System transfer function:
Modify the transfer function such that the response reaches steady state value after 8 seconds when unit step input is applied. Within 2% of steady state value can be considered "reached." Only the response speed should change and the form should not change to any input. Plot the response before and after modification.
Response reaches less than 2% of steady state value at around 4τ so we want τ to be 2 seconds. We want a slower response. We time time-scale the response by 0.05/2 = 0.025. Modified transfer function is then given by,
Response before and after modification as a function of time (unit step input so A = 1):
Plotting these equations: