Construction of Root Locus

The construction of a root locus involves several key steps to analyze and visualize the behavior of a system's poles with varying gain. The number of branches in the root locus equals the number of closed-loop poles and is symmetrical about the real axis.

For positive gain values, the root locus exists on the real axis to the left of an odd number of finite open-loop poles or zeros. The root locus starts at the open-loop poles and traces the paths of the closed-loop poles as the gain increases. It ends at the open-loop zeros, where the system poles stabilize as the gain continues to rise.

If a function approaches infinity as s approaches infinity, it has a pole at infinity. Conversely, if a function approaches zero, it has a zero at infinity. As the root locus extends towards infinity, it follows specific asymptotes. These asymptotes' equations are determined by the real-axis intercept and their angles, which indicate the paths the loci take from the poles to infinity.

To calculate these paths for a given system, one determines the real-axis intercept and the angles of the asymptotes. The angles are derived based on the number of poles and zeros and start repeating as the gain increases. The formula for the angle of asymptotes is given by,

where k is an integer, n is the number of poles, and m is the number of zeros.

The complete root locus plot begins at the open-loop poles and terminates at the open-loop zeros. This plot adheres to the rules governing the root locus method, ensuring that the system's stability and response can be thoroughly analyzed and predicted.

By following these steps, engineers can effectively use the root locus method to design and tune control systems, ensuring desired performance and stability across various operating conditions. This approach provides a clear visualization of how the system poles migrate with increasing gain, aiding in the robust design of control strategies.