User`s guide

5 ODE P ara meter Estimation (Grey-Box Modeling)
Under ideal conditions, this system is d escribed by the heat-diffusion
equation—which is a partial differential equation in space and time.
∂
∂
=
∂
∂
xt
t
xt(, ) (, )ξ
κ
ξ
ξ
2
2
To get a continuous-time state-space model, you can represent the
second-derivative using the following difference approx imatio n:
∂
∂
=
+
()
−+−
()
()
=⋅
2
22
2
xt
xt L xt xt L
L
kL
(, )
,(,),
ξ
ξ
ξξξ
ξ
ΔΔ
Δ
Δwhere
This transformation produces a state-space model o f order
n
L
L
=
Δ
,where
the state variables
xtk L(, )⋅Δ
are lumped representations for
xt(, )ξ
for the
following range of values:
kL k L⋅≤<+
()
ΔΔξ 1
The d imension of x depends on t h e spatial grid size
ΔL
in the approximation.
The heat-diffusion equation is mapped to the following continuous-time
state-space model structure to ide ntify the state-space matrices:


xt Fxt Gut Kwt
yt Hxt Dut wt
xx
() () () ()
() () () ()
()
=++
=++
=00
5-10