Question

Which of the following is an example for an unbounded input?

• A. Step function
• B. Sinusoidal function
• C. Ramp function
• D. Impulse function

Hint:

A bounded signal is one that you can draw inside a horizontal band of finite width.
A ramp signal keeps climbing up and cannot be enclosed in any finite horizontal band, making it unbounded.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question asks us to identify which of the standard test inputs is considered an "unbounded input".
In system dynamics and control, inputs are categorized as bounded or unbounded to analyze system stability, specifically Bounded-Input Bounded-Output (BIBO) stability.

Step 2: Key Formula or Approach:
An input signal $u(t)$ is defined as bounded if its magnitude remains less than some finite positive constant $M$ for all time $t \ge 0$:
\[ |u(t)| \le M \lt \infty \quad \forall t \ge 0 \] If the magnitude of $u(t)$ can grow to infinity as $t$ increases, then the input is unbounded:
\[ \lim_{t \to \infty} |u(t)| = \infty \]

Step 3: Detailed Explanation:
Let us analyze each option individually:
1. Step function: $u(t) = A$ for $t \ge 0$, where $A$ is a constant. The maximum value is $A$, which is finite. Thus, it is a bounded input.
2. Sinusoidal function: $u(t) = A \sin(\omega t)$ or $A \cos(\omega t)$. The magnitude $|u(t)|$ is always bounded between $-A$ and $+A$. Thus, it is a bounded input.
3. Ramp function: $u(t) = A \cdot t$ for $t \ge 0$. As time $t$ approaches infinity, the value of $u(t)$ also approaches infinity:
\[ \lim_{t \to \infty} A \cdot t = \infty \] Since there is no finite upper limit $M$ to constrain this function, the ramp function is an unbounded input.
4. Impulse function: $\delta(t)$ represents an extremely short-duration pulse with infinite amplitude at $t = 0$ but zero elsewhere, with a finite integral of 1. In the context of standard control input signals over time, it is transient and doesn't continuously grow. The ramp is the classic continuous unbounded signal.

Step 4: Final Answer
The ramp function is the correct example of an unbounded input, corresponding to option (C).