<s>
In	O
control	O
system	O
theory	O
,	O
and	O
various	O
branches	O
of	O
engineering	O
,	O
a	O
transfer	B-Algorithm
function	I-Algorithm
matrix	I-Algorithm
,	O
or	O
just	O
transfer	O
matrix	B-Architecture
is	O
a	O
generalisation	O
of	O
the	O
transfer	O
functions	O
of	O
single-input	O
single-output	O
(	O
SISO	O
)	O
systems	O
to	O
multiple-input	O
and	O
multiple-output	O
(	O
MIMO	O
)	O
systems	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
relates	O
the	O
outputs	O
of	O
the	O
system	O
to	O
its	O
inputs	O
.	O
</s>
<s>
This	O
results	O
in	O
all	O
the	O
elements	O
of	O
the	O
transfer	O
matrix	B-Architecture
being	O
in	O
units	O
of	O
impedance	O
.	O
</s>
<s>
This	O
is	O
needed	O
both	O
to	O
describe	O
transducers	B-Algorithm
that	O
make	O
connections	O
between	O
domains	O
and	O
to	O
describe	O
the	O
system	O
as	O
a	O
whole	O
.	O
</s>
<s>
If	O
the	O
matrix	B-Architecture
is	O
to	O
properly	O
model	O
energy	O
flows	O
in	O
the	O
system	O
,	O
compatible	O
variables	O
must	O
be	O
chosen	O
to	O
allow	O
this	O
.	O
</s>
<s>
A	O
MIMO	O
system	O
with	O
outputs	O
and	O
inputs	O
is	O
represented	O
by	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
Each	O
entry	O
in	O
the	O
matrix	B-Architecture
is	O
in	O
the	O
form	O
of	O
a	O
transfer	O
function	O
relating	O
an	O
output	O
to	O
an	O
input	O
.	O
</s>
<s>
This	O
may	O
be	O
written	O
more	O
succinctly	O
in	O
matrix	B-Architecture
operator	O
notation	O
as	O
,	O
</s>
<s>
where	O
is	O
a	O
column	O
vector	O
of	O
the	O
outputs	O
,	O
is	O
a	O
matrix	B-Architecture
of	O
the	O
transfer	O
functions	O
,	O
and	O
is	O
a	O
column	O
vector	O
of	O
the	O
inputs	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
it	O
is	O
convenient	O
to	O
express	O
the	O
transfer	O
matrix	B-Architecture
in	O
terms	O
of	O
the	O
Laplace	O
transform	O
(	O
in	O
the	O
case	O
of	O
continuous	O
time	O
variables	O
)	O
or	O
the	O
z-transform	B-Algorithm
(	O
in	O
the	O
case	O
of	O
discrete	O
time	O
variables	O
)	O
of	O
the	O
variables	O
.	O
</s>
<s>
which	O
indicates	O
that	O
the	O
variables	O
and	O
matrix	B-Architecture
are	O
in	O
terms	O
of	O
,	O
the	O
complex	O
frequency	O
variable	O
of	O
the	O
s-plane	O
arising	O
from	O
Laplace	O
transforms	O
,	O
rather	O
than	O
time	O
.	O
</s>
<s>
For	O
discrete	O
time	O
systems	O
is	O
replaced	O
by	O
from	O
the	O
z-transform	B-Algorithm
,	O
but	O
this	O
makes	O
no	O
difference	O
to	O
subsequent	O
analysis	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
is	O
particularly	O
useful	O
when	O
it	O
is	O
a	O
proper	O
rational	O
matrix	B-Architecture
,	O
that	O
is	O
,	O
all	O
its	O
elements	O
are	O
proper	O
rational	O
functions	O
.	O
</s>
<s>
In	O
systems	O
engineering	O
,	O
the	O
overall	O
system	O
transfer	O
matrix	B-Architecture
is	O
decomposed	O
into	O
two	O
parts	O
:	O
representing	O
the	O
system	O
being	O
controlled	O
,	O
and	O
representing	O
the	O
control	O
system	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
transfer	O
matrix	B-Architecture
of	O
a	O
two-port	B-Algorithm
network	I-Algorithm
can	O
be	O
defined	O
as	O
follows	O
,	O
</s>
<s>
The	O
z-parameters	O
are	O
not	O
the	O
only	O
way	O
that	O
transfer	O
matrices	O
are	O
defined	O
for	O
two-port	B-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
However	O
,	O
only	O
two	O
of	O
these	O
can	O
be	O
extended	O
beyond	O
two	B-Algorithm
ports	I-Algorithm
to	O
an	O
arbitrary	O
number	O
of	O
ports	O
.	O
</s>
<s>
These	O
two	O
are	O
the	O
z-parameters	O
and	O
their	O
inverse	O
,	O
the	O
admittance	B-Algorithm
parameters	I-Algorithm
or	O
y-parameters	B-Algorithm
.	O
</s>
<s>
which	O
can	O
be	O
considered	O
the	O
trivial	O
case	O
of	O
a	O
1×1	O
transfer	O
matrix	B-Architecture
.	O
</s>
<s>
To	O
correctly	O
predict	O
the	O
behaviour	O
of	O
the	O
circuit	O
,	O
the	O
currents	O
entering	O
or	O
leaving	O
the	O
ports	O
must	O
also	O
be	O
taken	O
into	O
account	O
,	O
which	O
is	O
what	O
the	O
transfer	O
matrix	B-Architecture
does	O
.	O
</s>
<s>
The	O
impedance	O
matrix	B-Architecture
for	O
the	O
voltage	O
divider	O
circuit	O
is	O
,	O
</s>
<s>
The	O
concept	O
of	O
impedance	O
can	O
be	O
extended	O
into	O
the	O
mechanical	O
,	O
and	O
other	O
domains	O
through	O
a	O
mechanical-electrical	O
analogy	O
,	O
hence	O
the	O
impedance	O
parameters	O
,	O
and	O
other	O
forms	O
of	O
2-port	B-Algorithm
network	I-Algorithm
parameters	O
,	O
can	O
be	O
extended	O
to	O
the	O
mechanical	O
domain	O
also	O
.	O
</s>
<s>
Expressing	O
the	O
behaviour	O
of	O
a	O
mechanical	O
component	O
as	O
a	O
two-port	B-Algorithm
or	O
multi-port	O
with	O
a	O
transfer	O
matrix	B-Architecture
is	O
a	O
useful	O
thing	O
to	O
do	O
because	O
,	O
like	O
electrical	O
circuits	O
,	O
the	O
component	O
can	O
often	O
be	O
operated	O
in	O
reverse	O
and	O
its	O
behaviour	O
is	O
dependent	O
on	O
the	O
loads	O
at	O
the	O
inputs	O
and	O
outputs	O
.	O
</s>
<s>
In	O
this	O
example	O
the	O
effort	O
and	O
flow	O
variables	O
are	O
torque	B-General_Concept
and	O
angular	O
velocity	O
respectively	O
.	O
</s>
<s>
The	O
transfer	O
matrix	B-Architecture
in	O
terms	O
of	O
z-parameters	O
will	O
look	O
like	O
,	O
</s>
<s>
A	O
gear	O
train	O
is	O
the	O
analogue	O
of	O
an	O
electrical	O
transformer	B-Algorithm
and	O
the	O
h-parameters	O
(	O
hybrid	O
parameters	O
)	O
better	O
describe	O
transformers	B-Algorithm
because	O
they	O
directly	O
include	O
the	O
turns	O
ratios	O
(	O
the	O
analogue	O
of	O
gear	O
ratios	O
)	O
.	O
</s>
<s>
The	O
gearbox	O
transfer	O
matrix	B-Architecture
in	O
h-parameter	O
format	O
is	O
,	O
</s>
<s>
is	O
the	O
reverse	O
direction	O
torque	B-General_Concept
ratio	O
of	O
the	O
gear	O
train	O
with	O
input	O
shaft	O
clamped	O
,	O
equal	O
to	O
the	O
forward	O
velocity	O
ratio	O
for	O
an	O
ideal	O
gearbox	O
,	O
</s>
<s>
In	O
robotics	O
and	O
mechatronics	O
,	O
actuators	B-Algorithm
are	O
required	O
.	O
</s>
<s>
These	O
usually	O
consist	O
of	O
a	O
transducer	B-Algorithm
converting	O
,	O
for	O
instance	O
,	O
signals	O
from	O
the	O
control	O
system	O
in	O
the	O
electrical	O
domain	O
into	O
motion	O
in	O
the	O
mechanical	O
domain	O
.	O
</s>
<s>
The	O
control	O
system	O
also	O
requires	O
sensors	O
that	O
detect	O
the	O
motion	O
and	O
convert	O
it	O
back	O
into	O
the	O
electrical	O
domain	O
through	O
another	O
transducer	B-Algorithm
so	O
that	O
the	O
motion	O
can	O
be	O
properly	O
controlled	O
through	O
a	O
feedback	O
loop	O
.	O
</s>
<s>
Other	O
sensors	O
in	O
the	O
system	O
may	O
be	O
transducers	B-Algorithm
converting	O
yet	O
other	O
energy	O
domains	O
into	O
electrical	O
signals	O
,	O
such	O
as	O
optical	O
,	O
audio	O
,	O
thermal	O
,	O
fluid	O
flow	O
and	O
chemical	O
.	O
</s>
<s>
Another	O
application	O
is	O
the	O
field	O
of	O
mechanical	O
filters	O
which	O
require	O
transducers	B-Algorithm
between	O
the	O
electrical	O
and	O
mechanical	O
domains	O
in	O
both	O
directions	O
.	O
</s>
<s>
A	O
simple	O
example	O
is	O
an	O
electromagnetic	O
electromechanical	O
actuator	B-Algorithm
driven	O
by	O
an	O
electronic	O
controller	O
.	O
</s>
<s>
This	O
requires	O
a	O
transducer	B-Algorithm
with	O
an	O
input	O
port	O
in	O
the	O
electrical	O
domain	O
and	O
an	O
output	O
port	O
in	O
the	O
mechanical	O
domain	O
.	O
</s>
<s>
This	O
might	O
be	O
represented	O
simplistically	O
by	O
a	O
SISO	O
transfer	O
function	O
,	O
but	O
for	O
similar	O
reasons	O
to	O
those	O
already	O
stated	O
,	O
a	O
more	O
accurate	O
representation	O
is	O
achieved	O
with	O
a	O
two-input	O
,	O
two-output	O
MIMO	O
transfer	O
matrix	B-Architecture
.	O
</s>
<s>
where	O
is	O
the	O
force	O
applied	O
to	O
the	O
actuator	B-Algorithm
and	O
is	O
the	O
resulting	O
velocity	O
of	O
the	O
actuator	B-Algorithm
.	O
</s>
<s>
An	O
example	O
of	O
a	O
two-port	B-Algorithm
acoustic	O
component	O
is	O
a	O
filter	O
such	O
as	O
a	O
muffler	O
on	O
an	O
exhaust	O
system	O
.	O
</s>
<s>
A	O
transfer	O
matrix	B-Architecture
representation	O
of	O
it	O
may	O
look	O
like	O
,	O
</s>
<s>
The	O
component	O
can	O
be	O
just	O
as	O
easily	O
described	O
by	O
the	O
z-parameters	O
,	O
but	O
transmission	O
parameters	O
have	O
a	O
mathematical	O
advantage	O
when	O
dealing	O
with	O
a	O
system	O
of	O
two-ports	B-Algorithm
that	O
are	O
connected	O
in	O
a	O
cascade	O
of	O
the	O
output	O
of	O
one	O
into	O
the	O
input	O
port	O
of	O
another	O
.	O
</s>
<s>
In	O
such	O
cases	O
the	O
overall	O
transmission	O
parameters	O
are	O
found	O
simply	O
by	O
the	O
matrix	B-Architecture
multiplication	O
of	O
the	O
transmission	O
parameter	O
matrices	O
of	O
the	O
constituent	O
components	O
.	O
</s>
<s>
The	O
power	O
conjugate	O
of	O
temperature	O
difference	O
is	O
not	O
thermal	O
power	O
,	O
but	O
rather	O
entropy	B-Algorithm
flow	O
rate	O
,	O
something	O
that	O
cannot	O
be	O
directly	O
measured	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
representation	O
of	O
linear	O
algebraic	O
equations	O
has	O
been	O
known	O
for	O
some	O
time	O
.	O
</s>
<s>
Poincaré	O
in	O
1907	O
was	O
the	O
first	O
to	O
describe	O
a	O
transducer	B-Algorithm
as	O
a	O
pair	O
of	O
such	O
equations	O
relating	O
electrical	O
variables	O
(	O
voltage	O
and	O
current	O
)	O
to	O
mechanical	O
variables	O
(	O
force	O
and	O
velocity	O
)	O
.	O
</s>
<s>
Kavanagh	O
in	O
1956	O
gave	O
the	O
first	O
completely	O
general	O
treatment	O
,	O
establishing	O
the	O
matrix	B-Architecture
relationship	O
between	O
system	O
and	O
control	O
and	O
providing	O
criteria	O
for	O
realisability	O
of	O
a	O
control	O
system	O
that	O
could	O
deliver	O
a	O
prescribed	O
behaviour	O
of	O
the	O
system	O
under	O
control	O
.	O
</s>
