Root Locus Asymptotes Calculation

Introduction

In control system design, the root locus plot is a graphical representation of the closed-loop system's pole locations as a function of a parameter, typically the gain. Asymptotes of the root locus provide valuable information about the system's behavior without requiring detailed calculations.

The root locus asymptotes are straight lines that the root locus approaches as the gain increases. These asymptotes are particularly useful for understanding the system's stability and response characteristics.

Root Locus Asymptotes Formula

The angles of the root locus asymptotes are determined by the poles and zeros of the open-loop transfer function. The formula for the angles of the asymptotes is:

The center of the asymptotes is given by the average of the finite poles and zeros of the open-loop transfer function.

How to Calculate Asymptotes

To calculate the root locus asymptotes, follow these steps:

1. Identify the number of finite poles (n) and zeros (m) of the open-loop transfer function.
2. Calculate the angles of the asymptotes using the formula θ = (180° + 360° × k) / n for k = 0 to (n-1).
3. Determine the center of the asymptotes by averaging the finite poles and zeros.
4. Plot the asymptotes on the root locus plot.

Worked Example

Consider an open-loop transfer function with poles at s = -1 and s = -2 and a zero at s = -3. We will calculate the root locus asymptotes.

1. Number of finite poles (n) = 2
2. Number of zeros (m) = 1
3. Calculate the angles of the asymptotes:
   • For k = 0: θ₁ = (180° + 0) / 2 = 90°
   • For k = 1: θ₂ = (180° + 360°) / 2 = 270°
4. Calculate the center of the asymptotes:

   Center = (Pole₁ + Pole₂ + Zero) / (n + m) = (-1 + -2 + -3) / (2 + 1) = -6 / 3 = -2

The root locus asymptotes are at 90° and 270° with a center at s = -2.