Question 6 of 8: Identifying a process from its Bode magnitude
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exams, 07-Elec-A2 Systems & Control, December 2016, 3 hours, closed book (approved Casio/Sharp calculator plus one signed, double-sided 8.5 × 11" formula sheet; a Laplace-transform table and the standard ζ–overshoot and ζ–resonant-peak design plots are supplied). Questions 1 and 2 are compulsory; a complete paper is five questions, so a candidate chooses three of Q3–Q8. Each question is worth 20 marks. All eight are worked below so the set is a complete study resource.
Reference texts: N. S. Nise, Control Systems Engineering (7th ed., Wiley) — Routh–Hurwitz (Ch. 6), root locus (Ch. 8), steady-state error and static error constants (Ch. 7), frequency response, Bode, Nyquist, gain/phase margins (Ch. 10), lead–lag and PID design (Ch. 9–11), state space and Mason’s rule (Ch. 3, 5); K. Ogata, Modern Control Engineering (5th ed., Prentice Hall) — dominant-poles modelling and second-order correlations. All block diagrams, root loci, pole–zero maps, Bode sketches, Nyquist plots and the realization diagrams below are redrawn as inline figures.
Reading the supplied design charts. The percent-overshoot chart uses $PO=100\,e^{-\zeta\pi/\sqrt{1-\zeta^2}}$ and the second-order model is $G_m(s)=K_{dc}\,\dfrac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$. Questions 4, 6 and 7 are read from printed Bode/response plots, so those values carry normal chart-reading tolerance; the s-domain results (Q1, Q2, Q3, Q5, Q8) are exact.
Question 6 — Identifying a process from its Bode magnitude [20]
Given. a magnitude curve flat at $\approx-6$ dB below $\omega\approx0.05$, a resonant peak of $\approx33.7$ dB at $\omega\approx4$, and a final roll-off of $-40$ dB/decade. Find. $z,\ p,\ \zeta,\ \omega_n,\ K_{dc}$ and the polynomial form. Approach. read the DC level and corner slopes, place $\omega_n$ at the peak, and back out $\zeta$ from the peak height.
Reconstructed model magnitude: DC $-6$ dB, $+20$ dB/dec above the zero corner $z\approx0.1$, a resonant peak of $33.7$ dB at $\omega_n\approx4$, then $-20$ then $-40$ dB/dec past the real pole $p\approx40$.
DC gain. The low-frequency asymptote sits at $\approx-6$ dB, so $K_{dc}=10^{-6/20}=\boxed{0.50}$.
Zero corner. The curve leaves the flat region and climbs at $+20$ dB/dec from about $\omega\approx0.1$ rad/s — that is where it sits $3$ dB above the $-6$ dB asymptote, the corner-frequency signature of a real zero — so $\boxed{z\approx0.1}$ rad/s.
Pole corner. After the peak the slope is $-20$ dB/dec, steepening to the final $-40$ dB/dec above $\omega\approx40$ rad/s (real pole), so $\boxed{p\approx40}$ rad/s. (Net high-frequency slope $=+20-40-20=-40$ dB/dec, consistent with three poles and one zero.)
Natural frequency. The resonant peak is centred at $\boxed{\omega_n\approx4}$ rad/s.
Damping from the peak. The peak rises to $33.7$ dB, but what fixes $\zeta$ is its height above the local asymptote, not above $0$ dB. At $\omega_n$ the non-resonant factors give $-6+20\log_{10}(\omega_n/z)=-6+20\log_{10}(4/0.1)\approx26$ dB (the real pole at $-40$ contributes almost nothing this low). Hence $M_r\approx33.7-26=7.7$ dB $=2.43$, and from $M_r=\dfrac{1}{2\zeta\sqrt{1-\zeta^2}}$, $\boxed{\zeta\approx0.21}$ — a clear but only moderately sharp resonance. This asymptote is what makes the read sensitive: placing the zero corner a couple of octaves too high lowers it and makes the pair look far more lightly damped than it is.
Polynomial form. With $K_{dc}=0.50,\ z=0.1,\ p=40,\ \omega_n=4,\ \zeta=0.21$: numerator $K_{dc}\omega_n^2\frac{p}{z}(s+z)$ and denominator $(s^2+2\zeta\omega_n s+\omega_n^2)(s+p)$ give $$\boxed{G_m(s)=\frac{3200\,s+320}{s^3+41.68\,s^2+83.2\,s+640}.}$$ Check: $G_m(0)=320/640=0.50$ (DC), high-frequency slope $-40$ dB/dec.
Pole–zero map of $G_m$: the pair at $-0.84\pm j3.91$ ($\omega_n=4$, $\zeta=0.21$), a real pole at $-40$, and the real zero at $-0.1$.
Parameter
Value
$K_{dc}$
$0.50$ ($-6$ dB)
zero $-z$ / pole $-p$
$-0.1$ / $-40$ rad/s
$\omega_n$ / $\zeta$
$4$ rad/s / $0.21$
$G_m(s)$
$\dfrac{3200s+320}{s^3+41.68s^2+83.2s+640}$
Check: every parameter here is read graphically from Figure Q6.1. Digitising the printed curve and least-squares fitting the model form gives $K_{dc}=0.49$, $z=0.094$, $p=42$, $\omega_n=4.09$, $\zeta=0.216$; the rounded reads above reproduce the printed magnitude to better than $1.8$ dB everywhere. Exact values depend on how finely the corners and the peak are read — $\zeta$ most of all, since it follows from the peak height above the local asymptote.
