Geoscience Reference
In-Depth Information
-
is the deviation of the global temperature with regard to the
x
=
T
T
G
E
equilibrium;
- G x
is the infrared flow radiated, supposing that it is proportional to
small temperature deviations.
λ
The energy stored by the planet is the product I G .T G , and its variation is
proportional to the balance of flows, hence
dT
. The equilibrium
G
I
=
Φ
G
R
dt
d
temperature T E , which being constant by definition, gives
T
=
0
, and
dt
results from the equation for the model:
d
I
x
=
λ
x
+
α
u
G
G
dt
At any time t , we can consider that the variable x ( t ) constitutes the state
of this system, and the expression above is known as a state space equation .
Here, it is a first order equation. The process defined thus is a linear time
invariant system (LTI).
To an equation of this kind, we can associate a transmittance (or transfer
function) 1 . This can be done by replacing the differential operator
d / with
the symbol s (the Laplace variable). The equation above then becomes
u
dt
, that can be processed like an ordinary algebraic equation.
I
s
x
=
λ
x
+
α
G
G
After factoring,
, it finally becomes:
(
I
s
+
λ
)
x
=
α
u
G
G
α
x
=
u
I
s
+
λ
G
G
α
+
The expression
is the transmittance associated to the
G
(
s
)
=
I
G s
λ
G
initial equation of state. It is a rational fraction in s . It symbolically
represents the process of transmission from a causal input u to the output x .
1 A presentation of these concepts, relevant to this work, is given in Chapter 3 entitled “LTI
systems and transmittances” of the topic Automatique Appliquée by Philippe de Larminat
[DEL 09].
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