Question

Selected data points of the step response of a stable first-order linear time-invariant (LTI) system are given below. The closest value of the time constant (in seconds) of the system is:

\[ \begin{array}{|c|c|} \hline \textbf{Time (sec)} & \textbf{Output} \\ \hline 0.6 & 0.78 \\ 1.6 & 2.8 \\ 2.6 & 2.98 \\ 10 & 3 \\ \infty & 3 \\ \hline \end{array} \]

• A. 1
• B. 3
• C. 4
• D. 2

Hint:

For a first-order LTI system, the time constant \( \tau \) can be estimated by looking at the time when the output reaches approximately 63\% of its steady-state value.

Answer 1

The Correct Option is A

For a first-order LTI system, the output follows the form:
\[ y(t) = 1 - e^{-\frac{t}{\tau}} \] where \( \tau \) is the time constant. As time approaches infinity, the output reaches its steady-state value (3 in this case). 
To estimate the time constant \( \tau \), we can use the given data points:
- At \( t = 0.6 \) sec, the output is 0.78, and at \( t = 1.6 \) sec, the output is 2.8.
- For a first-order system, the output reaches approximately 63% of its final value after \( t = \tau \). 
Since the output is approaching the steady-state value of 3, the time constant \( \tau \) can be estimated by observing that the system reaches around 63% of 3 around 1 second. The output is 0.78 at \( t = 0.6 \), and it moves closer to 3 as time increases, confirming that the time constant is closest to 1 second. Thus, the closest value for the time constant is \( \tau = 1 \) second.