Stability range from Bode, root locus and Routh–Hurwitz: 22-Elec-A2 Systems and Control, May 2015, Q4

Question 4 — Stability range from Bode, root locus and Routh–Hurwitz [10 + 10]

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.

Given. The PID term simplifies: $\left(1+\tfrac{1}{2s}\right)(1+0.5s)=\dfrac{0.5s^2+1.25s+0.5}{s}=\dfrac{0.5(s+2)(s+0.5)}{s}$, so $G_{open}(s)=\dfrac{5K_p(s+2)(s+0.5)}{s^2(s+20)(s-1)}$: poles $0,0,-20,+1$ (one RHP pole), zeros $-2,-0.5$.

Find. The stable range of $K_p$, twice (graphically and by Routh).

[Figure not reproduced: Figure Q4.2 redrawn — the branch from the RHP pole at $+1$ and the double-integrator branches migrate into the left half-plane, crossing the imaginary axis at $\pm j2.05$; the loop becomes stable only above that gain. See the official exam paper.]

Part 1 — Stability range from the figures [10]

Approach. With an open-loop RHP pole the system is stable only for sufficient gain (Nyquist needs the $-1$ point encircled once); the Bode gain margin measures the distance from the plotted gain to that boundary.

Read the Bode margin. The plot (drawn at $K_p=1$) shows $G_m=16.1$ dB at the phase-crossover $\omega=2.05$ rad/s. Thus the gain must be raised by $16.1$ dB, i.e. by a factor $10^{16.1/20}=6.4$, to reach marginal stability: $\boxed{K_{p,\min}\approx6.4}$.
Read the root locus. The branches originating at the origin and at the RHP pole $+1$ move left as $K_p$ increases and cross the $j\omega$-axis at $\pm j2.05$; for gains above the crossing all closed-loop poles lie in the LHP.
Range. Both figures agree: $\boxed{K_p\gt6.4}$ (no finite upper limit — higher gain only moves the poles deeper into the LHP).

Part 2 — Routh–Hurwitz verification [10]

Approach. Form the closed-loop characteristic polynomial $s^2(s+20)(s-1)+5K_p(s+2)(s+0.5)=0$ and require a positive first column.

Characteristic polynomial. $$s^4+19s^3+(5K_p-20)s^2+12.5K_p\,s+5K_p=0.$$
Routh array. $$\begin{array}{c|ccc}s^4&1&5K_p-20&5K_p\\s^3&19&12.5K_p&\\s^2&\frac{82.5K_p-380}{19}&5K_p&\\s^1&c_1&&\\s^0&5K_p&&\end{array}$$ with $c_1=12.5K_p-\dfrac{95K_p}{(82.5K_p-380)/19}$.
First-column conditions. $5K_p\gt0$ (all $K_p\gt0$); the $s^2$ entry $\gt0$ needs $K_p\gt4.61$; the $s^1$ entry $c_1\gt0$ needs $12.5(82.5K_p-380)\gt1805\Rightarrow K_p\gt6.36$. The binding condition is $\boxed{K_p\gt6.36}$.
Oscillation frequency. At $K_p=6.36$ the $s^2$-row auxiliary equation gives $s^2=-\dfrac{5K_p}{7.6}=-4.18$, i.e. $\omega_{osc}=\boxed{2.05\ \text{rad/s}}$ — exactly the Bode phase-crossover and the root-locus $j\omega$ crossing.

Quantity	Result
Stable range (figures)	$K_p\gt6.4$ (from $16.1$ dB above $K_p=1$)
Stable range (Routh)	$\boxed{K_p\gt6.36}$
$\omega_{osc}$ at the boundary	$2.05$ rad/s (all three methods agree)

Concept & Theory

Topic. Stabilising an open-loop-unstable plant — a lower gain limit.

Key relations. Bode $G_m=1/|L(j\omega_{pc})|$ read from the plotted gain; root-locus $j\omega$ crossing = Routh marginal gain; Routh first-column sign test. (Nise Ch. 6, 8, 10.)

Why this works. A right-half-plane open-loop pole forces the Nyquist plot to require one encirclement of $-1$; too little gain fails to produce it, so the closed loop is unstable below a threshold. Raising the gain drags the migrating branches across the $j\omega$-axis into the LHP, and all three tools locate that crossing at the same $2.05$ rad/s.

Common pitfall. Reading the Bode gain margin as an upper limit (the reflex for stable plants). With an unstable pole the margin is the gain the system must gain to become stable, and a negative phase margin at low gain is expected, not an error.

Source. Nise 7e, §6.4, §10.7–10.9.

Question 4 of 8: Stability range from Bode, root locus and Routh–Hurwitz

Question 4 — Stability range from Bode, root locus and Routh–Hurwitz [10 + 10]

Part 1 — Stability range from the figures [10]

Part 2 — Routh–Hurwitz verification [10]