Formulating an Energy Balance Model - Climate Change: Identification and Projections - page 43

Here, it is a first order equation. By dividing the numerator and the
denominator by , it can be written in the form: λ

G

S

G ( s ) =

Ts + 1

where is the static gain or equilibrium sensitivity of the output in

relation to the input u i , and where

S = α / λ

G

is the time constant of the
system. An equilibrium being a situation where all the derivatives are zero,
d/dt = 0, the static gain of a transmittance

T = I / λ

G G

is given by . G ( s ) G ( 0 )

A first order black box model can be defined by its transmittance

S

, without any reference to “physical” coefficients, here

G ( s ) = Ts

α , λ

G

+ 1

and I G . Parameters S and T characterize the step response of the process, that
is, its response to a hypothetical variation in a step input of u ( t ), as Figure
4.1 shows.

8

6

4

2

Step response
T ime constant

0

0 50 100 150 200

Time (years)

8

Figure 4.1. Step response of a transmittance: Gs () =

40s + 1

Observing only the external behavior of a system, it is impossible to
determine the value of its physical parameters unless there is a minimal
number of them. Otherwise, it would be necessary to arbitrarily fix one or
the other, whereas we imposed on ourselves the rule of not using any a
priori knowledge.