Stability of a system

The stability of a system refers to its ability to return to a stable state after being subjected to a disturbance. In the context of control systems, stability is an important characteristic as it determines whether the system will exhibit desirable behavior or become uncontrollable.

Roots with higher multiplicity

The roots of a system's characteristic equation play a significant role in determining the stability of the system. A root with a higher multiplicity means that it appears multiple times in the characteristic equation.

Imaginary axis

The imaginary axis in the complex plane represents values of the complex variable where the real part is zero. In the context of system stability, roots located on the imaginary axis can have a significant impact.

Unstable system

An unstable system is one that does not return to a stable state after being subjected to a disturbance. It exhibits unpredictable behavior and can potentially lead to catastrophic consequences in control applications.

Explanation

When roots with higher multiplicity appear on the imaginary axis, it indicates that the system has oscillatory modes with a specific frequency. These oscillations can lead to instability if not properly controlled. Here's a detailed explanation of why the correct answer is option 'B' (Unstable):

1. Oscillatory behavior: When roots with higher multiplicity appear on the imaginary axis, it implies that the system has oscillatory behavior. The presence of imaginary roots indicates the existence of sinusoidal modes in the system response.

2. Lack of damping: Oscillatory modes without any damping can lead to unbounded growth or decay of the system response. The absence of damping prevents the system from dissipating energy, leading to unstable behavior.

3. Exponential growth or decay: When roots with higher multiplicity are present on the imaginary axis, the system response exponentially grows or decays over time. This exponential behavior is characteristic of an unstable system.

4. Unpredictable response: In an unstable system, the response to a disturbance becomes unpredictable and can diverge to infinity. Such behavior is undesirable in control systems as it can lead to instability and loss of control over the system.

Therefore, when roots with higher multiplicity are located on the imaginary axis, the system is classified as unstable. It is crucial to introduce proper damping or control mechanisms to stabilize the system and ensure its desirable behavior.