## Textbooks & Solution Manuals

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

## Tip our Team

Our Website is free to use.
To help us grow, you can support our team with a Small Tip.

## Holooly Tables

All the data tables that you may search for.

## Holooly Help Desk

Need Help? We got you covered.

## Holooly Arabia

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

Products

## Textbooks & Solution Manuals

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

## Holooly Arabia

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

## Holooly Help Desk

Need Help? We got you covered.

## Q. 4.2

REALIZATION OF PROCESS AND DISTURBANCE TRANSFORM-DOMAIN TRANSFER FUNCTIONS.

Given the following transfer function model:

$y(s) = g(s) u(s) + g_{d}(s) d(s)$        (4.64a)

with

$g(s) = \frac{K}{\tau s + 1}$        (4.64b)

$g_{d}(s) = \frac{K_{d}}{\tau _{d} s + 1}$        (4.64c)

find an equivalent state-space representation.

## Verified Solution

Introducing Eqs. (4.64b,c) into Eq. (4.64a) the transfer function model may be rewritten as:

$y(s) = x_{1}(s) + x_{2}(s)$        (4.65a)

where

$x_{1}(s) = \frac{K}{\tau s + 1} u(s)$        (4.65b)

$x_{2}(s) = \frac{K_{d}}{\tau _{d} s + 1} d(s)$        (4.65c)

If we now rearrange Eqs. (4.65b,c) as was done in Example (4.1) we will have:

$(\tau s + 1) x_{1}(s) = K u(s)$        (4.66a)

$(\tau _{d} s + 1) x_{2}(s) = K_{d} d(s)$        (4.66b)

from where we may now deduce one possible set of differential equations which upon Laplace transformation would give rise to Eq. (4.66a,b) and which are, respectively:

$\tau \frac{dx_{1}}{dt} + x_{1}(t) = K u(t)$        (4.67a)

for the process, and

$\tau _{d} \frac{dx_{2}}{dt} + x_{2}(t) = K_{d} d(t)$        (4.67b)

for the disturbance. The process output is obtained as:

$y(t) = x_{1}(t) + x_{2}(t)$        (4.68)

Along with the initial conditions with $x_{1}$(0) = 0, and $x_{2}$(0) = 0, Eqs. (4.67a,b) and (4.68) constitute one possible state-space equivalent of the model given in Eq. (4.64.)

Remarks

In the special case for which $\tau = \tau _{d}$, we may add Eq. (4.66a) to Eq. (4.66b) to obtain (by means of Eq. (4.65a)):

$(\tau s + 1) y(s) = K u(s) + K_{d} d(s)$

and the differential equation extracted from here will simply be:

$\tau \frac{dy}{dt} + y(t) = K u(t) + K_{d} d(t)$

We may then set y(t) = x(t) and obtain an equivalent state-space representation:

$\tau \frac{dx}{dt} + x(t) = K u(t) + K_{d} d(t)$

$y(t) = x(t)$