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## Q. 4.3

REALIZATION OF A FIRST-ORDER-PLUS-TIME-DELAY TRANSFORM-DOMAIN TRANSFER FUNCTION.

Given the following transfer function model:

$y(s) = g(s) u(s)$        (4.69a)

with

$g(s) = \frac{Ke^{- \alpha s}}{\tau s + 1}$        (4.69b)

find an equivalent state-space representation.

## Verified Solution

Again, introducing Eq. (4.69b) into Eq. (4.69a), the transfer function model may be rewritten as:

$(\tau s + 1) y(s) = Ke^{-\alpha s} u(s)$         (4.70)

To obtain a differential equation which, upon Laplace transform. would give rise to Eq. (4.69), we now only need to recall the time-shift property of Laplace transform given in Eq. (3.24) in Chapter 3. This, along with what we have seen earlier in Example 4.1, helps us to establish that one such differential equation is:

$L \left\{f(t – a)\right\} = e^{-as} \bar f(s)$        (3.24)

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

We may then set y(t) = x(t) and rewrite this as:

$\tau \frac{dx}{dt} + x(t) = K u(t – \alpha)$        (4.71a)

$y(t) = x(t)$        (4.71b)

and this will then provide a state-space equivalent of the model given in Eq. (4.69).