5. TRANSFER FUNCTIONS AND BLOCK DIAGRAMS
The theory of Laplace transforms is quite extensive, but fortunately only
a small and isolated part is required for the initial study of dynamics and
control.
1. Laplace transforms – Pierre-Simon Laplace (1749-1827)
2. Transfer function models of dynamic systems
3. Linear, time invariant models
4. Block diagram form
● manipulation
● simplification
Topic 5: Transfer Functions and Block Diagrams 1
,In Topic 1 we saw that it is desirable to model a system as an input-
output relationship, e.g.:
output input
⑧ ~
(5.1)
tranfer function
In Topic 2 we found a generic description of the form:
(5.2)
using the Laplace transform we can express (5.2) in the form of (5.1)
If and are the respective Laplace transforms of and ,
and and are related by the DE (5.2) then
with Transfer function
fls) - is a ratio of polynomials
Topic 5: Transfer Functions and Block Diagrams 2
,5.1 Laplace transforms (again)
Using differential equations, it is difficult to model a system as a block
diagram convolution.
O. ->
just little
a
bit
The Laplace transform is defined as: before O
(5.3)
Where s = 0+yw is a complex variable
Knowing and that the integral in (5.3) exists, it is possible to find
the function )
the Laplace transform of .
The Laplace transform changes a function of time, , into a new
function of the complex variable such that integration and
differentiation are algebraic operations.
Topic 5: Transfer Functions and Block Diagrams 3
, 5.2 Laplace transform examples
5.2.1 The unit step input Use: transient response
steady-state error
something changes from value to another
this sharp change between
one
O &1, its a bit of square wave form
ult)
3 I
this edge here tell us
that we have got an It I
is for to
- <0
infinite
in it.
range of frequencies & 0 for
Laplace transform is found by substituting into (5.3)
1, t> 0
St
N pure integrator
-
9.Pile
I
-
"
df
-
e
- If you multiply signal (5.4)
a
=
S
S
O with it, you will be
integrating it.
For a step of magnitude , the transform of is
Topic 5: Transfer Functions and Block Diagrams 4
The theory of Laplace transforms is quite extensive, but fortunately only
a small and isolated part is required for the initial study of dynamics and
control.
1. Laplace transforms – Pierre-Simon Laplace (1749-1827)
2. Transfer function models of dynamic systems
3. Linear, time invariant models
4. Block diagram form
● manipulation
● simplification
Topic 5: Transfer Functions and Block Diagrams 1
,In Topic 1 we saw that it is desirable to model a system as an input-
output relationship, e.g.:
output input
⑧ ~
(5.1)
tranfer function
In Topic 2 we found a generic description of the form:
(5.2)
using the Laplace transform we can express (5.2) in the form of (5.1)
If and are the respective Laplace transforms of and ,
and and are related by the DE (5.2) then
with Transfer function
fls) - is a ratio of polynomials
Topic 5: Transfer Functions and Block Diagrams 2
,5.1 Laplace transforms (again)
Using differential equations, it is difficult to model a system as a block
diagram convolution.
O. ->
just little
a
bit
The Laplace transform is defined as: before O
(5.3)
Where s = 0+yw is a complex variable
Knowing and that the integral in (5.3) exists, it is possible to find
the function )
the Laplace transform of .
The Laplace transform changes a function of time, , into a new
function of the complex variable such that integration and
differentiation are algebraic operations.
Topic 5: Transfer Functions and Block Diagrams 3
, 5.2 Laplace transform examples
5.2.1 The unit step input Use: transient response
steady-state error
something changes from value to another
this sharp change between
one
O &1, its a bit of square wave form
ult)
3 I
this edge here tell us
that we have got an It I
is for to
- <0
infinite
in it.
range of frequencies & 0 for
Laplace transform is found by substituting into (5.3)
1, t> 0
St
N pure integrator
-
9.Pile
I
-
"
df
-
e
- If you multiply signal (5.4)
a
=
S
S
O with it, you will be
integrating it.
For a step of magnitude , the transform of is
Topic 5: Transfer Functions and Block Diagrams 4
