Root Locus Technique in Control System | Root Locus Plot

Developed by Walter R. Evans, the root locus technique plots the closed-loop poles of a feedback system as a selected real gain varies. For a loop transfer function K G(s)H(s), the roots satisfy the characteristic equation shown below.

Open-loop poles and zeros determine the locus geometry. Closed-loop pole locations help estimate continuous-time stability and transient modes for the model, but performance also depends on zeros, input-output paths and nonlinear limits. Unmodelled inductance, capacitance and other uncertainty require robustness analysis rather than a different gain branch on the same plot. The root locus technique in control system design helps select gain and compensator poles or zeros by showing how the model’s closed-loop poles move.
Its main practical uses are listed below.

Advantages of Root Locus Technique

1. It gives a visual relation between a varying gain and the closed-loop pole locations.
2. It identifies gain ranges that place all modelled continuous-time closed-loop poles in the left half-plane.
3. It supports compensator design by showing how added poles and zeros reshape the locus.

The following terms apply to the standard positive-gain root locus of a real-coefficient rational loop transfer function.

1. Characteristic Equation Related to Root Locus Technique: For loop transfer K G(s)H(s), closed-loop poles satisfy 1 + K G(s)H(s) = 0. Expressing K as a function of s and solving dK/ds = 0 gives candidate multiple-root points, which must also satisfy the root-locus conditions and a permitted gain.
2. Breakaway Points: At a real-axis breakaway point, two or more branches meet and leave the real axis as gain changes. The characteristic equation 1 + K G(s)H(s) = 0 has a multiple root there. A stationary value of K is only a candidate: it need not be a maximum, and candidates outside valid real-axis locus segments or with invalid gain are rejected.
3. Break-in Point: At a break-in point, branches enter a valid real-axis segment. Find it from the same multiple-root condition, then check the angle and gain rules. A useful geometric case is: a valid real-axis segment may lie between adjacent zeros.
4. Centroid: When the loop has more finite poles than finite zeros, the branches that end at infinity approach asymptotes through a real-axis centroid. It equals the sum of open-loop poles minus the sum of open-loop zeros, divided by the pole excess. The centroid is denoted by σ_A.
   
   Here, N is the number of open-loop poles and M is the number of finite open-loop zeros.
5. Asymptotes of Root Loci: If N exceeds M, N – M branches end at infinity and approach straight-line asymptotes through the centroid. These lines describe large-gain branch directions, not the departure direction from every breakaway point.
6. Angle of Asymptotes: For positive gain and a positive leading coefficient, the asymptote angles follow the formula below.
   
   Use p = 0, 1, 2, … , N – M – 1.
   N is the number of open-loop poles.
   M is the number of finite open-loop zeros.
7. Angle of Arrival or Departure: Apply the root-locus angle condition near a complex pole to calculate departure angle, or near a complex zero to calculate arrival angle. Sum the angles contributed by all other poles and zeros with a consistent sign convention.
8. Intersection of Root Locus with the Imaginary Axis: The Routh-Hurwitz criterion can find gain values at which continuous-time closed-loop roots lie on the imaginary axis. The auxiliary equation from the relevant zero row gives the crossing frequency when the standard Routh conditions apply.
9. Gain Margin: Gain margin is a frequency-domain measure of how much loop gain can change before a specified Nyquist stability boundary is reached. A root-locus imaginary-axis crossing can identify the corresponding critical positive gain for suitable systems. The formula below uses the page’s stated convention.
   
10. Phase Margin: Read phase margin from the open-loop frequency response at a gain-crossover frequency. Root-locus geometry alone does not provide it. Under the stated convention:
    
11. Symmetry of Root Locus: For a real-coefficient characteristic equation, the root locus is symmetric about the real axis because complex roots occur in conjugate pairs.

After a point s satisfies the angle condition, its gain follows from the magnitude condition:

1. Magnitude Criterion: At a point on the root locus, apply the magnitude condition shown below.
   
   This gives K for that point under the transfer-function convention used in the equation.
2. Pole-Zero Distances: The same magnitude condition can be evaluated as a ratio of distances from s to the open-loop poles and zeros:
   

Root Locus Plot

A root locus plot shows the modelled closed-loop pole locations for each permitted value of K. For a continuous-time system, gain ranges with all poles in the open left half-plane are asymptotically stable, subject to hidden modes and modelling assumptions. The plot does not by itself prove optimal performance or robustness.
Two basic construction rules are:

1. Real-axis segments: For the standard positive-gain locus, a point on the real axis belongs to the locus when the number of real open-loop poles and zeros to its right is odd.
   Complex poles and zeros do not enter this particular real-axis count.
2. Number of branches: The number of root-locus branches equals the number N of open-loop poles. If N exceeds M, then N – M branches terminate at infinity along the asymptotes.

Procedure to Plot Root Locus

The following procedure sketches the standard positive-gain root locus plot for a suitable real-coefficient rational single-loop model.

1. Find and plot all finite open-loop poles and zeros on the complex plane.
2. Start one branch at each open-loop pole when K = 0. As K tends to infinity, branches end at finite open-loop zeros or at infinity. When pole count exceeds zero count, the number ending at infinity equals the difference between the number of poles & number of zeros of G(s)H(s).
3. Mark valid real-axis segments using the odd-count rule for the chosen gain sign.
4. Calculate candidate breakaway and break-in points from dK/ds = 0, then reject points that fail the locus or gain conditions.
5. If N exceeds M, plot the centroid and N – M asymptotes at their calculated angles.
6. Calculate departure and arrival angles where needed, and find any imaginary-axis crossing with Routh-Hurwitz or direct substitution.
7. Use the magnitude condition to determine K at selected points, then verify computed closed-loop roots.

This procedure gives a sketch of the root locus plot. Numerical plotting and direct closed-loop pole calculation should verify the sketch, especially near repeated roots and stability boundaries.

1. Calculate gain margin from the open-loop frequency response if it is a required specification.
2. Calculate phase margin from the open-loop frequency response.
3. Use a Routh array to verify the continuous-time stability range where its assumptions apply.