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

Step 1: Understanding system stability.
For a linear time-invariant (LTI) control system to be stable, its natural response must decay to zero with time. This behavior depends on the location of the system poles in the s-plane.
Step 2: Relation between pole location and stability.
If the real part of all poles is negative, the exponential terms in the system response decay with time, ensuring stability. Poles on or to the right of the imaginary axis result in sustained or growing oscillations, leading to instability or marginal stability.
Step 3: Analyzing the options.
(A) On the imaginary axis: This leads to sustained oscillations and represents marginal stability, not stable behavior.
(B) In the right half of s-plane: Poles here cause exponential growth, making the system unstable.
(C) In the left half of s-plane: Correct — poles with negative real parts ensure exponential decay and stable system response.
(D) At the origin: This indicates marginal stability and does not guarantee stability.
Step 4: Conclusion.
A stable LTI system must have all its poles strictly located in the left half of the s-plane.