User`s guide

3 Linear Model Identification
Are Grey-Box Models Similar to State-Space Models with
Structured Parameterization?
Structured parameterization state-space models are similar to grey-box
modeling. However, the state-space models are simpler to e stimate than
grey-box models. To learn more about grey-box m odels, see Chapter 5, “ODE
Parameter Estimation (Grey-Box Mod el ing) ”.
Example – Estimating Structured Discrete-Time State-Space
Models
In this example, yo u estimate the unknown parameters (
θθθθθ
12345
,,,,
)
in the following discrete-time model:
xt xt ut et
yt
( ) () () ()
()
+=
⎡
⎣
⎢
⎤
⎦
⎥
+
⎡
⎣
⎢
⎤
⎦
⎥
+
⎡
⎣
⎢
⎤
⎦
⎥
=
1
1
01
1
1
2
3
4
5
θ
θ
θ
θ
θ
00
0
0
0
[]
+
=
⎡
⎣
⎢
⎤
⎦
⎥
xt et
x
() ()
()
Suppose that the nominal values of the unknown parameters (
θθθθθ
12345
,,,,
)
are
-1, 2, 3, 4,and5,respectively.
The discrete-time state-space model structure is defined by the following
equation:
x kT T Ax kT Bu kT Ke kT
ykT CxkT DukT ekT
x
( ) () () ()
() () () ()
()
+= + +
=+ +
=0 xx0
To construct and estimate the parameters of this discrete-time state-space
model:
1 Construct the parameter matrices and initia li ze the pa r ameter values
using the nominal parameter values:
A = [1,-1;0,1];
B = [2;3];
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