Root Locus Construction, Gain Selection for a Target Damping Ratio, and Gain Margin: 17-Phys-B5 Systems and Control, December 2017, Q7

Question 7: Root Locus Construction, Gain Selection for a Target Damping Ratio, and Gain Margin (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.

Figure Q7.1 — root locus for $G(s)=100/[(s+1)(s+2)(s+5)]$: real-axis branches merging at the breakaway point $-1.465$ (poles $-1,-2$) and the far branch leaving $-5$; the complex branches cross the $j\omega$-axis at $K_{crit}=1.26$, with the $\zeta=0.5$ design point at $K_{op}\approx0.165$.

Given. $G(s)=100/[(s+1)(s+2)(s+5)]$, unity feedback, proportional gain $K_p$; target closed-loop damping ratio $\zeta=0.5$.

Find. 1) Asymptotes, centroid, breakaway, $j\omega$-crossing ($\omega_{osc},K_{crit}$). 2) $K_{op}$ for $\zeta=0.5$, and Gain Margin there. 3) 2nd-order model $K_{dc},\omega_n,G_m(s)$ and step specs.

Approach. Apply the standard root-locus construction rules to poles $-1,-2,-5$ (no zeros); find the $\zeta=0.5$ design point by intersecting the constant-damping ray with the locus via the angle condition; confirm the imaginary-axis crossing (for $K_{crit}$/Gain Margin) via Routh–Hurwitz.

Part 1) — asymptotes, centroid, breakaway, $j\omega$-crossing. Three poles, no zeros $\Rightarrow$ 3 asymptotes at $\theta=\dfrac{(2k{+}1)180^\circ}{3}= \boxed{60^\circ,180^\circ,300^\circ}$, centroid $\sigma_a=\dfrac{-1-2-5}{3}=\boxed{-2.67}$. Real-axis locus exists on $(-2,-1)$ and $(-\infty,-5)$ (odd pole count to the right). Breakaway: solving $d/ds\big[(s+1)(s+2)(s+5)\big]=0$ gives $s=-1.465$ (inside the $(-2,-1)$ segment $\Rightarrow$ genuine breakaway) and $s=-3.869$ (NOT on any real-axis locus segment, hence rejected): $$\boxed{\sigma_{break}=-1.465}.$$ Routh–Hurwitz on $s^3+8s^2+17s+(10{+}100K_p)=0$: the $s^1$ row vanishes at $K_p=1.26$, giving the auxiliary equation $8s^2+126=0\Rightarrow s^2=-15.75\Rightarrow \boxed{\omega_{osc}=\sqrt{17}\approx4.123\ \text{rad/s}}$, so $\boxed{K_{crit}=1.26}$.
Part 2) — $K_{op}$ for $\zeta=0.5$ and Gain Margin. The $\zeta=0.5$ ray is $\omega=-\sigma\tan(60^\circ)$; applying the angle condition (sum of angles from the three poles $=180^\circ$) along this ray locates the dominant point at $$s_{dom}=-1.0625+j1.8403,\qquad\omega_n=|s_{dom}|=\boxed{2.125\ \text{rad/s}}.$$ The magnitude criterion there gives $$\boxed{K_{op}=\frac{|(s_{dom}{+}1)(s_{dom}{+}2)(s_{dom}{+}5)|}{100}\approx0.1653}.$$ The Gain Margin (ratio of the critical gain to the operating gain) is $$\boxed{G_M=\frac{K_{crit}}{K_{op}}=\frac{1.26}{0.1653}\approx7.62\ \text{V/V}\ (17.6\ \text{dB})}.$$
Part 3) — 2nd-order model and step-response estimate. At $K_{op}=0.1653$ the third (real) closed-loop pole sits at $-5.875$ (fast compared to the dominant pair, confirming the 2nd-order approximation is reasonable). The closed-loop DC gain is $$K_{dc}=\frac{100K_{op}}{10+100K_{op}}=\frac{16.53}{26.53}\approx\boxed{0.623},$$ so with $\zeta=0.5,\ \omega_n=2.125$: $$\boxed{G_m(s)=\frac{0.623\times4.516}{s^2+2.125s+4.516}}.$$ $$PO=100e^{-0.5\pi/\sqrt{0.75}}\approx\boxed{16.3\%},\qquad T_{settle(\pm2\%)}=\frac{4}{0.5\times2.125}\approx\boxed{3.77\ \text{s}},\qquad T_{rise(0-100\%)}\approx\boxed{1.14\ \text{s}},\qquad e_{ss(step\%)}=(1-0.623)\times100\approx\boxed{37.7\%}.$$ The large steady-state error is expected: this is a Type 0 loop under pure proportional control, so a modest $K_{op}$ chosen for damping (not accuracy) leaves a substantial offset.

Final results — Question 7
Item	Result
Asymptotes, centroid	$60^\circ,180^\circ,300^\circ$; $-2.67$
Breakaway	$-1.465$
$\omega_{osc}$, $K_{crit}$	$\sqrt{17}\approx4.12$ rad/s, $1.26$
$K_{op}$ ($\zeta=0.5$), Gain Margin	$0.1653$, $7.62$ V/V ($17.6$ dB)
2nd-order model	$K_{dc}=0.623$, $\omega_n=2.125$ rad/s
Step specs	$PO\approx16.3\%$, $T_{settle(2\%)}\approx3.77$ s, $T_{rise}\approx1.14$ s, $e_{ss}\approx37.7\%$

Topic: Root-locus construction rules and gain selection for a target damping ratio.

Key relations: asymptote angles $(2k{+}1)180^\circ/n$; centroid $\sigma_a=\sum p_i/n$; breakaway from $dK/ds=0$ restricted to genuine real-axis locus segments; angle condition $\sum\angle(s^*{-}p_i)=180^\circ$ locates a design point on the locus for a target $\zeta$; magnitude criterion $K=1/|G(s^*)|$ then reads off the operating gain there.

Why this works: every point ON the root locus satisfies the closed-loop characteristic equation for SOME real positive gain — the angle condition alone identifies WHERE on the s-plane a given $\zeta$-ray intersects the locus, and the magnitude condition then supplies WHICH gain puts a closed-loop pole exactly there.

Common pitfall: accepting BOTH roots of $dK/ds=0$ as breakaway points without checking they lie on an actual real-axis locus segment (only odd pole/zero counts to the right qualify); and forgetting that Gain Margin is a RATIO of gains ($K_{crit}/K_{op}$), not a difference.

Source: Nise, Control Systems Engineering, Ch. 8 (root locus) and Ch. 9 (design via root locus); Ogata, Modern Control Engineering, Ch. 6.

Question 7 of 8: Root Locus Construction, Gain Selection for a Target Damping Ratio, and Gain Margin

Question 7: Root Locus Construction, Gain Selection for a Target Damping Ratio, and Gain Margin (20 marks)