Question 6 of 8: Root Locus of a System with a Right-Half-Plane Pole — Conditional Stability

Question 6: Root Locus of a System with a Right-Half-Plane Pole — Conditional Stability (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.

Given. $G(s)=1/[(s-5)(s^2+6s+40)]$, unity negative feedback, proportional gain $K_p$; open-loop poles at $s=+5$ (RHP!) and $s=-3\pm j5.568$.

Find. 1) $K_{crit}$ and $\omega_{osc}$. 2) Root-locus features (real-axis segments, break-away/in, asymptotes, angle of departure, centroid). 3) Validity of a $\zeta=0.707$ dominant-pole model.

Approach. Expand the characteristic equation and Routh-array it (part 1); apply the standard root-locus construction rules (part 2); check the achievable damping ratio across the ENTIRE stable gain range against $\zeta=0.707$ (part 3).

1. Part 1) — marginal-stability gains. Expanding, $(s-5)(s^2+6s+40)=s^3+s^2+10s-200$, so the characteristic equation is $s^3+s^2+10s+(K_p-200)=0$. The Routh array: $$\begin{array}{c|cc}s^3&1&10\\s^2&1&K_p-200\\ s^1&\dfrac{1\times10-1\times(K_p-200)}{1}=210-K_p&0\\s^0&K_p-200\end{array}$$ Both first-column entries after row $s^2$ must be positive: $210-K_p>0$ AND $K_p-200>0$, so the system is stable ONLY in the band $\boxed{200\lt K_p\lt210}$ — a direct consequence of the open-loop RHP pole (enough gain is needed to pull the real-axis branch left of the origin, but too much gain drives the complex pair unstable). Two distinct marginal gains therefore exist: at $K_p=200$ the $s^0$ row vanishes, i.e. a root sits exactly at the origin (verified: $s^3+s^2+10s=s(s^2+s+10)=0$ at $K_p=200$, real-axis crossing, $\omega=0$); at $K_p=210$ the $s^1$ row vanishes, and the auxiliary equation $s^2+(K_p-200)=s^2+10=0$ gives $$\boxed{K_{crit}=210,\qquad \omega_{osc}=\sqrt{10}\approx3.162\ \text{rad/s}}$$ (confirmed: at $K_p=210$ the cubic factors exactly as $(s+1)(s^2+10)$).
2. Part 2) — root-locus construction. Real axis: only one real open-loop pole ($s=5$), so the ENTIRE real axis $s<5$ satisfies the odd-poles-and-zeros-to-the-right rule and is part of the locus for all $K_p>0$ — the real branch starts at $s=5$ ($K_p=0$) and moves continuously left as $K_p$ increases (through the origin at $K_p=200$, reaching $s=-1$ at $K_p=210$), never meeting another real branch. Break-away/break-in: $dK_p/ds=0$ where $K_p=-(s^3+s^2+10s-200)$ gives $3s^2+2s+10=0$, discriminant $4-120<0$ — NO REAL ROOTS, confirming there is no break-away/break-in point (consistent with the single unbroken real branch above). Asymptotes (3 branches, no zeros): centroid $\sigma_a=(5+(-3+5.568j)+(-3-5.568j))/3=\boxed{-1/3}$; angles $(2k+1)\times180^\circ/3=\boxed{60^\circ,180^\circ,300^\circ}$. Angle of departure from $-3+j5.568$: with the other two poles at $5+0j$ and $-3-j5.568$, the angles subtended are $145.2^\circ$ and $90^\circ$, so $$\theta_{dep}=180^\circ-(145.2^\circ+90^\circ)=\boxed{-55.2^\circ}$$ (and $+55.2^\circ$ from the conjugate pole, by symmetry).
3. Part 3) — is $\zeta=0.707$ a valid dominant-pole assumption? NO. Across the ENTIRE stable band $200\lt K_p\lt210$, the complex-pair damping ratio only ranges from $\zeta=0.5/|{-0.5+j3.12}|=0.158$ (at $K_p=200$) down to $\zeta=0$ (at $K_p=210$, purely oscillatory) — it never comes close to $0.707$ for ANY stabilizing gain. Moreover, even where the complex pair IS dominant-looking, the third (real) pole sits at comparable distance from the imaginary axis (e.g. at $K_p=205$: real pole at $-0.51$ vs. complex-pair real part $-0.24$ — not the $\gtrsim5\times$ separation a dominant-pole model requires). So a $\zeta=0.707$ second-order approximation is NOT justified here on either count: the achievable damping is far too light, and the neglected real pole is not far enough away.