For the transfer functions

G_{P}(s)=\frac{3 e^{-2 s}}{(15 s+1)(10 s+1)} \quad G_{v}=G_{m}=1

(a) Derive an analytical expression for the step response to a unit step change. Evaluate the step-response coefficients, \left\{S_{i}\right\}, for a sampling period of \Delta t=1.

(b) What value of model horizon N should be specified in order to ensure that the step-response model covers a period of at least 99 \% of the open-loop settling time? (That is, we require that N \Delta t \geq t_{99} where t_{99} is the 99 \% settling time.)