Pole and System Stability

The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.

Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's response.

Repeated poles, occurring more than once in the denominator, indicate a more complex system behavior. These poles suggest either oscillatory behavior or slower decay rates, leading to terms involving tⁿe^σt in the time-domain response, where n is the multiplicity of the pole, and σ is the real part of the pole.

Complex poles have both real and imaginary parts, resulting in oscillatory components in the system's response. These poles typically appear in conjugate pairs, σ±jω, leading to responses involving sine and cosine terms modulated by an exponential decay,

e^σt(cos⁡(ωt)+jsin⁡(ωt)).

The stability of a Linear Time-Invariant (LTI) system is determined by the locations of its poles in the s-plane. For Bounded Input, Bounded Output (BIBO) stability, all poles must lie in the left half-plane (LHP), ensuring every impulse response decays over time. Repeated poles in the LHP contribute to stability but with more gradual decay due to the increased order of the system's response.

Conversely, poles in the right half-plane (RHP) lead to instability, as these poles cause exponential growth in the system's response, resulting in unbounded output even for bounded inputs.

Proper rational functions have a numerator degree less than or equal to the denominator degree and follow stability rules similar to strictly proper functions. Improper rational functions, where the numerator degree exceeds the denominator degree, are inherently not BIBO stable. This is because such functions imply that the output can become unbounded for bounded input signals, violating the principle of boundedness.

In summary, the poles of a transfer function—whether simple, repeated, or complex—are pivotal in understanding the system's response and stability. The location of these poles in the s-plane determines whether the system exhibits stable behavior or becomes unstable, with proper and improper rational functions providing additional layers of stability.