User`s guide

Estimating Linear Grey- Box Models
Description of the SISO System
This example is based on a discrete, singl e- input and single -output (SI SO)
system represented by the following state-space equations:
xkT T
par par
xkT ukT wkT
y kT par
( ) () () ()
()
+=
⎡
⎣
⎢
⎤
⎦
⎥
+
⎡
⎣
⎢
⎤
⎦
⎥
+
=
12
10
1
0
334
00
par x kT e kT
xx
[]
+
=
()()
()
where w and e are i ndependent w hite-noise terms w ith cov ariance matrices
R1 and R2,respectively.R1=E{ww’} is a 2–by-2 matrix and R2=E{ee’} is a
scalar. par1, par2, par3,andpar4 represent the unknown parameter values
to be estim ated.
Assume that you know the variance of the measurement noise R2 to be 1.
R1(1,1) i s unk n own and is treated a s an addition a l param eter par5.The
remaining elem ents of R1 are known to be zero.
Estimating the Parameters of an idgrey Model
You can repre sent the system described in “Description of the SISO System”
on page 5-13 as a n
idgrey (grey-box) model using an M-file. Then, you can
use this M-file and the
pem comm and to estimate the model parameters based
on initial p arameter guesses.
To run th is e xa mple, you must load an input-output data set and represent
it as an
iddata or idfrd object called data. For more information about
this operation, see “Representing Time- and Frequency-Domain Data Using
iddata Objects” on page 1-47 or “Representing Frequency-Response Data
Using idfrd Objects” on page 1-67.
To estimate the parameters o f a grey-box model:
5-13