Control System Problem

In an open-loop system, such as a basic thermostat, the poles of the transfer function influence the system's response but do not determine its stability. However, when feedback is introduced to form a closed-loop system, such as an advanced thermostat that adjusts heating based on room temperature, stability is governed by the new poles of the closed-loop transfer function.

When forming a closed-loop system, issues can arise if the poles cross into the unstable region, leading to potential temperature fluctuations. Identifying the poles of the open-loop transfer function is relatively straightforward and remains constant despite changes in system gain. In contrast, the poles of the closed-loop transfer function vary with adjustments in system gain and require more complex calculations involving the factoring of the denominator.

Although the zeros and poles of transfer functions are generally known, pinpointing the poles of a specific function that changes with system gain is more challenging. The transient response and overall stability of a system are closely linked to these poles. Without considering specific gain values, the system's performance remains unclear.

The root locus method offers a visual approach to understanding how the poles of a system vary with changes in system gain. By plotting the possible locations of the closed-loop poles on the s-plane, the root locus method provides insights into how the system's stability and transient response will evolve as the gain changes. This method allows engineers to predict and adjust the system's behavior to ensure stability and desired performance.

In summary, while open-loop system poles are easily identified and stable, the poles of a closed-loop system depend on the system gain and require more detailed analysis. The root locus method is a valuable tool for visualizing these changes, aiding in the design and tuning of stable closed-loop systems.