Stability of a System

A system is said to be stable if it is able to return to its equilibrium state after experiencing a disturbance. In the context of control systems, stability is a crucial property that determines the behavior and performance of the system. It is important to analyze the location of the poles and zeros of the system to determine its stability.

Poles and Zeros

In the theory of control systems, the transfer function is commonly used to represent the relationship between the input and output of a system. The poles and zeros of the transfer function play a significant role in determining the stability of the system.

Poles: The poles of a transfer function are the values of s (complex numbers) for which the transfer function becomes infinite. They are also known as the characteristic roots of the system. The poles determine the dynamic behavior of the system.

Zeros: The zeros of a transfer function are the values of s for which the transfer function becomes zero. They represent the input values that result in zero output. The zeros affect the steady-state behavior of the system.

Stability Criteria

The location of the poles and zeros of a system provides valuable information about its stability. The stability of a system can be determined using the following criteria:

1. Bounded-Input Bounded-Output (BIBO) Stability: A system is said to be BIBO stable if all its poles lie in the left half of the complex plane. In other words, the real part of all the poles should be negative. This criterion ensures that the system response remains bounded for any bounded input.

2. Lyapunov Stability: Lyapunov stability is a more general concept that defines stability based on the properties of a Lyapunov function. It requires the poles of the system to have negative real parts and be strictly stable.

Answer Explanation

The correct answer to the given question is option A: "Its poles lie at the left of the imaginary axis." This answer aligns with the BIBO stability criterion. If all the poles of a system are located in the left half of the complex plane, it indicates that the real part of the poles is negative.

When the poles have negative real parts, it implies that the system is stable and will return to its equilibrium state after experiencing a disturbance. The negative real parts ensure that the system response remains bounded for any bounded input.

Therefore, the location of the poles plays a crucial role in determining the stability of a system. Option A correctly identifies this criterion for stability.