Stability of linear systems: Whether a linear system is stable or unstable depends primarily on what factors?

Introduction / Context:Stability analysis determines whether system outputs remain bounded for bounded inputs. For linear time-invariant (LTI) systems, stability is a key concept tested in control theory.

Given Data / Assumptions:

• We are considering linear, time-invariant systems.
• Bounded-input bounded-output (BIBO) stability definition is applied.

Concept / Approach:

BIBO stability requires that the impulse response of the system be absolutely integrable. This depends solely on the system’s poles or impulse response, not on any particular input signal.

Step-by-Step Solution:

Verification / Alternative check:

Testing with sinusoidal, exponential, or step inputs confirms: if system is stable, all outputs remain bounded; if unstable, any bounded input can produce unbounded output.

Why Other Options Are Wrong:

• Input function alone cannot determine stability; it only excites the system.
• Both (a) and (b) is misleading, since input cannot rescue an unstable system.
• Either (a) or (b) is logically incorrect.

Common Pitfalls:

• Confusing stability with performance (e.g., overshoot or steady-state error).
• Believing certain “gentle” inputs can make unstable systems behave stably.

Final Answer:

It is a property of the system only.