imaginary axis than every other pole and zero, its residues dominate the inverse transform, so the full response is well approximated by a second-order model with the same DC gain.

Common pitfall: Forgetting to carry $K_{dc}$ into the numerator (the model must reproduce $G_{cl}(0)$, which is $0.724$ here, not 1), and reading $\omega_r$ as if it equalled $\omega_n$ — the two differ by $\sqrt{1-2\zeta^2}$ and the resonant peak only exists for $\zeta\lt 0.707$.

Source: Nise, Control Systems Engineering, 8th ed., Ch. 4 (Time Response) and Ch. 10; Dorf & Bishop, Modern Control Systems, 13th ed., Ch. 5.

Question 4 of 8: Second Order Dominant Poles Model, Step Response Specifications (20 marks)

Question 4: Second Order Dominant Poles Model, Step Response Specifications (20 marks)