Impact of fuel supply impedance and fuel staging on gas turbine ...

5.2 System Identification
5.2 System Identification
To accomplish this task a routine based on a non-recursive least square method
<strong>and</strong> the linear regression property are used (see [73] for more details). The
predictor <strong>of</strong> Eqn. (5.4) can be described as
where θ denotes the unknown parameters
<strong>and</strong> ϕ the regression vector
ŷ n|θ = ϕ T n θ, (5.6)
[h (1)
0 h(1) 1 ...h(1) M h(2) 0
...h (2)
M h(3) 0
...h (3)
M ]T
[x n (1) , x (1)
n−1 ...x(1) n−M x(2) n ...x (2)
n−M x(3) n ... x (3)
n−M ]T .
Using the predictor in Eqn. (5.6) the prediction error can be estimated:
ǫ n,θ = y n − ϕ T n θ. (5.7)
A least-square criterion for the linear regression can now be defined as
1
V (θ)=
N − M+ 1
N∑
n=M
1 [
yn − ϕ T n
2
θ] 2
, (5.8)
where N denotes the number <strong>of</strong> time steps considered. Eqn. (5.8) can be minimized
analytically due to its quadratic characteristics in θ:
[
] −1
1 N∑
ˆθ=
ϕ n ϕ T 1 N∑
n
ϕ n y n . (5.9)
N − M+ 1
N− M+ 1
n=M
n=M
The least-square estimate ˆθ can be calculated, if the inverse on the right side
<strong>of</strong> the latter equation exists. The first sum <strong>of</strong> Eqn. (5.9) can be estimated as the
3M× 3M auto-correlation matrix <strong>of</strong> the input signals:
1
Γ xx =
N − M+ 1
N∑
n=M
x (i )
n x(j ) ; k = 0,..., M; i , j = 1,2,3. (5.10)
n−k
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