Question 8 of 8: Three 2nd-Order Dominant-Pole Models — s-Domain, Open-Loop and Closed-Loop Frequency Response

Question 8: Three 2nd-Order Dominant-Pole Models — s-Domain, Open-Loop and Closed-Loop 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.

[Figure not reproduced: Figure Q8.3 — closed-loop magnitude response of the SAME system (exact frequency sweep of $100/(s^3+10s^2+25s+100)$), matching the source figure's shape: flat near unity at low $\omega$, a sharp resonant peak, then roll-off. Recomputed peak: $M_r=2.116$ V/V at $\omega_r=3.235$ rad/s. See the official exam paper.]

Given. $G(s)=100/[s(s+5)^2]$, $K_p=1$; closed-loop poles $-0.7791\pm j3.3524,\ -8.4418$ (verified: these are exactly the roots of $s^3+10s^2+25s+100=0$, the characteristic equation of this loop).

Find. 1) $G_{m1}(s)$ from the exact poles. 2) $G_{m2}(s)$ from the open-loop Bode (PM, $\omega_{cp}$). 3) $G_{m3}(s)$ from the closed-loop resonant peak ($M_r,\omega_r$). 4) Compare; use the best model for $e_{ss(step\%)},T_{rise},PO$.

Approach. Model 1 reads $\zeta,\omega_n$ straight off the given complex pair once the real pole is confirmed non-dominant. Model 2 uses the page-3 formula sheet's $\Phi_m$-vs-$\zeta$ relation with $\omega_n\approx\omega_{cp}$. Model 3 solves the given $M_r/K_{dc}$ formula together with the standard resonant-frequency relation $\omega_r=\omega_n\sqrt{1-2\zeta^2}$ simultaneously for $\zeta,\omega_n$.

1. Part 1) — model from the exact poles. The real pole $-8.4418$ is $8.4418/0.7791\approx10.8\times$ farther from the imaginary axis than the complex pair's real part — comfortably past the usual $5\times$ rule of thumb, so a dominant-pole model IS justified. With $K_{dc}=G_{cl}(0)=100/100=1$ (Type 1 open loop, exact), $$\omega_n=\sqrt{0.7791^2+3.3524^2}=\boxed{3.442\ \text{rad/s}},\qquad \zeta=\frac{0.7791}{3.442}=\boxed{0.2264},$$ $$\boxed{G_{m1}(s)=\frac{11.845}{s^2+1.558s+11.845}}.$$
2. Part 2) — model from the open-loop Bode. Reading Figure Q8.2: $\omega_{cp}=2.961$ rad/s, $PM=28.7^\circ$. Inverting the exam's own accurate $\Phi_m$-vs-$\zeta$ formula gives $\zeta=\boxed{0.257}$ (the sheet's linear shortcut, $\Phi_m\approx100\zeta$, would give $0.287$ — used only as a cross-check); taking $\omega_n\approx\omega_{cp}=\boxed{2.961\ \text{rad/s}}$ (the simplification implied by the formula sheet, which supplies no separate $\omega_{cp}$-vs-$\omega_n$ relation), $$\boxed{G_{m2}(s)=\frac{8.77}{s^2+1.52s+8.77}}.$$
3. Part 3) — model from the closed-loop resonant peak. Reading Figure Q8.3: $M_r=2.116$ V/V at $\omega_r=3.235$ rad/s (with $K_{dc}=1$, matching the plot's own low-frequency value of $1.0$). Solving $M_r=1/(2\zeta\sqrt{1-\zeta^2})$ and $\omega_r=\omega_n\sqrt{1-2\zeta^2}$ simultaneously: $$\boxed{\zeta=0.244,\qquad \omega_n=3.446\ \text{rad/s}},\qquad \boxed{G_{m3}(s)=\frac{11.87}{s^2+1.679s+11.87}}.$$
4. Part 4) — comparison and step-response estimate. Models 1 and 3 agree closely ($\zeta\approx0.23$–$0.24$, $\omega_n\approx3.44$–$3.45$ rad/s), since both are built from properties of the ACTUAL 3rd-order system (exact poles; exact closed-loop peak). Model 2 under-estimates $\omega_n$ ($2.96$ vs. $\approx3.44$) because the simple $\omega_{cp}\approx\omega_n$ substitution is a rougher approximation for this loop's moderate damping. Model 1 is the most accurate (built directly from the EXACT given poles, with no intermediate reading/approximation), so it is used for the estimate: because the loop is Type 1 with $G_{cl}(0)=1$ EXACTLY (independent of any model), $$\boxed{e_{ss(step\%)}=0\%}.$$ With $\zeta=0.2264,\omega_n=3.442$: $\theta=\cos^{-1}\zeta=1.342$ rad, $\omega_d=\omega_n\sqrt{1-\zeta^2}=3.354$ rad/s, so $$T_{rise(0-100\%)}=\frac{\pi-\theta}{\omega_d}=\boxed{0.537\ \text{s}},\qquad PO=100e^{-\zeta\pi/\sqrt{1-\zeta^2}}=\boxed{48.2\%}$$ (the dominant-pole model is a reasonable but not perfect estimate here, since the neglected real pole, while $\gtrsim10\times$ farther out, still measurably slows the rise).