APPENDIX
FDI FEEDBACK MATRIX DESIGN
D
This appendix presents the algorithm for the design of the feedback matrix in the failure detection and
identification (FDI) method of model-based diagnostics in Chapter 11. The feedback matrix determines
the output error residual vector and, in particular, its direction in the output vector space due to failure
or degradation in the performance of the component for which the FDI is designed. In Chapter 11,a
model for the system in which the failed/failing component is incorporated is repeated below ¼6p
_
x ¼ Ax + Bu + f
i
where f
i
¼failure event vector
The FDI is based on the state estimator
^
x, which is given by the solution to the following equation:
_
^
x ¼ A
^
x + Bu + Dy
^
y
ðÞ
where y ¼Cx
^
y ¼ C
^
x
An algorithm for designing the D matrix is outlined below in which the inputs are the failure event
vector f
i
and λ
d
, which is the eigenvalue for the detection space. For an N-dimensional state vector,
the design requires N eigenvalues to be chosen, one of which is λ
d
. The remaining eigenvalues are
incorporated in a matrix that is denoted P and is part of the design algorithm.
P ¼ λ
d
λ
2
λ
3
⋯λ
N
½
T
These eigenvalues determine the dynamic response of the state estimator and must be negative or, if
complex, must be in complex conjugate pairs with negative real parts. Normally, the choice for λ
n
is for
all to be negative real numb ers. With these parameters and inputs, the algorithm consists of the follow-
ing steps:
a) C
f
¼ Cf
i
b) C
p
¼ C
T
f
C
f
1
C
T
f
c) C
s
¼ INðÞC
f
C
p
where I(N) ¼N-dimensional identity matrix
d) C
i
¼ C
s
C
e) D
d
¼λ
d
f
i
+ Af
i
ðÞC
P
f) G
p
¼ A D
d
C
g) K ¼ place G
T
p
, C
T
i
, P
h) D
P
¼ K
T
C
s
i) D ¼ D
d
+ D
P
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