Products
Rewards
from HOLOOLY

We are determined to provide the latest solutions related to all subjects FREE of charge!

Enjoy Limited offers, deals & Discounts by signing up to Holooly Rewards Program

HOLOOLY

HOLOOLY
TABLES

All the data tables that you may search for.

HOLOOLY
ARABIA

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

HOLOOLY
TEXTBOOKS

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

HOLOOLY
HELP DESK

Need Help? We got you covered.

## Q. 20.1

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.)

## Verified Solution

a) The unit step response is

$Y(s)=G_{p}(s) U(s)=\left(\frac{3 e^{-2 s}}{(15 s+1)(10 s+1)}\right)\left(\frac{1}{s}\right)=3 e^{-2 s}\left[\frac{1}{s}-\frac{45}{15 s+1}+\frac{20}{10 s+1}\right]$

Therefore,

$y(t)=3 S(t-2)\left[1+2 e^{-(t-2) / 10}-3 e^{-(t-2) / 15}\right]$

For $\Delta t=1$,

$S_{i}=y(i \Delta t)=y(i)=\{0,0,0.0095,0.036,0.076,0.13 \ldots\}$

b) Evaluate the expression for $y(t)$ in part (a)

$y(t)=0.99(3) \approx 2.97 \text { at } t=87$

Thus, $N=87$, for $99 \%$ complete response.