Question

For a stable linear time-invariant system, the location of all poles must be:

• A. On the imaginary axis
• B. In the right half of s-plane
• C. In the left half of s-plane
• D. At the origin

Hint:

Remember: {Left half-plane = Stable}, {Imaginary axis = Marginally stable}, {Right half-plane = Unstable}.

Answer 1

The Correct Option is C

In control systems, particularly for linear time-invariant (LTI) systems, the stability of the system is determined by the location of the poles of its transfer function in the complex s-plane.

1. Understanding Poles: In the context of a transfer function, poles are the values of s that make the denominator of the transfer function zero. The poles are critically important for assessing system behavior, particularly stability.

2. Stable System Criteria: For a system to be stable, all poles of its transfer function must lie in the left half of the s-plane. This is because:

   • Pole locations in the left half of the s-plane correspond to exponential decay, which signifies stability.
   • If any pole lies on the right half of the s-plane, it will induce exponential growth, leading to instability.
   • If poles lie on the imaginary axis, they result in sustained oscillations, indicating a marginally stable system which is not considered stable in strict terms.

3. Justification of Options:

   • On the imaginary axis: This corresponds to marginal stability, not absolute stability.
   • In the right half of s-plane: Poles here indicate instability due to exponential growth.
   • In the left half of s-plane: This indicates that all exponential terms decay over time, ensuring stability.
   • At the origin: Indicates a pole at zero which can be part of a marginally stable system but does not ensure overall stability.

4. Conclusion: The correct answer is that for a stable linear time-invariant system, the location of all poles must be in the left half of the s-plane.