Root Locus: Construction Rules

Root locus plots are the roadmap of a control system's dynamic response. By showing how the closed-loop poles move in the complex plane as a system's gain changes from zero to infinity, they allow you to predict stability, damping, and oscillation speed without solving the characteristic equation repeatedly. Mastering the systematic construction rules transforms this powerful analysis from a tedious calculation into an efficient sketching skill, enabling rapid design decisions about controller gain selection.

Core Concept: The Foundation and Real-Axis Segments

The root locus is a graphical technique for analyzing how the roots of a closed-loop system's characteristic equation change with variation of a single parameter, typically the gain $K$. We start from the open-loop transfer function, expressed as $KG(s)H(s)$, where its poles and zeros form the skeleton of the plot.

Rule 1: Locus Start and End Points. The branches of the root locus begin at the open-loop poles (where $K=0$) and end at the open-loop zeros (where $K\to \infty$). If the system has more poles $n$ than zeros $m$, the $n-m$ excess branches will travel to zeros located at infinity. This rule immediately tells you the number of separate locus branches you will be sketching.

Rule 2: Real-Axis Segments. A point on the real axis lies on the root locus if the number of real open-loop poles and zeros to its right is odd. To apply this, test any point on the real axis. Count all real-axis poles and zeros that are to the right of this test point. If the total is 1, 3, 5..., the point is on the locus. This rule quickly defines which sections of the real axis are part of the root locus plot.

Determining Asymptotic Behavior and Centroid

When branches go to infinity (because $n>m$), they do so along straight-line asymptotes. These asymptotes provide the ultimate direction of the branches for very high gain.

Rule 3: Asymptote Angles. The angles of these asymptotes, relative to the positive real axis, are given by: $${\theta }_k=\frac{(2k+1)180^{\circ }}{n-m},k=0,1,2,...,n-m-1$$ For example, if $n-m=3$, the asymptote angles are $60^{\circ }$, $180^{\circ }$, and $300^{\circ }$ (or $-60^{\circ }$).

Rule 4: Asymptote Centroid (Intersection Point). All asymptotes radiate from a common point on the real axis called the centroid, ${\sigma }_a$. It is calculated from the poles and zeros: $${\sigma }_a=\frac{\sum \text{(real parts of poles)}-\sum \text{(real parts of zeros)}}{n-m}$$ This is a real number. The centroid is the balancing point of the pole-zero configuration and indicates where the asymptotes originate.

Calculating Breakaway and Break-In Points

Branches on the real-axis segment between two poles will repel each other and break away into the complex plane. Conversely, branches on the real-axis segment between two zeros will attract each other and break in from the complex plane to meet at a real axis point.

Rule 5: Finding Breakaway/Break-in Points. These points, where multiple branches meet and depart from/arrive at the real axis, satisfy the equation: $$\frac{dK}{ds}=0$$ where $K$ is expressed from the characteristic equation $1+KG(s)H(s)=0$. A simpler, equivalent method is to solve: $$\sum \frac{1}{\sigma -p_i}=\sum \frac{1}{\sigma -z_i}$$ where $\sigma$ is the trial point on the real axis, $p_i$ are the open-loop poles, and $z_i$ are the open-loop zeros. Not all solutions are valid; you must check that the candidate point $\sigma$ lies on a real-axis segment of the locus.

Angles of Departure and Arrival

When a branch begins at a complex pole (not on the real axis), it leaves at a specific angle. Similarly, a branch ending at a complex zero arrives at a specific angle. These angles ensure the locus obeys the fundamental angle condition.

Rule 6: Angle of Departure from a Complex Pole. To calculate the departure angle ${\theta }_{p,dep}$ from a specific complex pole, apply the angle condition to a point infinitely close to that pole. The formula is: $${\theta }_{p,dep}=180^{\circ }-\sum \angle (p-\text{other poles})+\sum \angle (p-\text{zeros})$$ Here, you sum the angles from the pole in question to all other poles and zeros. This calculation gives the direction the locus "shoots out" from the complex pole.

Rule 7: Angle of Arrival at a Complex Zero. The logic is similar for the arrival angle ${\theta }_{z,arr}$ at a complex zero: $${\theta }_{z,arr}=180^{\circ }+\sum \angle (z-\text{poles})-\sum \angle (z-\text{other zeros})$$ This tells you the direction from which the locus branch approaches the complex zero.

Crossing the Imaginary Axis

The most critical point on the root locus is often where it crosses the imaginary axis, as this defines the gain at which the system becomes unstable (poles move into the right-half plane).

Rule 8: Imaginary Axis Crossings. These points are found by substituting $s=j\omega$ into the characteristic equation $1+KG(s)H(s)=0$. This results in two equations: one for the real part and one for the imaginary part. Solve these simultaneously for the frequency $\omega$ (where the crossing occurs) and the corresponding critical gain $K_{crit}$. The Routh-Hurwitz criterion applied to the characteristic equation is an alternate, often simpler, method to find $K_{crit}$ and then solve for $\omega$.

Common Pitfalls

1. Misapplying the Real-Axis Rule: The most common error is forgetting that the rule counts only real-axis poles and zeros. Complex poles/zeros do not affect the count for real-axis segments. Always verify your real-axis segments by testing a point between each pole and zero.
2. Incorrect Centroid Calculation: The centroid formula uses the algebraic sum of real parts. Ensure you subtract the sum of the zero real parts from the sum of the pole real parts. Also, remember the centroid is always a real number; if your calculation yields a complex number, you've made an error.
3. Assuming All Breakaway Solutions are Valid: Solving $\frac{dK}{ds}=0$ yields candidate points. You must check that these points lie on an actual real-axis segment of the root locus (using Rule 2). A point not on the locus is extraneous and must be discarded.
4. Confusing Angle of Departure/Arrival Formulas: The formulas are similar but have critical sign differences. A reliable method is to derive them from scratch using the angle condition: at any point on the locus, the sum of angles from all zeros minus the sum of angles from all poles must equal $180^{\circ }+360^{\circ }k$. Apply this to a point infinitesimally close to the complex pole or zero.

Summary

• The root locus graphically traces the migration of closed-loop poles as gain $K$ varies, beginning at open-loop poles ($K=0$) and terminating at open-loop zeros ($K\to \infty$).
• Real-axis segments belong to the locus if the count of real poles/zeros to the right is odd. Asymptotes guide branches going to infinity, defined by their angles and centroid on the real axis.
• Breakaway and break-in points on the real axis are found where $dK/ds=0$, and angles of departure/arrival from complex poles/zeros ensure the angle condition is met locally.
• The imaginary axis crossing, found via the $s=j\omega$ substitution or Routh-Hurwitz, identifies the critical gain for stability, completing a sketch that predicts system behavior across all possible gains.