Root-Locus Analysis & Compensator Design

For negative feedback with

$$L(s)=K L_0(s),\qquad K\ge0,$$

closed-loop poles solve

$$1+K L_0(s)=0.$$

The root locus is every solution as $K$ changes. It visualizes a one-parameter family of closed loops; it is not the open-loop pole-zero plot and it does not show zeros moving.

Branches begin at open-loop poles and end at finite zeros or infinity. A desired pole must satisfy the angle condition before any positive gain can place it there.

Angle and magnitude conditions #

At a locus point $s$,

$$K L_0(s)=-1.$$

Therefore

$$\angle L_0(s)=(2q+1)180^\circ,$$

and after the angle condition is met,

$$K=\frac{1}{|L_0(s)|}.$$

The angle condition selects geometry; the magnitude condition selects gain. Testing magnitude alone can return a gain for a point that is not on the locus.

Construction rules and why they hold #

Let $n$ be open-loop poles and $m$ finite zeros.

• There are $n$ branches, starting at poles when $K=0$.
• $m$ branches end at finite zeros; $n-m$ go to infinity when $n>m$.
• On the real axis, a point lies on the positive-$K$ locus when the number of real poles and zeros to its right is odd. Each contributes $0^\circ$ or $180^\circ$ to the angle condition.
• Asymptote angles for branches going to infinity are

$$\theta_q=\frac{(2q+1)180^\circ}{n-m},\qquad q=0,\ldots,n-m-1,$$

with centroid

$$\sigma_a=\frac{\sum p_i-\sum z_i}{n-m}.$$

• Breakaway and break-in candidates satisfy $dK(s)/ds=0$ for $K(s)=-1/L_0(s)$, but only candidates on valid real-axis segments with $K\ge0$ belong to the locus.

Departure from complex poles and arrival at complex zeros follow from the angle condition after subtracting all other pole and zero angles.

Worked gain family #

For

$$L(s)=\frac{K}{s(s+1)(s+3)},$$

the characteristic equation is

$$s^3+4s^2+3s+K=0.$$

Routh gives first column

$$1,\quad4,\quad\frac{12-K}{4},\quad K,$$

so the stable gain interval is $0<K<12$. At $K=12$, the auxiliary polynomial $4s^2+12$ gives crossing at $s=\pm j\sqrt3$. The root locus must tell the same story: three branches start at $0,-1,-3$, two break away into a complex pair, and the pair crosses the imaginary axis at that gain.

This cross-check is powerful. A sketch inconsistent with Routh has a construction or sign error.

Translating time requirements into a region #

For a dominant complex pole $s=-\sigma\pm j\omega_d$,

$$\zeta=\frac{\sigma}{\sqrt{\sigma^2+\omega_d^2}},\qquad \omega_n=\sqrt{\sigma^2+\omega_d^2}.$$

Constant damping-ratio lines are rays from the origin; constant natural-frequency curves are circles; a settling-envelope target often becomes a vertical line $\sigma\ge\sigma_{min}$. These are second-order approximations. After choosing a locus point, calculate the complete response and inspect zeros, extra poles, residues, and actuator effort.

Reshaping the locus #

Changing gain moves poles only along the existing locus. If the desired region is not reached, add controller dynamics deliberately.

• A lead/PD-like zero contributes positive angle near a desired pole and pulls branches toward it; a high-frequency pole makes the controller proper.
• A lag pole-zero pair near the origin raises low-frequency gain relative to crossover but can add phase lag and a slow mode.
• An integrator adds a pole at zero, changes system type, and changes the entire locus.
• A notch is not just a pole-zero decoration; it shapes a resonance and must be checked for cancellation fragility.

At a proposed desired pole $s_d$, compute the angle deficiency of the uncompensated loop. Place compensator pole/zero contributions to supply it, then use the magnitude condition for gain. Finally check all closed-loop poles and frequency margins.

Root locus in discrete time #

The algebraic rules apply to $L(z)$, but stability is inside the unit circle rather than left of a vertical axis. Damping and natural-frequency grids can be mapped through $z=e^{sT_s}$. A visually small movement near $z=1$ can represent a large change in continuous time when $T_s$ changes, so always state the sample period.

What root locus does not prove #

• Good closed-loop pole locations do not ensure low sensitivity or adequate gain/phase margins.
• A pole map does not show control effort, noise transmission, or saturation.
• Near pole-zero cancellations can hide fragile internal modes.
• Dominant-pole assumptions can fail when residues or zeros are important.
• One scalar gain cannot explore a genuinely multi-parameter or MIMO design space.

Design verification #

For the selected controller, verify:

1. internal closed-loop poles and lack of unstable cancellation;
2. time-domain metrics for reference and disturbances;
3. Bode/Nyquist margins and sensitivity peaks;
4. control magnitude/rate and noise amplification;
5. uncertainty in plant poles, zeros, gain, and delay;
6. nonlinear and sampled implementation behavior.

Root locus is a geometric design language. Its strength is showing why gain alone can or cannot move the dynamics where they need to go.