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```					              Transfer Functions

1.   Transfer functions
2.   Standard process inputs
3.   First-order systems
5.   Integrating systems
Transfer Functions
l   The transfer function
» Represent relation between input U(s) & output Y(s) in the
Laplace domain
U(s)            Y(s)
» Usually denoted as G(s)                G(s)
Input           Output
» Y(s) = G(s)U(s)
Transfer
» Only applicable to linear models!     function
l   Deviation variables
» Defined as difference between variable and its steady-state
value

» Transfer functions always specified in terms of deviation
variables
» Y’(s) = G(s)U’(s)
» Usually often omit primes for notational simplicity
Transfer Function Example
l   Stirred tank heater

l   Initial conditions:

l   Subtract steady-state equation
Transfer Function Example cont.
l   Laplace transform

l   Rearrange noting that T’(0) = 0

l   Definitions

l   Transfer functions – 1st-order system
Properties of Transfer Functions

» Y(s) = G1(s)U1(s)+ G2(s)U2(s)

l   Multiplicative property
» Y2(s) = G1(s)G2(s)U(s)

l   ODE equivalence
Standard Process Inputs
l   Step input

l   Ramp input

l   Rectangular pulse input   l   Sinusoidal input
System Order
l   General transfer function

l   System order
» Order of the denominator polynomial D(s)
» Generally equal to the number of ODEs from which
G(s) was derived
l   First-order system

l   Second-order system
First-Order System
l   Standard form

l   Stirred tank heater

l   Step response
Ramp Response
Sinusoidal Response

l   First-order system:

l   Sinusoidal input:

Integrating Systems
l   Liquid storage tank

l   Deviation model

l   Laplace domain

l   Step response

» Integrating systems do not have a steady-state gain

```
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