Second-order dominant-pole models three ways: 22-Elec-A2 Systems and Control, December 2014, Q3

Question 3 — Second-order dominant-pole models three ways [5 + 5 + 5 + 5]

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. Closed-loop $T(s)=\dfrac{100}{s^3+10s^2+25s+100}$, $T(0)=1$; the factored poles above; from Q1 the open-loop margins ($\text{PM}=28.7^\circ$, $\omega_{cp}=2.96$ rad/s); from Figure Q1.3 a resonant peak $M_r\approx2.15$ at $\omega_r\approx4$ rad/s.

Find. $G_{m1},G_{m2},G_{m3}$ and the step-response specs from the most accurate model.

Closed-loop poles: the complex pair at $-0.78\pm j3.35$ sits $10.8\times$ nearer the axis than the real pole at $-8.44$ — strong dominance.

Part 1 — $G_{m1}$ from pole locations [5]

Dominance test. $|\mathrm{Re}\,p_3|/|\mathrm{Re}\,p_{1,2}| =8.44/0.779=10.8\gt5$, so the complex pair dominates and a 2nd-order model is valid.
Model parameters. $\omega_n=|p_{1,2}|=\sqrt{0.779^2+3.352^2}=\boxed{3.442\ \text{rad/s}}$; $\zeta=\dfrac{0.779}{3.442}=\boxed{0.226}$. Preserving $T(0)=1$, $$G_{m1}(s)=\frac{11.85}{s^2+1.558s+11.85}.$$

Part 2 — $G_{m2}$ from the open-loop Bode plot [5]

Damping from PM. Inverting $\text{PM}=\arctan\dfrac{2\zeta}{\sqrt{\sqrt{1+4\zeta^4}-2\zeta^2}}$ at $\text{PM}=28.7^\circ$ gives $\boxed{\zeta\approx0.257}$.
Natural frequency from $\omega_{cp}$. $\omega_{cp}=\omega_n\sqrt{\sqrt{1+4\zeta^4}-2\zeta^2}$ with $\omega_{cp}=2.96$ gives $\omega_n\approx3.16$ rad/s, so $$G_{m2}(s)=\frac{9.99}{s^2+1.62s+9.99}.$$

Part 3 — $G_{m3}$ from the closed-loop Bode plot [5]

Damping from the resonant peak. $M_r=\dfrac{1}{2\zeta\sqrt{1-\zeta^2}}=2.15\Rightarrow\boxed{\zeta\approx0.240}$.
Natural frequency from $\omega_r$. $\omega_r=\omega_n\sqrt{1-2\zeta^2}=4$ gives $\omega_n\approx4.25$ rad/s, so $$G_{m3}(s)=\frac{18.1}{s^2+2.04s+18.1}.$$

Part 4 — Comparison & step specifications [5]

Approach. $G_{m1}$ is built from the exact poles, so it is the reference; $G_{m2}$ and $G_{m3}$ agree on $\zeta\approx0.24$–$0.26$ and $\omega_n\approx3.2$–$4.3$ — the graphical reads bracket the exact model. Use $G_{m1}$.

Step error. Type 1 loop with $T(0)=1\Rightarrow e_{ss(step\%)}=\boxed{0}$.
Rise time (0–100%). $T_r=\dfrac{\pi-\theta}{\omega_d}$ with $\theta=\arccos\zeta=76.9^\circ$ and $\omega_d=\omega_n\sqrt{1-\zeta^2}=3.35$ rad/s: $T_r=\dfrac{\pi-1.342}{3.35}=\boxed{0.537\ \text{s}}$.
Overshoot. $PO=100\,e^{-\zeta\pi/\sqrt{1-\zeta^2}}=100\,e^{-0.729}=\boxed{48\%}$ (large, because $\zeta$ is small).

Dominant-model ($G_{m1}$) unit-step response: fast rise, $\approx48\%$ overshoot, settling toward unity.

Model	$\zeta$	$\omega_n$ (rad/s)	Source
$G_{m1}$	$0.226$	$3.44$	exact poles (reference)
$G_{m2}$	$0.257$	$3.16$	open-loop PM, $\omega_{cp}$
$G_{m3}$	$0.240$	$4.25$	closed-loop $M_r$, $\omega_r$
Step specs ($G_{m1}$): $e_{ss(step\%)}=0$, $T_{rise}=0.54$ s, $PO=48\%$

Concept & Theory

Topic. Reducing a higher-order loop to a dominant second-order model.

Key relations. Dominance when the extra pole is $\ge5\times$ farther from the axis; $\text{PM}\leftrightarrow\zeta$ and $\omega_{cp}\leftrightarrow\omega_n$; $M_r=1/(2\zeta\sqrt{1-\zeta^2})$, $\omega_r=\omega_n\sqrt{1-2\zeta^2}$. (Nise Ch. 4, 10.)

Why this works. Three independent windows — pole map, open-loop phase margin, closed-loop resonance — all describe the same pair, so they must give consistent $(\zeta,\omega_n)$. Agreement to a few percent validates the reduction.

Common pitfall. Forgetting to renormalise $K_{dc}$ after dropping the far pole (the numerator must equal $\omega_n^2$ to keep DC gain 1), and reading $\omega_r$ as $\omega_n$ (they differ by $\sqrt{1-2\zeta^2}$).

Source. Nise 7e, §4.6–4.8, §10.10.