Question 2 of 8: Stability via Root Locus, Bode and Routh–Hurwitz
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 98-Phys-B5 Systems & Control, National Examination
May 2016 — 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/lag design, state-space representation, steady-state errors); K. Ogata, Modern
Control Engineering, 5th ed. (root-locus and frequency-response compensator design,
controllability/observability, Nyquist stability).
Question 2: Stability via Root Locus, Bode and Routh–Hurwitz (20 marks, compulsory)
Given. $G(s)=(s+50)^2/(s+4)^3$ in a unity-feedback loop with proportional
gain $K_p$; a Root-Locus sketch (Figure Q2.2, triple pole at $-4$, double zero at $-50$) and a
Bode magnitude/phase plot of $G(s)$ (Figure Q2.3, magnitude falling from about $+25$ dB at
low frequency, phase dipping below $-180^{\circ}$ between roughly $\omega=10$ and
$35$ rad/s before recovering).
Find. Every value of $K_{crit}$ at which the closed loop is marginally
stable, the corresponding $\omega_{osc}$ at each, and the resulting safe/unsafe gain ranges.
Figure — root locus of $(s+4)^3+K(s+50)^2=0$, computed directly from
the printed $G(s)$ (matches the shape of the source's Figure Q2.2: a loop that leaves the
triple pole at $-4$, bulges out to about $\mathrm{Re}=-65$, $\mathrm{Im}=\pm82$, and returns
toward the double zero at $-50$, crossing the imaginary axis twice near the origin).
Approach. Because a triple real pole and a double real zero sit only $46$
units apart on the real axis, the two complex-conjugate root-locus branches leave the pole,
sweep out into the complex plane, and curve back toward the zero — crossing the imaginary
axis twice (once destabilizing, once re-stabilizing) rather than once. Solve for both
crossings algebraically via Routh–Hurwitz, then cross-check each with the magnitude
criterion and with the Bode phase plot, exactly as the three sub-parts request.
Part 3) — Routh array for the exact crossings. The characteristic
equation is $(s+4)^3+K(s+50)^2=0$, i.e.
$$s^3+(12+K)s^2+(48+100K)s+(64+2500K)=0.$$
The Routh array's $s^1$ row is $b_1=\dfrac{(12+K)(48+100K)-(64+2500K)}{12+K}$; setting the
numerator to zero, $100K^2-1252K+512=0$, gives TWO positive roots
$$K_{crit}=\frac{313\pm23\sqrt{161}}{50}\ \Rightarrow\ \boxed{K_1=0.423,\quad K_2=12.10.}$$
At each, the auxiliary equation from the $s^2$ row, $(12+K)s^2+(64+2500K)=0$, gives the
oscillation frequency $\omega_{osc}=\sqrt{(64+2500K)/(12+K)}$:
$\omega_1=\boxed{9.50\ \text{rad/s}}$ at $K_1$ and $\omega_2=\boxed{35.46\ \text{rad/s}}$ at
$K_2$ — confirmed by direct root-finding of the cubic at each $K$, which returns the pair
$\pm j9.504$ and $\pm j35.464$ exactly.
Part 1) — Magnitude Criterion cross-check (Root Locus). At the
crossover point $s^{*}=j\omega_{osc}$, $K=1/|G(s^{*})|$ with $G(s)=(s+50)^2/(s+4)^3$:
at $\omega_1=9.50$, $|G(j9.50)|=2.363\Rightarrow K=1/2.363=\boxed{0.423}$; at $\omega_2=35.46$,
$|G(j35.46)|=0.0827\Rightarrow K=1/0.0827=\boxed{12.10}$ — both reproduce Part 3's Routh
values exactly. Reading the two crossovers on Figure Q2.2 (both lie extremely close to the
origin because the real-axis span of the plot, $-140$ to $+20$, compresses the pole's own
$-4$ almost onto the imaginary-axis gridline) and interpreting the safe-gain ranges: the locus
leaves the pole in the STABLE left half-plane, crosses into the RIGHT half-plane at $K_1$,
loops through its widest excursion, and crosses back into the LEFT half-plane at $K_2$ before
converging on the double zero at $-50$ as $K\to\infty$. The safe operating ranges are therefore
$\boxed{0 \lt K_p \lt 0.423}$ (light gain) and $\boxed{K_p>12.10}$ (heavy gain); the closed loop is
UNSTABLE for the intermediate band $0.423 \lt K_p \lt 12.10$.
Part 2) — Bode-plot cross-check. Marginal stability under
proportional control occurs exactly where $\angle G(j\omega)=-180^{\circ}$ (any real $K>0$
leaves the phase unchanged). Evaluating $\angle G(j\omega)=2\angle(j\omega+50)-3\angle(j\omega+4)$
gives exactly $-180^{\circ}$ (mod $360^{\circ}$) at $\omega=9.504$ and at $\omega=35.464$,
matching Figure Q2.3's phase trace, which starts near $0^{\circ}$, dips through a minimum of
about $-193^{\circ}$ near $\omega\approx18$ rad/s (crossing $-180^{\circ}$ on the way down
and again on the way back up), and recovers toward $-90^{\circ}$ at high frequency (consistent
with the $2$ zeros $-$ $3$ poles $=-1$ net relative degree). The magnitude at those two phase
crossings is $|G(j9.504)|=7.47$ dB and $|G(j35.464)|=-21.65$ dB, whose reciprocals in
linear units reproduce $K_1$ and $K_2$ from Parts 1 and 3 to within rounding.
