Question

A measuring instrument has transfer function \[ G_m(s)=\frac{1.05}{2s+1}\,e^{-s}. \] At \(t=0\), a step of \(+1\) unit is applied. Find the time taken for the output to increase by \(1\) unit.

Hint:

For a FOPDT device, the output reaches a value \(y^*\) at time \(t=L-\tau\ln(1-y^*/K)\) (for a unit input), remembering to include the pure time delay \(L\).

Answer 1

Correct Answer: 6.99

Step 1: The step response of a FOPDT model \(K\,\dfrac{1}{\tau s+1}e^{-Ls}\) is \[ y(t)=K\left[1-e^{-(t-L)/\tau}\right]u(t-L). \] Here \(K=1.05,\ \tau=2,\ L=1\). Step 2: Set \(y(t)=1\) and solve for \(t\ge L\): \[ 1=1.05\Big(1-e^{-(t-1)/2}\Big) \;\Rightarrow\; e^{-(t-1)/2}=1-\frac{1}{1.05}=\frac{1}{21}. \] \[ \frac{t-1}{2}=\ln 21 \;\Rightarrow\; t=1+2\ln 21=1+2(3.044522)=7.089\ \text{time units}. \] Rounded to two decimals: \(\boxed{7.09}\).