Second-order dominant-pole models: 22-Elec-A2 Systems and Control, December 2016, Q4

Question 4 — Second-order dominant-pole models [20]

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.

Figure Q4.1 — unity-feedback loop, process $150/[s(s^3+15s^2+62s+72)]$.

Item (1) — dominant-pole model $G_{m1}$

Given. factored closed-loop poles $-8.35,\,-6.07$ and the pair $-0.29\pm j1.70$. Find. the appropriate second-order model.

$G_{cl}$ pole map: the pair $-0.29\pm j1.70$ lies $\sim21\times$ closer to the jw-axis than the real poles $-6.07,-8.35$, so it dominates the response.

Identify the dominant pair. Real part of the pair is $-0.29$; the real poles at $-6.07,-8.35$ are more than $20\times$ further left, so they decay far faster and are negligible. Keep the pair $s^2+0.58s+2.96$.
Match DC gain. $G_{cl}(0)=150/150=1$, so scale the numerator to keep unity DC gain: $$\boxed{G_{m1}(s)=\frac{2.96}{s^2+0.58s+2.96}}.$$
Parameters. $\omega_{n1}=\sqrt{2.96}=1.72$ rad/s, $\zeta_1=\dfrac{0.58}{2(1.72)}=0.169$, giving $PO\approx58\%$.

Items (2)–(3) — models from the frequency responses

[Figure not reproduced: Figure Q4.2 redrawn. Closed loop (dashed) is flat at $0$ dB with a resonant peak $\approx9$ dB near $\omega_r\approx1.67$; open loop (solid) rises at low $\omega$ (integrator) and crosses $0$ dB at $\omega_{cp}\approx1.53$. See the official exam paper.]

$G_{m2}$ from the open-loop response. Read gain crossover $\omega_{cp}\approx1.53$ rad/s and phase margin $\Phi_m\approx22^\circ$. Using $\zeta\approx\Phi_m/100=0.22$ and $\omega_n=\omega_{cp}/\sqrt{\sqrt{1+4\zeta^4}-2\zeta^2}\approx1.60$: $$G_{m2}(s)=\frac{2.57}{s^2+0.71s+2.57},\quad \zeta_2=0.22,\ \omega_{n2}=1.60.$$
$G_{m3}$ from the closed-loop response. Read resonant peak $M_r\approx9.1$ dB $=2.85$ at $\omega_r\approx1.67$. From $M_r=\dfrac{1}{2\zeta\sqrt{1-\zeta^2}}$ get $\zeta_3\approx0.178$, and $\omega_{n3}=\dfrac{\omega_r}{\sqrt{1-2\zeta^2}}\approx1.72$: $$G_{m3}(s)=\frac{2.96}{s^2+0.61s+2.96},\quad \zeta_3=0.178,\ \omega_{n3}=1.72.$$
Contrast. $G_{m1}$ and $G_{m3}$ nearly coincide ($\omega_n=1.72$, $\zeta\approx0.17$) because the closed-loop resonance is set by the same dominant pair. $G_{m2}$ (from the open-loop margin) gives a slightly higher damping ($\zeta=0.22$) and lower $\omega_n$, because the phase-margin correlation is only approximate and the two neglected real poles still bleed a little phase near crossover. All three predict a lightly-damped, oscillatory step response of comparable speed.

Model	$\omega_n$	$\zeta$	Source
$G_{m1}$	1.72	0.169	s-domain dominant pair
$G_{m2}$	1.60	0.22	open-loop $\Phi_m,\omega_{cp}$
$G_{m3}$	1.72	0.178	closed-loop $M_r,\omega_r$

Check: $G_{m2}$ and $G_{m3}$ rest on graphical reads ($\omega_{cp},\Phi_m,M_r,\omega_r$) from Figure Q4.2; the numbers above are the exact model values the printed curves correspond to, quoted to chart-reading tolerance.

Topic. Reducing a high-order loop to a dominant second-order model, from poles or from frequency response.

Key relations. dominance ratio ($\ge5\times$ real-part separation); $\zeta\approx\Phi_m/100$; $M_r=1/(2\zeta\sqrt{1-\zeta^2})$; $\omega_r=\omega_n\sqrt{1-2\zeta^2}$.

Why this works. Poles far into the LHP contribute fast-decaying modes that vanish long before the dominant pair settles, so the transient is governed by the pair alone; the frequency-domain correlations then let you recover $\zeta,\omega_n$ from a margin or a resonant peak without factoring the polynomial.

Common pitfall. Applying $\omega_r\approx\omega_n$ blindly — for very light damping $\omega_r\lt\omega_n$; and the $\zeta\approx\Phi_m/100$ rule degrades above $\zeta\approx0.6$. Source. Ogata Ch. 7–8; Nise Ch. 4 & 10.

Question 4 of 8: Second-order dominant-pole models

Question 4 — Second-order dominant-pole models [20]

Item (1) — dominant-pole model $G_{m1}$

Items (2)–(3) — models from the frequency responses