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 sec, of the system is:

Time (sec)	0.6	1.6	2.6	10	∞
Output	0.78	1.65	2.18	2.98	3

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

For first-order systems, use the step response formula \( y(t) = A(1 - e^{-t/\tau}) \) and substitute a known output value to solve for the time constant \( \tau \).

Answer 1

The Correct Option is B

The final value of the step response is 3. The step response of a first-order LTI system is:

y(t) = A(1 - e^-t/τ)

where A = 3 is the final value and τ is the time constant. We can estimate τ using a data point.

Using the value at t = 1.6, we have:

    y(1.6) = 1.65 = 3(1 - e-1.6/τ)
    ⇒ 1.65 / 3 = 1 - e-1.6/τ
    ⇒ e-1.6/τ = 1 - 0.55 = 0.45
    ⇒ -1.6 / τ = ln(0.45)
    ⇒ τ = 1.6 / -ln(0.45) ≈ 1.6 / 0.798 ≈ 2.0