Question 6 of 8: Root Locus, Gain Selection and Dominant-Pole Validity

Question 6: Root Locus, Gain Selection and Dominant-Pole Validity (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. $G(s)=(s+3)(s+6)/[s^2(s+2)]$: double pole at $s=0$, pole at $s=-2$; zeros at $s=-3,-6$.

Find. Full root-locus geometry; $K_{p5\%}$ and the corresponding transient specs; a discussion of dominant-pole validity.

Approach. Standard root-locus construction rules on $s^2(s+2)+K_p(s+3)(s+6)=0$; Routh–Hurwitz to check for any imaginary-axis crossing; numerically track the complex branch pair to find the $K_p$ giving the desired damping ratio.

1. Asymptotes, real-axis segments. $n=3$ poles, $m=2$ zeros $\Rightarrow$ 1 asymptote at $180^\circ$. Real-axis test: the segment $(-3,-2)$ has one pole ($s=-2$) and no zero to its right — odd — on the locus (this hosts the branch from the single pole at $s=-2$ travelling directly, and only, to the zero at $s=-3$). The segment $(-\infty,-6)$ has 3 poles and 2 zeros to its right ($=5$, odd) — also on the locus (the $-\infty$-bound asymptotic branch).
2. Breakaway (at the double pole) and break-in point. The double pole at $s=0$ breaks away immediately for any $K_p\gt0^+$ (angle-of-departure symmetric, $\pm90^\circ$) into a complex-conjugate pair. Solving $dK/ds=0$ for $K(s)=-s^2(s+2)/[(s+3)(s+6)]$ gives one further REAL, positive-$K$ root: $$\boxed{s_{break-in}=-12.823,\quad K=26.55}$$ — the complex pair, having travelled out from the origin, returns to the real axis here and splits into two real branches: one heading right to the zero at $s=-6$, the other continuing left along the $180^\circ$ asymptote.
3. Imaginary-axis crossing. Characteristic equation $s^3+(2+K_p)s^2+9K_ps+18K_p=0$; Routh $s^1$-row numerator is $9K_p^2/(2+K_p)$ — strictly POSITIVE for every $K_p\gt0$. No finite $K_{crit}$ exists: the system is stable for all $K_p\gt0$ (a Type-2 system whose double pole at the origin never migrates into the right half-plane for this particular zero configuration).
4. Part 2) — $K_{p5\%}$ and transient specs. $PO=5\%\Rightarrow \zeta=0.6901$. Solving for the point on the locus at this damping ratio (numerically, since $K_p$ appears nonlinearly): $$\boxed{K_{p5\%}=13.696},\qquad \text{dominant poles }s=-6.427\pm6.740j,\ \ \omega_n=9.313\ \text{rad/s},$$ with the third (real) closed-loop pole at $s=-2.843$ (close to the zero at $-3$, as expected from the $(-3,-2)$ real-axis segment). Then $$T_{settle(5\%)}=\frac3{\zeta\omega_n}=0.467\ \text{s},\qquad T_{rise(0-100\%)}\approx\frac{0.8+2.5\zeta}{\omega_n}=0.271\ \text{s (standard approximation)}, \qquad \boxed{e_{ss(step)}=0}$$ (zero steady-state error to a step is automatic here: $G(s)$ has a double pole at the origin, i.e. a Type-2 system, so $e_{ss}(step)=0$ for ANY stabilizing $K_p$).
5. Part 3) — dominant-pole model vs. actual response. The third pole ($s=-2.843$) is only $0.44\times$ the dominant pair's real part ($-6.427$) — the OPPOSITE of the usual "far away and negligible" assumption; here the non-dominant pole is actually CLOSER to the origin than the "dominant" pair, and it sits right next to the zero at $-3$. Simulating the true third-order step response at $K_p=13.696$ gives $PO_{actual}=0\%$ (no measurable overshoot at all) against the idealized $5\%$ prediction — the nearby real pole and zero pair substantially slow and de-oscillate the true response relative to the 2nd-order estimate. This is a clear illustration that the "$\zeta,\omega_n$ from the dominant complex pair" shortcut requires genuine pole separation to be trustworthy, and should always be checked (as here) against the full closed-loop simulation or at least the location of the remaining poles/zeros.