Transfer Function Poles and System Response

In control systems and signal processing, a transfer function, often denoted as $H(s)$, describes the relationship between the output and input of a linear time-invariant (LTI) system in the complex frequency domain (using the Laplace transform variable $s$).

The poles of a transfer function are the specific values of the complex frequency $s$ for which the transfer function's denominator equals zero. These are essentially the roots of the characteristic equation of the system.

Significance of Poles in System Behavior

The location of the poles in the complex $s$-plane is critical because it dictates the system's stability and the nature of its transient response. Each pole contributes a term of the form $A e^{p_i t}$ to the system's impulse response, where $p_i$ is the value of the pole.

• If a pole is located at $s = p$, the system's response will include a component related to $e^{pt}$.
• When $p$ is a complex number, $p = \sigma + j\omega$, this term becomes $e^{(\sigma + j\omega)t} = e^{\sigma t} e^{j\omega t}$. This represents a sinusoidal response with an amplitude that either grows ($ \sigma > 0 $), decays ($ \sigma < 0 $), or remains constant ($ \sigma = 0 $), depending on the real part ($\sigma$).

Response Behavior at Poles

Mathematically, poles are singularities of the transfer function $H(s)$. This means that as the complex frequency $s$ approaches a pole value $p$, the magnitude of the transfer function $|H(s)|$ tends to grow without bound. In simpler terms, the system's response magnitude can become extremely large, approaching infinite, at or very near these specific complex frequencies (the poles). This is particularly evident in concepts like resonance. Therefore, the poles represent the complex frequencies where the system's response exhibits this unbounded characteristic.