What is defined as the characteristic equation in control systems?

The characteristic equation in control systems is fundamentally linked to the stability and dynamic behavior of the system. It is derived from the transfer function, which describes the input-output relationship of the system.

Specifically, the characteristic equation is obtained by setting the denominator of the transfer function equal to zero. This is crucial because the values that satisfy this equation determine the poles of the system. The locations of these poles in the complex plane are paramount in assessing system stability: if any poles are in the right half-plane, the system is deemed unstable, while poles in the left half-plane indicate stability.

In contrast, the numerator of the transfer function relates to the zeros of the system and does not provide information about stability. The overall transfer function is a comprehensive representation of the system's input-output dynamics but does not specifically indicate stability characteristics unless the denominator is examined. The heat transfer equation, while important in thermal systems, is not related to the control systems concept of a characteristic equation.

Thus, defining the characteristic equation as the denominator of the transfer function set to zero highlights its critical role in determining a system's stability and dynamic response characteristics.