Root locus — Evans' method, rules for sketching

WHAT is the root locus?

WHY this equation? The closed-loop transfer function is

Poles are where the denominator vanishes, i.e. $1 + KG(s)H(s) = 0$. Everything below flows from this one line.

Deriving the two conditions from scratch

Write $L(s) = KG(s)H(s) = K\dfrac{N(s)}{D(s)}$. Characteristic equation:

Now $-1$ is a complex number with magnitude 1 and angle $180°$ (or any odd multiple). Splitting $L(s)=-1$ into magnitude and angle:

The sketching rules (each DERIVED, not memorized)

Let $n$ = number of open-loop poles (roots of $D$), $m$ = number of open-loop zeros (roots of $N$). Assume $n \ge m$.

Rule 1 — Start and end. Rewrite $1 + K\frac{N}{D}=0$ as $D(s) + KN(s) = 0$.

• $K \to 0$: $D(s)=0$ → branches start at open-loop poles.
• $K \to \infty$: divide by $K$, $N(s)=0$ → branches end at open-loop zeros. There are $n$ branches; $m$ end on finite zeros, the remaining $n-m$ go to infinity.

Rule 2 — Symmetry. Coefficients are real, so complex poles come in conjugate pairs → the locus is symmetric about the real axis.

Rule 3 — Real-axis segments. A real point lies on the locus if the number of real poles+zeros to its right is odd. WHY? Angle contributions from complex-conjugate pairs cancel; each real singularity to the right contributes $180°$. Odd count → total is an odd multiple of $180°$ → angle condition met.

Rule 4 — Asymptotes. The $n-m$ escaping branches approach straight lines with angles

meeting the real axis at the centroid

WHY the centroid? For large $|s|$, $G(s)H(s)\approx \frac{1}{(s-\sigma_a)^{n-m}}$ after matching the first two terms of the polynomial division — the far-field looks like a single pole of multiplicity $n-m$ at the centroid.

Rule 5 — Breakaway/break-in points. Where branches leave or rejoin the real axis, two poles coincide → $K$ is at a local extremum. Solve

WHY? At a double root the polynomial has a repeated root, so $K(s)$ turns around: $dK/ds=0$.

Rule 6 — $j\omega$-axis crossing. Use Routh–Hurwitz on $D(s)+KN(s)$ to find the $K$ that makes a whole row zero; the auxiliary equation gives the crossing frequency $\omega$. This is the critical gain for stability.

Rule 7 — Angle of departure/arrival (from complex pole/zero $p$):

WHY? Apply the angle condition to a test point infinitesimally close to $p$.

Worked Example 1 — $G(s)H(s)=\dfrac{1}{s(s+2)}$

Poles: $s=0,\,s=-2$ ($n=2$). Zeros: none ($m=0$).

Step — Real axis segment. Why? Between $0$ and $-2$, exactly one pole ($s=0$) sits to the right of any interior point → odd → segment $[-2,0]$ is on the locus. ✔

Step — Asymptotes. $n-m=2$ so $\theta_a = 90°, 270°$. Centroid $\sigma_a = \frac{(0)+(-2)-0}{2} = -1$. Why centroid at $-1$? It's the average of the poles. Branches shoot straight up and down from $s=-1$.

Step — Breakaway. $K=-D/N = -s(s+2)=-(s^2+2s)$. Set $dK/ds = -(2s+2)=0 \Rightarrow s=-1$. Why here? Symmetry demanded the breakaway between the two real poles; the algebra confirms $s=-1$.

Step — Gain at breakaway. $K = -(-1)(-1+2) = 1$. So at $K=1$ we get a double pole at $-1$; beyond that, poles become complex with real part $-1$ (constant damping).

Insight: This system is stable for all $K>0$ — the vertical asymptote never crosses into the right half plane.

Worked Example 2 — $G(s)H(s)=\dfrac{1}{s(s+1)(s+2)}$

Poles $0,-1,-2$ ($n=3$), $m=0$.

Asymptotes. $\theta_a=60°,180°,300°$; $\sigma_a=\frac{-3}{3}=-1$. Why 3 asymptotes? $n-m=3$ branches all escape.

$j\omega$ crossing (Routh). Char. eqn: $s^3+3s^2+2s+K=0$. Routh array first column needs $\frac{3\cdot2 - K}{3}>0 \Rightarrow K<6$. At $K=6$: auxiliary $3s^2+6=0 \Rightarrow s=\pm j\sqrt2$. Why this matters: the locus crosses the imaginary axis at $\omega=\sqrt2$ when $K=6$ — that is the maximum stable gain.

Flashcards

Connections

• Characteristic equation & closed-loop poles
• Routh–Hurwitz stability criterion
• PID / PD controller design — adding zeros to reshape the locus
• Bode plot & frequency response — magnitude/phase alternative view
• Nyquist criterion — encirclement view of $L(s)=-1$
• Damping ratio and settling time — reading $\zeta,\omega_n$ off pole locations
• GNC control loops — where gain tuning lands in autopilots

Concept Map

Hinglish (regional understanding)