Root-Locus Method

A cruise control system in a car is designed to maintain a specified speed automatically by adjusting the gas pedal. The system continuously measures the vehicle's speed and makes fine adjustments to the pedal to achieve this goal. The root locus method is particularly useful for understanding how the cruise control system's behavior changes under varying conditions, such as when the car goes uphill, downhill, or faces strong wind resistance.

This system can be represented by a block diagram, with its transfer function providing a mathematical model. To determine the locations of the system's poles for different gas pedal forces, the quadratic formula is applied to the denominator of the transfer function. As the pedal force changes, one pole of the system moves to the right while the other moves to the left. These poles eventually converge at a specific point before diverging into the complex plane, influencing the closed-loop poles of the system.

The root locus method visually illustrates how variations in pedal force impact the system's response. At low pedal forces, the system is overdamped, meaning it returns to the desired speed without oscillating but may take longer. At a specific force, the system is critically damped, achieving the fastest return to the desired speed without overshooting. At high pedal forces, the system becomes underdamped, resulting in oscillations around the desired speed before settling.

Importantly, the root locus for this system never crosses into the right half-plane of the s-plane, ensuring that the system remains stable regardless of the pedal force applied. This stability is a crucial feature for the reliable operation of the cruise control system.

The root locus method is not only useful for analyzing second-order systems but also proves valuable for higher-order systems, providing insights into system behavior and aiding in the design of robust control mechanisms. By leveraging root locus analysis, engineers can optimize the performance of complex systems like cruise control, ensuring they remain stable and responsive under various operating conditions.