The measurement or sensor element in a feedback system is important to the accuracy of the system [6]. The dynamic response of the sensor is important. Most sensor elements possess a transfer function
$$
H(s)=\frac{k}{\tau s+1}
$$
Suppose that a position-sensing photo detector has $\tau=4 \mu$ s and $0.999<k<1.001$. Obtain the step response of the system, and find the $k$ resulting in the fastest response-that is, the fastest time to reach $98 \%$ of the final value.