Control systems – classify inputs by boundedness From the following standard test inputs (given by their Laplace transforms), identify which input is unbounded as time increases: 1, 1/S, 1/S^2, and 1/(S^2 + 1).

Introduction / Context:
In control engineering, standard test inputs—impulse, step, ramp, and sinusoid—are used to probe system behavior. A key classification is whether an input is bounded in amplitude over time. This question asks you to spot which of the commonly used inputs grows without limit as time progresses.

Given Data / Assumptions:

• Impulse input: Laplace transform 1.
• Step input: Laplace transform 1/S.
• Ramp input: Laplace transform 1/S^2.
• Unit sine input: Laplace transform 1/(S^2 + 1).
• Boundedness refers to the magnitude of the time-domain signal for t ≥ 0.

Concept / Approach:
An input is called bounded if |u(t)| remains finite for all t ≥ 0. The ramp increases linearly with time (u(t) = t for t ≥ 0), so it is unbounded. The step is a constant (bounded). The sinusoid oscillates between fixed limits (bounded). The ideal impulse, though having a Dirac spike at t = 0, is a generalised function with finite area; in the usual boundedness discussion for test signals, “unbounded over time” identifies signals whose magnitude grows without limit as t increases—namely the ramp.

Step-by-Step Solution:

Verification / Alternative check:
Plot u(t) = t, u(t) = 1, and u(t) = sin(t). Only u(t) = t diverges as t → ∞, confirming unboundedness.

Why Other Options Are Wrong:

• 1/S: Step is finite for all t.
• 1/(S^2+1): Sinusoid has bounded amplitude.
• 1: Ideal impulse is a distribution with finite area; the question targets signals that grow without limit over time, which the impulse does not.

Common Pitfalls:
Confusing mathematical idealisations (impulse spike) with long-time unbounded growth; unbounded “over time” points to ramp behavior.

Final Answer:
1/S^2