Question 5 of 8: Lead Controller Design in Frequency Domain
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 17-Phys-B5 Systems and Control, National Exam, May
2019 — a three-hour closed-book examination with one double-sided handwritten formula/notes
sheet permitted and an approved calculator. Questions 1 and 2 are compulsory; candidates choose
three of the remaining six (3–8). Every question is nonetheless answered in full below so
the paper remains a complete study resource. All eight questions carry equal value (20 marks
each).
Reference texts. N. S. Nise, Control Systems Engineering, 7th ed.
(transient-response specifications, root locus, Routh–Hurwitz, frequency-response and lead
compensator design, state-space representation, steady-state errors); K. Ogata, Modern
Control Engineering, 5th ed. (root-locus construction, PID/rate-feedback compensator design,
controllability/observability, Mason's Gain Formula on signal-flow graphs).
Question 5: Lead Controller Design in Frequency Domain
(20 marks)
Given. $G(s)=100(s+0.8)/[(s+0.5)(s+1)^2(s+15)]$; standard lead form
$G_c(s)=K_c(\tau s+1)/(\alpha\tau s+1)$, $\alpha\lt1$ (zero at $-1/\tau$, pole at $-1/(\alpha\tau)$
further out — this is the correct reading of the source formula; see Verify note).
Find. $\phi_{m,u}$, $\omega_{gc,u}$ and estimated uncompensated response;
$K_{pos,u}$, $K_{pos,c}$, $K_c$; the lead parameters $\tau,\alpha$ and $G_c(s)$.
Approach. Read the uncompensated Bode data analytically from $G(s)$ (the printed Figure Q5.1 is not needed once $G(s)$ is known exactly); set $K_c$ from the steady-state
error spec; pick a target $(\zeta,\omega_n)$ from the PO/$T_{settle}$/$T_{rise}$ specs and design
the lead network to supply the missing phase at that crossover.
Check
The printed lead-controller equation
on the source page reads $G_c(s)=K_c(\tau s+1)/(\alpha\tau s+1)$ — zero at $-1/\tau$ closer
to the origin than the pole at $-1/(\alpha\tau)$ ($\alpha\lt1$) — the STANDARD textbook lead
form (Nise Ch. 9, Ogata Ch. 10). A literal $(\alpha\tau s+1)/(\tau s+1)$ reading inverts zero and pole into a LAG network and drives the closed loop unstable
at a much lower crossover; the standard form used here gives a fully stable, correctly-behaving
design (confirmed below by simulation).
Part a) — uncompensated system. $|G(j\omega)|=1$ (0 dB) at
$\omega_{gc,u}=2.4066$ rad/s, where $\angle G(j\omega_{gc,u})=-150.6^\circ$, so
$\phi_{m,u}=29.36^\circ$. Using the exact phase-margin–damping relation
$\phi_m=\tan^{-1}\!\left(2\zeta/\sqrt{\sqrt{1+4\zeta^4}-2\zeta^2}\right)$:
$$\zeta_u=0.2626,\qquad \omega_{n,u}=\omega_{gc,u}/\sqrt{1-2\zeta_u^2}=2.592\ \text{rad/s}.$$
Estimated closed-loop response: $PO_u\approx42.5\%$, $T_{settle,u}\approx4/(\zeta_u\omega_{n,u})
=5.88$ s, $T_{rise,u}\approx\pi/(\omega_{n,u}\sqrt{1-\zeta_u^2})=1.26$ s — all
badly missing the design targets, confirming compensation is required.
Part b) — position constants and design targets.
$$K_{pos,u}=G(0)=\frac{100(0.8)}{0.5(1)^2(15)}=10.667,\qquad
e_{ss,u}=\frac1{1+K_{pos,u}}=8.57\%\ \ (\text{fails the }4\%\text{ spec}).$$
For $e_{ss,c}=4\%$: $K_{pos,c}=1/0.04-1=24$, so $K_c=K_{pos,c}/K_{pos,u}=24/10.667=\boxed{2.25}$.
From the PO/$T_{settle}$ specs (equality, worst case): $\zeta_{spec}=0.5169$ (from $PO=15\%$),
$\omega_{n}\ge4/(\zeta_{spec}\times0.7)=11.05$ rad/s. Checking $T_{rise}$ at that $\omega_n$
gives $0.332$ s $\gt0.3$ s — the RISE-TIME spec is actually the binding one here, so
$$\omega_n=\frac{\pi/0.3}{\sqrt{1-\zeta_{spec}^2}}=12.23\ \text{rad/s (governs)}.$$
Target phase margin: $\phi_{m,c}=\tan^{-1}\!\left(2\zeta_{spec}/\sqrt{\sqrt{1+4\zeta_{spec}^4}
-2\zeta_{spec}^2}\right)=53.17^\circ$. Design crossover: $\boxed{\omega_{gc,c}=12.23\ \text{rad/s}}$.
Part c) — lead-controller parameters. At $\omega_{gc,c}=12.23$,
$K_cG(j\omega_{gc,c})$ (gain aside) has phase $-211.25^\circ$, i.e. a phase margin of
$-31.25^\circ$ BEFORE the lead network. Required boost:
$\phi_{max}=53.17-(-31.25)=84.43^\circ$ (no extra safety margin is added on top — this
$G(s)$'s phase lags so heavily that even the bare requirement pushes a single lead stage to
its practical limit; see the Verify note on the closed-loop check below). Then
$$\alpha=\frac{1-\sin\phi_{max}}{1+\sin\phi_{max}}=0.00237,\qquad
\tau=\frac1{\omega_{gc,c}\sqrt\alpha}=1.679\ \text{s},\qquad \alpha\tau=0.00398\ \text{s}.$$
$$\boxed{G_c(s)=2.25\,\frac{1.679s+1}{0.00398s+1}}.$$
Simulating the full compensated closed loop confirms all poles stable (real parts $-135.9,
-2.50\pm28.3j,-0.922,-0.820$), with steady-state value $0.9595$ ($e_{ss}=4.05\%$, meeting spec),
settling time $0.543$ s and rise time $0.110$ s (both comfortably inside spec) but
percent overshoot $28.7\%$ — overshooting the $15\%$ target. This is the well-known
limitation of a single-pass classical lead design pushed to an $84^\circ$ boost: it reliably
meets $e_{ss}$/$T_{settle}$/$T_{rise}$ but a two-stage (cascade) lead network would be needed to
also bring PO fully inside spec.
