Root Locus Method

Root locus is a control-systems method for seeing where the closed-loop poles can move as a gain changes. In the standard continuous-time negative-feedback setting, that gain is usually written as $K \ge 0$, and the plot lives in the complex $s$-plane.

This matters because pole location is tied to system behavior. For a continuous-time linear system, poles in the left half-plane are associated with stable modes, so root locus gives a fast way to judge how changing gain can help or hurt stability.

If the open-loop transfer factor is written as $K G(s)H(s)$, the closed-loop poles are the solutions of

$$1 + K G(s)H(s) = 0$$

So the root locus is the set of all closed-loop pole locations as $K$ varies.

What the Root Locus Plot Shows

The plot is not showing arbitrary points. It shows the possible closed-loop pole locations for one specific feedback model and one gain range.

Two facts give most of the intuition:

• Branches start at the open-loop poles when $K = 0$.
• Branches end at the open-loop zeros or go to infinity as $K \to \infty$.

That makes the practical question simple: if you turn the gain up, where do the poles go?

Why Students Use Root Locus

Think of root locus as a motion diagram for poles. You are not solving a brand-new problem for every value of $K$. You are tracking the path the poles follow as the gain changes continuously.

That is why the method is useful in design. Instead of testing many separate gains one at a time, you can see the overall trend on one plot.

Worked Example: Root Locus for $L(s) = \frac{K}{s(s+2)}$

Take the open-loop transfer factor

$$L(s) = \frac{K}{s(s+2)}$$

with unity negative feedback. The closed-loop characteristic equation is

$$1 + \frac{K}{s(s+2)} = 0$$

Multiply through by $s(s+2)$:

$$s(s+2) + K = 0$$

so

$$s^2 + 2s + K = 0$$

Now solve for the closed-loop poles:

$$s = \frac{-2 \pm \sqrt{4 - 4K}}{2} = -1 \pm \sqrt{1-K}$$

This formula already shows the main root-locus behavior.

When $K = 0$, the poles are at $s = 0$ and $s = -2$. Those are the open-loop poles, so they are the starting points of the locus.

When $0 < K < 1$, both poles stay on the real axis:

$$s = -1 \pm \sqrt{1-K}$$

As $K$ increases, those poles move toward each other along the real axis.

When $K = 1$, they meet at

$$s = -1$$

For $K > 1$, the square root becomes imaginary, so the poles become a complex-conjugate pair:

$$s = -1 \pm i\sqrt{K-1}$$

Now the real part stays fixed at $-1$, and the poles move up and down vertically.

This gives you the whole story at a glance:

• The branches start at $0$ and $-2$.
• They meet at $-1$.
• After that, they leave the real axis as a complex pair.
• There are no finite zeros, so the branches go to infinity.

Because the real part stays negative for every $K > 0$, this particular closed-loop system stays in the left half-plane for all positive gains. That conclusion depends on this specific example and the continuous-time setting.

Common Root Locus Mistakes

Mixing up open-loop and closed-loop poles

The root locus comes from the closed-loop characteristic equation. Open-loop poles and zeros guide the sketch, but the locus itself shows where the closed-loop poles can move.

Forgetting the feedback sign

The standard form above uses negative feedback and usually $K \ge 0$. If the feedback sign or gain range changes, the locus changes too.

Reading stability without stating the setting

For a continuous-time system, poles in the left half-plane indicate asymptotic stability. A discrete-time system uses a different stability region, so the same visual rule does not carry over unchanged.

Treating the plot like a time-response graph

The root locus tells you where poles are. It does not directly give overshoot, settling time, or response size unless you connect pole location to a particular model and approximation.

When the Root Locus Method Is Used

Root locus is used when you want to tune a gain and understand how that tuning changes the pole locations of a linear feedback system.

That comes up often in introductory control design, especially when you want a gain that keeps poles in a stable region or shifts them toward a faster or slower response. Even when software draws the plot for you, the idea still matters because it tells you what the plot is actually saying.

How to Start Any Root Locus Problem

Before sketching anything, answer these questions:

1. What is the characteristic equation?
2. Where are the open-loop poles and zeros?
3. Are you using the standard negative-feedback setup with $K \ge 0$?

If those three points are unclear, the plot is easy to misread.

Try Your Own Version

Try the same process with

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

Write the closed-loop characteristic equation, solve for the poles, and track what happens as $K$ increases. If you can identify where the two branches start and when they stop being purely real, the main idea of root locus has clicked.