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Optimal Auxiliary Input Design for Fault Detection of Systems with Model Uncertainty
where Ol(i) = (a l (;) ... ale;) bl(i) .. . bl(;»T ( . l' 'P'1' ,q tO, 1) and Our;) = (a u1 (;) , .. . , aut;) bur;) . . . bU(;»)T P , 1 , , q (i = 0, 1) represent the lower and upper bounds of parameters, respectively.
1993) showed that the fault detection can be accelerated by introducing a suitably designed auxiliary input. They, however, did not care of the side effect of the additional auxiliary input to the system behavior. The auxiliary input may affect the original system output so much, and this situation is not suitable in certain cases.
As shown in the papers of Zhang (Zhang, 1989), Kerestecioglu and Zarrop (Kerestecioglu and Zarrop 1991) , (Kerestecioglu and Zarrop, 1994), Uosaki et al. (Uosaki, Takata and Hatanaka, 1993), and Hatanaka and Uosaki (Hatanaka and Uosaki, 1995), (Hatanaka and Uosaki, 1999), a suitably designed auxiliary input {u*(t)} can be added to accelerate the fault detection and fault diagnosis. The auxiliary input should make the difference clearer between two models corresponding to normal and fault modes. To do this, an auxiliary input will be introduced to enlarge the distance between the models measured by the Kullback discrimination information (KDI) .
Considering this side effect, the optimal auxiliary input design problem for fault detection and fault diagnosis considering side effect is recently adressed (Uosaki, Takata and Hatanaka, 1993), (Hatanaka and Uosaki, 1995), (Hatanaka and Uosaki, 1999) . The auxiliary inputs are chosen to enlarge the distance measured by the Kullback discrimination information (KDI) between the system models corresponding to the normal and the fault mode, but not to deviate much from the normal mode without auxiliary input. The optimal auxiliary inputs are obtained by 'onestep-at-a-time' maximization of the time increment of the KDI in time domain (Uosaki, Takata and Hatanaka, 1993) and maximization of the time-average of the KDI in frequency domain (Hatanaka and Uosaki, 1995), respectively.
The KDI for discrimination in favor of the model M; over the model M j is defined by
(3)
All the design approaches above are developed under the assumption that the system models corresponding to normal and fault modes are known exactly to the designers. However, the system models are hardly known without uncertainties in practice. Hence, the optimal auxiliary input design for fault detection is considered here in the case of the models with uncertainty.
where pj(y t lut-1) is the probability density function of yt = (y(t),· · · ,y(I»T given u t - 1 = (u(t1), · · · , u(I»T under the model M j (j = 0, 1), respectively. The models of normal and fault modes with auxiliary input {u*(t)} are given by
2. PROBLEM STATEMENT
P
M: : y(t) = L a~;)y(t - k) k=1
Consider the following two stable autoregressive models with exogenous input (ARX models),
q
+ Lb~;)(u(t - k) + u*(t - k» k=1
P
Mi : y(t)
=L
a~i)y(t - k)
(i
k=1 b~i)U(t - k)
k=1 = O(i)T 4>(t)
+ c:(i)(t)
+ c:(i)(t),
(i = 0, 1) (1)
... , a(i) b(i) . .. , b(i»T and where O(i) = (a(i) l' P , l' q 4>(t) =(y(t-l),·· · , y(t-p) , u(t-l), · · · , u(t-q»T , and c:(') (t) is independently normally distributed (i = 0, 1) . Here, with mean zero and variance models Mo and M1 represent models for normal mode and fault mode, respectively, and it is supposed that system fault occurs at (unknown) time instant T . It is assumed here that the model parameters 0(0) and 0(1) have uncertainty, i.e., true parameters O(i) are in the following regions e(i) , respectively.
It can be seen as in (4) that the introduction of an auxiliary input {u· (t)} makes the system model different from the original model Mo , and hence the system behaviors such as the output process {y(t)} will change. The large deviations from the original system behaviors due to the introduction of the auxiliary input is not suitable in some practical cases. To avoid this, the auxiliary input should be chosen not to affect the model so much that is, the KDI IdMo : M~ ; yt ,(ut - 1,U*t-1)] (distance between the normal mode models without the auxiliary input (original model) and with
0';
(i
= 0, 1)
(4)
The auxiliary input {u·(t)} can be chosen to make the distance between the models of normal and fault modes larger, i.e., to make the time increment of the KDI IdM{ : M~ ; yt, (u t - 1, u· t - 1 )], larger.
q
+L
= 0, 1)
(2)
488
tl. 1dM l : Mo; yt, ut-I] =a(1)u 2(t - 1) + 2{1(1)U(t - 1) +
the auxiliary input) should not be so large. Thus the problem is then summarized as follows: (PO) "Find an auxiliary input {u·(t)} such that it maximizes the time increment of KDI between models M~ (normal mode with auxiliary input) and M; (fault mode with auxiliary input) under the restrictions the KDI between models Mo (normal mode without auxiliary input) and M~ (normal mode with auxiliary input) and the input should not exceed a certain bounds when the system parameters have uncertainty as in (2)."
(7)
,),(1)
with c5b(1)2
= _1_
a(l)
20-02
'
c5 (1)
o
+
o
q
= ~(" c5b(I)U(t 20-2 L..... k
{1(1)
k)
k=2
L" c5a~l) /I.(I)(t - k», k=1
It can be rewritten as
lu(t)
(ii)
+ u·(t)l
~
= _1_[(~ c5b(l)U(t _ 20-2 L..... k
,),(1)
(PI) "Find an auxiliary input {u·(t)} maximizing tl.It [M'l '. M"0" yt (u t - 1 , u· t - 1 ») under the constraints (i) tl.1dMo: M~; yt,(u t - 1 ,U· t - 1») ~ L
o
+
k=2
(L c5b~I)U(t - k»(L c5a~l) /I.(l)(t - k» q
"
k=2
k=l
+ tr[Et-1 + "t-1 ("t-1 )T c5a(I)c5a(I)T) t-p(l) ""t-p(l) ""t-,,(I)
C
when the parameters have uncertainty as (2)." 0
2
o-i - 0-5 + -1 log 0-~ + --=--..,,-~
In
2 0-5 20-5 /I.(l)(t) = Edy(t)lu t - 1 ), t-1 t - 1 1 t-l) /l.t-,,(I) = E 1 [Yt-p u ,
3. AUXILIARY INPUT DESIGN
1 t-1 - /l.t-p(l) t-1) t-p(l) = E 1 [( Yt-p X(yt-l t-p _ "t-l ,...t-p(l) )Tlut-1) ,
£a(l) -- a(l) _ k
c5b~l)
00
c5a(1) --
Pi(yt-1Iut-1)Pi(y(t)lyt-1,ut-1)
-00
X
bkO),
(c5a(l) •.. l'
,
(k
= 1, ... ,q),
c5a(I)T p
While, the time increment of the KDI between normal mode models with and without the auxiliary input is given by
Pi(y(t)l yt-1, ut-I) t-1 log Po(y(t)lyt-1, ut-I) dy(t)dy
= EdidM; : Mo; y(t), ut-Ill
= b~l) -
(k -- 1, ... , p) ,
a(O) k'
uk
!
(5)
tl.ldMo: M~; yt,(u t - 1 ,u· t - 1 »)
where
q
= _1_(" b(O)u.(t _
20-2 L..... o k=1
!
k
k»2
~ftl.ldMo : M~; u·(t - 1»)
00
=
(8)
Et-
The time increment of the KDI can be written as
=
k»2
PI (y(t)l yt-1, ut-I)
(9)
Based on this, the optimal auxiliary input can be obtained by solving the following mathematical programming problem when the system parameters have no uncertainties.
-00
(P2')
"Find u·(t - 1) maximizing tl.lt[M{: M~; yt,(u t - 1,U· t - 1»)
=a(I)'(u(t -
and El implies the conditional expectation with the probability density function PI (yt- 1 Iu t - 2 ) using the assumptions of system linearity and normality of £(t).
1)
+ u·(t -
1»2
+2{1(l)'(u(t - 1) + u·(t - 1» +
~ftl.ldM{ : M~; u·(t -1») subject to
After some manipulations (see (Hatanaka and Uosaki, 1989) for details), the time increment of the KDI between the normal mode and fault mode models is obtained as
489
,),(1)'
(10)
1993) showed that the fault detection can be accelerated by introducing a suitably designed auxiliary input. They, however, did not care of the side effect of the additional auxiliary input to the system behavior. The auxiliary input may affect the original system output so much, and this situation is not suitable in certain cases.
As shown in the papers of Zhang (Zhang, 1989), Kerestecioglu and Zarrop (Kerestecioglu and Zarrop 1991) , (Kerestecioglu and Zarrop, 1994), Uosaki et al. (Uosaki, Takata and Hatanaka, 1993), and Hatanaka and Uosaki (Hatanaka and Uosaki, 1995), (Hatanaka and Uosaki, 1999), a suitably designed auxiliary input {u*(t)} can be added to accelerate the fault detection and fault diagnosis. The auxiliary input should make the difference clearer between two models corresponding to normal and fault modes. To do this, an auxiliary input will be introduced to enlarge the distance between the models measured by the Kullback discrimination information (KDI) .
Considering this side effect, the optimal auxiliary input design problem for fault detection and fault diagnosis considering side effect is recently adressed (Uosaki, Takata and Hatanaka, 1993), (Hatanaka and Uosaki, 1995), (Hatanaka and Uosaki, 1999) . The auxiliary inputs are chosen to enlarge the distance measured by the Kullback discrimination information (KDI) between the system models corresponding to the normal and the fault mode, but not to deviate much from the normal mode without auxiliary input. The optimal auxiliary inputs are obtained by 'onestep-at-a-time' maximization of the time increment of the KDI in time domain (Uosaki, Takata and Hatanaka, 1993) and maximization of the time-average of the KDI in frequency domain (Hatanaka and Uosaki, 1995), respectively.
The KDI for discrimination in favor of the model M; over the model M j is defined by
(3)
All the design approaches above are developed under the assumption that the system models corresponding to normal and fault modes are known exactly to the designers. However, the system models are hardly known without uncertainties in practice. Hence, the optimal auxiliary input design for fault detection is considered here in the case of the models with uncertainty.
where pj(y t lut-1) is the probability density function of yt = (y(t),· · · ,y(I»T given u t - 1 = (u(t1), · · · , u(I»T under the model M j (j = 0, 1), respectively. The models of normal and fault modes with auxiliary input {u*(t)} are given by
2. PROBLEM STATEMENT
P
M: : y(t) = L a~;)y(t - k) k=1
Consider the following two stable autoregressive models with exogenous input (ARX models),
q
+ Lb~;)(u(t - k) + u*(t - k» k=1
P
Mi : y(t)
=L
a~i)y(t - k)
(i
k=1 b~i)U(t - k)
k=1 = O(i)T 4>(t)
+ c:(i)(t)
+ c:(i)(t),
(i = 0, 1) (1)
... , a(i) b(i) . .. , b(i»T and where O(i) = (a(i) l' P , l' q 4>(t) =(y(t-l),·· · , y(t-p) , u(t-l), · · · , u(t-q»T , and c:(') (t) is independently normally distributed (i = 0, 1) . Here, with mean zero and variance models Mo and M1 represent models for normal mode and fault mode, respectively, and it is supposed that system fault occurs at (unknown) time instant T . It is assumed here that the model parameters 0(0) and 0(1) have uncertainty, i.e., true parameters O(i) are in the following regions e(i) , respectively.
It can be seen as in (4) that the introduction of an auxiliary input {u· (t)} makes the system model different from the original model Mo , and hence the system behaviors such as the output process {y(t)} will change. The large deviations from the original system behaviors due to the introduction of the auxiliary input is not suitable in some practical cases. To avoid this, the auxiliary input should be chosen not to affect the model so much that is, the KDI IdMo : M~ ; yt ,(ut - 1,U*t-1)] (distance between the normal mode models without the auxiliary input (original model) and with
0';
(i
= 0, 1)
(4)
The auxiliary input {u·(t)} can be chosen to make the distance between the models of normal and fault modes larger, i.e., to make the time increment of the KDI IdM{ : M~ ; yt, (u t - 1, u· t - 1 )], larger.
q
+L
= 0, 1)
(2)
488
tl. 1dM l : Mo; yt, ut-I] =a(1)u 2(t - 1) + 2{1(1)U(t - 1) +
the auxiliary input) should not be so large. Thus the problem is then summarized as follows: (PO) "Find an auxiliary input {u·(t)} such that it maximizes the time increment of KDI between models M~ (normal mode with auxiliary input) and M; (fault mode with auxiliary input) under the restrictions the KDI between models Mo (normal mode without auxiliary input) and M~ (normal mode with auxiliary input) and the input should not exceed a certain bounds when the system parameters have uncertainty as in (2)."
(7)
,),(1)
with c5b(1)2
= _1_
a(l)
20-02
'
c5 (1)
o
+
o
q
= ~(" c5b(I)U(t 20-2 L..... k
{1(1)
k)
k=2
L" c5a~l) /I.(I)(t - k», k=1
It can be rewritten as
lu(t)
(ii)
+ u·(t)l
~
= _1_[(~ c5b(l)U(t _ 20-2 L..... k
,),(1)
(PI) "Find an auxiliary input {u·(t)} maximizing tl.It [M'l '. M"0" yt (u t - 1 , u· t - 1 ») under the constraints (i) tl.1dMo: M~; yt,(u t - 1 ,U· t - 1») ~ L
o
+
k=2
(L c5b~I)U(t - k»(L c5a~l) /I.(l)(t - k» q
"
k=2
k=l
+ tr[Et-1 + "t-1 ("t-1 )T c5a(I)c5a(I)T) t-p(l) ""t-p(l) ""t-,,(I)
C
when the parameters have uncertainty as (2)." 0
2
o-i - 0-5 + -1 log 0-~ + --=--..,,-~
In
2 0-5 20-5 /I.(l)(t) = Edy(t)lu t - 1 ), t-1 t - 1 1 t-l) /l.t-,,(I) = E 1 [Yt-p u ,
3. AUXILIARY INPUT DESIGN
1 t-1 - /l.t-p(l) t-1) t-p(l) = E 1 [( Yt-p X(yt-l t-p _ "t-l ,...t-p(l) )Tlut-1) ,
£a(l) -- a(l) _ k
c5b~l)
00
c5a(1) --
Pi(yt-1Iut-1)Pi(y(t)lyt-1,ut-1)
-00
X
bkO),
(c5a(l) •.. l'
,
(k
= 1, ... ,q),
c5a(I)T p
While, the time increment of the KDI between normal mode models with and without the auxiliary input is given by
Pi(y(t)l yt-1, ut-I) t-1 log Po(y(t)lyt-1, ut-I) dy(t)dy
= EdidM; : Mo; y(t), ut-Ill
= b~l) -
(k -- 1, ... , p) ,
a(O) k'
uk
!
(5)
tl.ldMo: M~; yt,(u t - 1 ,u· t - 1 »)
where
q
= _1_(" b(O)u.(t _
20-2 L..... o k=1
!
k
k»2
~ftl.ldMo : M~; u·(t - 1»)
00
=
(8)
Et-
The time increment of the KDI can be written as
=
k»2
PI (y(t)l yt-1, ut-I)
(9)
Based on this, the optimal auxiliary input can be obtained by solving the following mathematical programming problem when the system parameters have no uncertainties.
-00
(P2')
"Find u·(t - 1) maximizing tl.lt[M{: M~; yt,(u t - 1,U· t - 1»)
=a(I)'(u(t -
and El implies the conditional expectation with the probability density function PI (yt- 1 Iu t - 2 ) using the assumptions of system linearity and normality of £(t).
1)
+ u·(t -
1»2
+2{1(l)'(u(t - 1) + u·(t - 1» +
~ftl.ldM{ : M~; u·(t -1») subject to
After some manipulations (see (Hatanaka and Uosaki, 1989) for details), the time increment of the KDI between the normal mode and fault mode models is obtained as
489
,),(1)'
(10)