- Email: [email protected]
Fuzzy Artmap Neural Network and its Application to Fault Diagnosis of Integrated Navigation Systems
Copyright © IF AC Automatic Control in Aerospace, Seoul, Korea, 1998
FUZZY ARTMAP NEURAL NETWORK AND ITS APPLICATION TO FAULT DIAGNOSIS OF INTEGRATED NA VIGA TION SYSTEMS
H. Y. Zhangt, C. W. Chant, K. C. Cheungt, Y. J. Yet
t
Department of Automatic Control, Beijing University of Aeronautics and Astronautics Beijing 100083, P. R China, Email: [email protected] The authors are currently with the Department of Mechanical Engineering, University of Hong Kong
t Department of Mechanical Engineering, University of Hong Kong Pokfulam road Hong Kong, Email: [email protected]
Abstract: In this paper, a new approach which combines Chi-Square Test and Fuzzy ARTMAP neural network mapping is proposed to Fault Diagnosis of Integrated Navigation System. State Chi-Square Test on whole state vector and individual state is performed to detect faults. Then the pattern of State Chi-Square Test result is used to isolate and identify faults by using two separated Fuzzy-ARTMAP neural networks. The fault magnitude and fault occurring time can be identified. This may be useful for compensating the effect of fault on navigation accuracy. Simulation results show the effectiveness of the proposed method. Copyright CO 1998 IFAC Keywords: Fault Diagnosis, Integrated Navigation Systems, Fuzzy Neural Networks
1. INTRODUCITON
Chi-Square Test is usually employed to detect faults of navigation systems but not suitable for fault isolation and diagnosis (Da, 1994). Ren Da and Ching-Fan Lin (1995) proposed using ARTMAP neural network in fault diagnosis of navigation systems. ARTMAP (Carpenter, et al., 1991) is a class of neural network architectures that perform incremental supervised learning of recognizing categories and multidimensional maps in response to input vectors presented in arbitrary order. However ARTMAP can only deal with binary values of inputs, therefore it is not convenient to use in navigation systems. In this paper, we propose to use Fuzzy ARTMAP(Carpenter, et al., 1992} for fault diagnosis of navigation systems. Fuzzy ARTMAP combines fuzzy logic and adaptive resonance theory (ART) by exploiting a close formal similarity between the computation of fuzzy subsethood and ART category choice, resonance and learning. ·Fuzzy ARTMAP also realizes a new minmax learning law that conjointly minimizes
Fault tolerant design is an important approach to enhance the mission reliability of navigation systems . Fault detection arid diagnosis are key parts offault tolerant design. Kerr(1987} pointed out some special features of navigation systems to which the existing fault detection and diagnosis metho.ds may not be suitable. * Navigation systems are time varying systems while many methods can only apply to time invariant systems. * Navigation systems are described by high order linear models while many methods can only apply to low order models. * Navigation systems usually adopt reduced order Kalman filter, therefore in normal case the residual may not be a white noise while many metllods test the whiteness of residual for fault detection. * Navigation systems have low signal to noise ratio while many methods can only apply to high signal to noise ratio case.
243
predictive error and maximizes generalization. In navigation systems, signal is always noise contaminated and it is more suitable to use fuzzy logic to process the signal than using crisp set theory. In this paper, the fault of navigation systems is detected by using State Chi-Square Test (SCSn, after that fault isolation is accomplished using Fuzzy ARTMAP neural network as pattern recognition. Besides, the fault magnitude and occurring time can also be identified by using another Fuzzy ARTMAP neural network. This is very useful in compensating the effect of fault thus increasing the navigation accuracy. The paper is organized as follows: First section is the Introduction, Second section is State Chi-Square Test, Third section is Fuzzy ARTMAP neural network and its learning algorithm, Forth section is simulation of fault detection and diagnosis of INS/GPS integrated navigation system using State Chi Square Test and Fuzzy ARTMAP. Final section is conclusion.
It can be proven that when system is normal (with no fault) and model is perfect the residual (innovation) r. is a zero mean white Gaussian noise and its covariance matrix U. is given in (3-4). When fault happens, r. is no longer a zero mean white Gaussian . noise. We can make a binary hypothesis test on r.
Ho: E{rk}=O, E{Y.rJ}=U. (nofault) = J.l * 0, Covfr k} = Uk (with fault)
HI: Efr k}
We have the following conditional probability density function
2. STATE CHI-SQUARE TEST Consider a discrete time system with faults X k +1 {
=
Zk
k+I.kX.
+bP.e
+r.w.
(6) (1)
=H.x. +cp~ + v.
where, x ERn, Z ER"', .vectors b and c are unknown faults, B and tP are occurring time of faults. System noise w. and measurement
Because J.l is unknown, we use the maximal likelihood estimate
p to
replace
v. are Gaussian and have the following statistics
(7)
E[W d =0 E[W.W!] = Q.Oil E[v.] = 0 Elv. Vj ] = R.oil { E[w.vi]=o
A. reaches its maximal value. Substituting (7) into (6) and neglecting the coefficient 112 ,we can construct a fault detection function (8)
where Oil is the Kronecker Delta function. If the system has no faults, the Kalman filter gives the optimal estimation of the system states. The Kalman filter equations are listed below. Xhl\.l:
=
k+I,kx. 1k
J.l . When
noise
It can be proven that A.
has a Chi Square distribution with m degrees of freedom, m is the dimension of z . The fault detection criterion is
(3-1)
=hl,kP.Ik~+I,k +r.Q.rJ rhl =Zhl - Hhlx.+ 11k U hl =HhlPhlIkHi+1 +Rk+1
(3-3) (3-4)
where TD • is a
Khl = PhIIkH:'P;~1
(3-5)
by selecting a false alann rate PI and using the Chi
(3-6)
Square distribution table
P hllt
Xhllk+1
=
P hllk + 1 =
X hllk
+ Khlrhl
[1 - KhlH hI ]p.. ;;.
(9)
(3-2)
(3-7)
2.1 Residual Chi Square Test
244
threshold which can be determined
where X 2 (A,m) is Chi-square distribution function with m degrees of freedom. For example. when m=3. Pf = a = 10-3 , TD = 1627. Residual Chi- Square
using one state propagator. Because there is no measurement update in state propagator. the modeling error. system noise and initial estimation error will make the state propagation deviate from the true value gradually.
x;
Test is not very useful for detecting soft fault ( fault magnitude grows gradually). This can be seen from the expression of residual (3-3). When fault is small, it can not be detected and the undetected fault in the previous output will contaminate the present prediction estimate Xh l/ ••
Therefore even in no fault case P. will not be zero mean, which will decrease the sensitivity of fault detection. In (Da, 1994) Ren Da proposed using two state propagators. When one is used for fault detection, another. one is reset by the state estimate and covariance p. ofKalman filter. In next state update cycle, the functions of the two propagators are exchanged. The above test is based on vector P. that is the overall effect on all states. Therefore the sensitivity of the test is not very high. We can improve it by test on the components of P. When there is no fault, we have
therefore the fault effects contained in the two tenns on the right hand side of (3-3) will cancel each other somehow. This will make the detection based on r. ineffective. 2.2
x.
State Chi-Square Test (SCST)
Construct a state propagator (shadow filter)
(10)
where
P•.i
is the ith component of
P.,
T.,;; is the
(i,i) element of T• .
x;
Clearly, is free of measurement faults. Define the estimation error of kalman filter and state propagator respectively
(18)
i.e .• A•.1 has
degree of freedom. we can perform the Chi-Square Test on individual A•.1 • The SCST result of every
(i2)
state forms a pattern which is mainly detennined by the fault location and can be classified using Fuzzy AR1MAP, so that the fault can be isolated and identified.
and define (13)
The covariance of
Pi
is
3. FUZZY AR1MAP NEURAL NETWORK AND ITS LEARNING ALGORITIIM
Fuzzy AR1MAP learns to classify inputs by a Fuzzy set of features. Fig. 1. shows the architecture of Fuzzy ARTMAP network. When there is no fault,
P. is a zero mean Gauss-
ian stochastic vector. When fault happens.
map field Fah
x; is still
x.
unbiased but is not, therefore P. is not zero mean any more. Similar to the residual Chi-Square Test, we can construct a detection function
A. = PiT.-1p.
(15)
It can be proven that when the initial values Po = poiU, p. = p.iU . We have
The detection procedure is the same as that of residual Chi-Square Test. There is shortcoming of
Fig. 1 Fuzzy ARTMAP architecture
245
When the Jth category is chosen, Y J = 1; and During supervised learning , ART"
receives a
Yj =
stream {a (P)} of input patterns, and ART;, receives
°
for j"* J. The ·
F; active vector x obeys
the equation
a stream {b (p) } of input pattern. where {b (p)} is modules are linked by a
FIIb
map field
X ={ lAWJ
whi~h
match tracking signal if {a (P)} is not
outputs a
matched to {b (p)}. In field
Fa" ,
a vector A is
fonned
coding
A = (a,a c ),
by
a~ = 1- a j
•
complement
a; and a: are component of a and a
if F2 is inactive (23) if the Jth F2 node is chosen
1
the correct prediction given {a(P)}. These two
3.2
Resonance or reset:
resonance occurs if
11AWJI , (pO or ph) -Ill-~P
C
(24)
respectively. Complement coding nonnalizes the input vector while preserves amplitude infonnation. The same as for B. For ART" and ART, ' Field
called a vigilance parameter pe (0,1). The larger
F;
the
otherwise mismatch reset occurs, where p
, receives both bottom-up input from Fo and
p, the closer the WJ to I. When
is
resonance
input from a field, F;, that represents
occurs, learning of WJ takes place to move WJ
the active category. The Fo active vector is denoted
closer to I. WJ is updated according to the equation
top-down
1 = (J I ,I2'···,I M) with each component 1; in the interval [0,1], i=I, ... ,M. The F; active vector is denoted
X = (XI' x 2' ••• , XM) and the F2 active
17 E (0,1) . Fast learning corresponds to where setting 17 = 1. When mismatch reset occurs, a new index J is chosen, by (22). The search process continues until the chosen J satisfies (24).
vector Y = (YI ' Y2 ' ... , Y M) . The number of nodes in each field is arbitrary.
Associated with each F2
category node j( j= 1, 2, ... ,N) is an adaptive weigh~ vector Wj =[w j" wj2 , ••• ,wjM] . Formapfield FaIJ, the output vector is X aIJ =[x~,xt,···,x:], apd
Ujab = [wt;, w~, ... , w~] represents
the
3.3 Map field activation: The map field is activated whenever one of the ART" or ART;, categories is active. The output vector x llb of
weight
vector from the jth node of F2" to Fe.
Fe obeys
3.1
I AW! if the JIh F; rrxk is a:tive cnd l{ is a:tive x" _ 11f if the JIh F; rrxk is a:tive cnd l{ is inative - {I if the JIh F; rrxk is iTllXtiw! cnd l{ is cx:tive
Category choice:
For each input I and F2
node j, the choice function, Tj
,
is defined by
11AWI
T .(J) = - - ' J a+IWjl
°
(19)
if the JIh F; rrxk is iTllXtiw! cnd l{ is iTllXtiw! If 1x lib 1< Plib 1yb 1, Plib is the
Match tracking: where the fuzzy "and" operator A
vigilance I?arameter of match field, it means ART"
is defined by
and ART, are mismatched, then p" is increased until the resonance condition (24) no longer holds. ART" restart to search and activate
(20)
and the nonn 1-1 is defined by
another node of F;"
condition and match (IXllbl=ly'AWJllbl~PllbIY'I>. w;(O)
M
Ipl=Llp;1
which satisfies the resonance
(21)
;=1
when Jth node of ART"
for any M-dimensional vectors p and q, a is a small positive number. The chosen category J satisfies
category
k,
W~
condition is set to 1,
predicts the
ART;,
= 1 for all time.
4. SIMULATION EXAMPLE (22) An integrated navigation system which consists of a strap-down inertial navigation system and a
246
z(t) = H(t)X(t) +n(t)
Global Positioning Systems(SINS/GPS) is used in this simulation. The state variable vector X ER 18 where
z(t)=[ZI,Z2 ,Z3f,
n(t)=[nE,nN,nu]T,
H=[J3x3,03xI2,-J3x3] ' . Assume that the flight trajectory is level from south to north starting from 40 degrees latitude with velocity v = 300m/sec. where OrE ,l;'N' Oru - poslUon errors,
(NE'
/NN,
/Nu -
velocity errors' ;E ' ;N ' ;U - platfonn . angular errors, V V y ' V z - accelerometer biases, Jt ,
& :r '
& y'
&z -
Markov process parameters of gyro
errors, t5E , t5N , t5u - Markov process parameters of GPS errors. (E,N,U) represent east/northlup navigation coordinates. (x,y,z) represent body coordinates. System equation is described as X(t) = F(t)X(t) + GOJ(t)
Some parameters for simulation can be referred to Cheng(l994). The sampling period is one second. In the simulation, step-function faults of different components are introduced at 700th time step. First we use Chi-Square Test to detect the faults ,then we use Fuzzy ARTMAPl network to isolate faults and use Fuzzy ARTMAP2 network to identify the fault magnitude and fault occurring time. In the simulation, the following faults and the corresponding modes of Fuzzy AMfRAP are used as shown in table 1.
F and G can be found in Cheng(1994), or BarItzhack, et al.(l986) where a slightly different navigation coordinate system is adopted. Measurement equation is Table 1 Faults locations, magnitudes and corresponding modes of Fuzzy ARTMAP
...~!!~~..2U~.~~.L I......................~~~.~~~~.~f..f.~~.~~............__...~....B:!~..~~}...~.~~.~.. ..B:!~..~~.~ ..~.~~.. x accelerometer yaccelerometer
800,900,1000,1100, 1200, 1300(,ug) 800,900,1000, 1100,1200,1300(,ug)
1000 0100
1000 Cb 0100 Cb
x gyro
0.3,0.6,0.9,1.2, 1.5,1.8 (O/h)
0010
0010 Cb
y gyro
0.3,0.6,0.9, 1.2, 1.5,1.8
C /h)
0001
0001 Cb
Cb
=001 or 010 or 011 or 100 or 10 1 or 111 (corresponding to different fault magnitude)
Take the faults of x-gyro as an example, first we calculate the detection function values Al.; (SCST
By using Fuzzy ARTMAP 1 neural network, this pattern (denoted as s) is compared to all p trained patterns s· (k=1, .. .,p). If
results) using (18) for 18 states. As shown in Fig. 2 we have a pattern fonned by the SCST results (for clarity we have connected the points of Al.; by lines and different type line corresponds to different fault magnitude). Also we can see that different fault magnitudes only change a little of the pattern. We can take this effect as noise.
s
then is classified into category R and x-gyro fault is detected and isolated, where TR is a threshold Once a fault of sensor J is isolated, we can use Fuzzy ARTMAP2 neural network to identify fault magnitude and its occurring time. Suppose a sensor J has m different operational processes(each with different fault magnitudes and occurring times) and assume the faults are detected at time step k b •
0.8 0.6
Therefore we have m patterns as shown in Fig.3(m=3). Fuzzy ARTMAP2 stores p trained patterns in the data base through simulation, every pattern represents a certain fault magnitude and occurring time. We compare the SCST result AJ Fig. 2
with these stored patterns. If we have
SCST results for 18 states when x-gyro has different magnitude faults
247
Table 2 Then it means the fault of x-gyro is in category ~ The magnitude and occurring time of fault are then identified from category R. In Table 2 and 3 we list the simulation results . As ,can be seen from these tables, the results are quite accurate.
Fault identification of x-gyro, fault occurring time=700sec
hC/h) kt(sec.) ,..·..··..·ftC/h) ..·..0·:4'..·..·......·· ·............ ·o·j..·....·......· ....·........·68·5·............. 0.8 1.15 1.4 1.7
0.6 1.2 1.5 1.8
690 705 705 703
Table 3 Fault identification of x-accelerometer, fault occurring time=700sec (pg)
h(pg)
k/ (seC)
810 ' 930 1020 1210 1290
800 900 1000 1200 1300
694 696 698 700 701
Fig.3: SCST modes under different fault magnitudes and occurring times where
ft
is the true fault magnitude,
h is
self-organizing neural network, Neural Networks, Vol.4, pp565-588. Carpenter, G. A., S. Grossberg, N. Markuzon. J. H. Reynolds, and D. B. Rosen(1992), Fuzzy ARTMAP: A Neural Network Architecture for Incremental Supervised Learning of Analog Multidimensional Maps, IEEE Trans. on Neural Networks, Vol. 3, No. 5, September. Cheng H. L.(l994), Doctoral Dissertation, Department of Automatic Control, Beijing University of Aeronautics and Astronautics, May. Da R.(1994), Failure Detection of Dynamic Systems with the State Chi-Square Test, Journal of Guidance, Control and Dynamics, Vo1.l7, No. 2, March-April. Da R. and.Ching-Fang Lin(1995), Failure Diagnosis System using ARTMAP Neural Networks, Journal of Guidance, Control and Dynamics, Vol. 18, No. 4, July-August. Kerr, T. H.(l987), Decentralized Filtering and Redundancy Management for Multisensor Navigation. IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-23, No. 1, January, pp83-119.
the
identified fault magnitude, k/ is the identified faUlt occurring time.
5. CONCLUSION In this paper we have applied State Chi-Square Test and Fuzzy ARTMAP neural network to fault diagnosis of integrated navigation systems. The former one can be used to detect faults while the latter one can be used to isolate and identify faults. Further research is directed to compensate the fault effect on navigation accuracy by using the identified fault magnitude and occurring time. The influence of flight trajectories on the SCST resulted patterns should be investigated in the future. Acknowledgment The work was supported by the Research Grants Council of Hong Kong.
REFERENCES Bar-Itzhack, Itzhack Y. and Y. Medan(l986), GPS aided low cost strapdown INS for attitude determination. AIAA Guidance, Navigation and Control Conference, AIAA No.2149, pp516-521. Carpenter, G. A., S. Grossberg and J. H. Reynolds(1991), ARTMAP: Supervised real time learning and classification of nonstationary data by a
248
FUZZY ARTMAP NEURAL NETWORK AND ITS APPLICATION TO FAULT DIAGNOSIS OF INTEGRATED NA VIGA TION SYSTEMS
H. Y. Zhangt, C. W. Chant, K. C. Cheungt, Y. J. Yet
t
Department of Automatic Control, Beijing University of Aeronautics and Astronautics Beijing 100083, P. R China, Email: [email protected] The authors are currently with the Department of Mechanical Engineering, University of Hong Kong
t Department of Mechanical Engineering, University of Hong Kong Pokfulam road Hong Kong, Email: [email protected]
Abstract: In this paper, a new approach which combines Chi-Square Test and Fuzzy ARTMAP neural network mapping is proposed to Fault Diagnosis of Integrated Navigation System. State Chi-Square Test on whole state vector and individual state is performed to detect faults. Then the pattern of State Chi-Square Test result is used to isolate and identify faults by using two separated Fuzzy-ARTMAP neural networks. The fault magnitude and fault occurring time can be identified. This may be useful for compensating the effect of fault on navigation accuracy. Simulation results show the effectiveness of the proposed method. Copyright CO 1998 IFAC Keywords: Fault Diagnosis, Integrated Navigation Systems, Fuzzy Neural Networks
1. INTRODUCITON
Chi-Square Test is usually employed to detect faults of navigation systems but not suitable for fault isolation and diagnosis (Da, 1994). Ren Da and Ching-Fan Lin (1995) proposed using ARTMAP neural network in fault diagnosis of navigation systems. ARTMAP (Carpenter, et al., 1991) is a class of neural network architectures that perform incremental supervised learning of recognizing categories and multidimensional maps in response to input vectors presented in arbitrary order. However ARTMAP can only deal with binary values of inputs, therefore it is not convenient to use in navigation systems. In this paper, we propose to use Fuzzy ARTMAP(Carpenter, et al., 1992} for fault diagnosis of navigation systems. Fuzzy ARTMAP combines fuzzy logic and adaptive resonance theory (ART) by exploiting a close formal similarity between the computation of fuzzy subsethood and ART category choice, resonance and learning. ·Fuzzy ARTMAP also realizes a new minmax learning law that conjointly minimizes
Fault tolerant design is an important approach to enhance the mission reliability of navigation systems . Fault detection arid diagnosis are key parts offault tolerant design. Kerr(1987} pointed out some special features of navigation systems to which the existing fault detection and diagnosis metho.ds may not be suitable. * Navigation systems are time varying systems while many methods can only apply to time invariant systems. * Navigation systems are described by high order linear models while many methods can only apply to low order models. * Navigation systems usually adopt reduced order Kalman filter, therefore in normal case the residual may not be a white noise while many metllods test the whiteness of residual for fault detection. * Navigation systems have low signal to noise ratio while many methods can only apply to high signal to noise ratio case.
243
predictive error and maximizes generalization. In navigation systems, signal is always noise contaminated and it is more suitable to use fuzzy logic to process the signal than using crisp set theory. In this paper, the fault of navigation systems is detected by using State Chi-Square Test (SCSn, after that fault isolation is accomplished using Fuzzy ARTMAP neural network as pattern recognition. Besides, the fault magnitude and occurring time can also be identified by using another Fuzzy ARTMAP neural network. This is very useful in compensating the effect of fault thus increasing the navigation accuracy. The paper is organized as follows: First section is the Introduction, Second section is State Chi-Square Test, Third section is Fuzzy ARTMAP neural network and its learning algorithm, Forth section is simulation of fault detection and diagnosis of INS/GPS integrated navigation system using State Chi Square Test and Fuzzy ARTMAP. Final section is conclusion.
It can be proven that when system is normal (with no fault) and model is perfect the residual (innovation) r. is a zero mean white Gaussian noise and its covariance matrix U. is given in (3-4). When fault happens, r. is no longer a zero mean white Gaussian . noise. We can make a binary hypothesis test on r.
Ho: E{rk}=O, E{Y.rJ}=U. (nofault) = J.l * 0, Covfr k} = Uk (with fault)
HI: Efr k}
We have the following conditional probability density function
2. STATE CHI-SQUARE TEST Consider a discrete time system with faults X k +1 {
=
Zk
k+I.kX.
+bP.e
+r.w.
(6) (1)
=H.x. +cp~ + v.
where, x ERn, Z ER"', .vectors b and c are unknown faults, B and tP are occurring time of faults. System noise w. and measurement
Because J.l is unknown, we use the maximal likelihood estimate
p to
replace
v. are Gaussian and have the following statistics
(7)
E[W d =0 E[W.W!] = Q.Oil E[v.] = 0 Elv. Vj ] = R.oil { E[w.vi]=o
A. reaches its maximal value. Substituting (7) into (6) and neglecting the coefficient 112 ,we can construct a fault detection function (8)
where Oil is the Kronecker Delta function. If the system has no faults, the Kalman filter gives the optimal estimation of the system states. The Kalman filter equations are listed below. Xhl\.l:
=
k+I,kx. 1k
J.l . When
noise
It can be proven that A.
has a Chi Square distribution with m degrees of freedom, m is the dimension of z . The fault detection criterion is
(3-1)
=hl,kP.Ik~+I,k +r.Q.rJ rhl =Zhl - Hhlx.+ 11k U hl =HhlPhlIkHi+1 +Rk+1
(3-3) (3-4)
where TD • is a
Khl = PhIIkH:'P;~1
(3-5)
by selecting a false alann rate PI and using the Chi
(3-6)
Square distribution table
P hllt
Xhllk+1
=
P hllk + 1 =
X hllk
+ Khlrhl
[1 - KhlH hI ]p.. ;;.
(9)
(3-2)
(3-7)
2.1 Residual Chi Square Test
244
threshold which can be determined
where X 2 (A,m) is Chi-square distribution function with m degrees of freedom. For example. when m=3. Pf = a = 10-3 , TD = 1627. Residual Chi- Square
using one state propagator. Because there is no measurement update in state propagator. the modeling error. system noise and initial estimation error will make the state propagation deviate from the true value gradually.
x;
Test is not very useful for detecting soft fault ( fault magnitude grows gradually). This can be seen from the expression of residual (3-3). When fault is small, it can not be detected and the undetected fault in the previous output will contaminate the present prediction estimate Xh l/ ••
Therefore even in no fault case P. will not be zero mean, which will decrease the sensitivity of fault detection. In (Da, 1994) Ren Da proposed using two state propagators. When one is used for fault detection, another. one is reset by the state estimate and covariance p. ofKalman filter. In next state update cycle, the functions of the two propagators are exchanged. The above test is based on vector P. that is the overall effect on all states. Therefore the sensitivity of the test is not very high. We can improve it by test on the components of P. When there is no fault, we have
therefore the fault effects contained in the two tenns on the right hand side of (3-3) will cancel each other somehow. This will make the detection based on r. ineffective. 2.2
x.
State Chi-Square Test (SCST)
Construct a state propagator (shadow filter)
(10)
where
P•.i
is the ith component of
P.,
T.,;; is the
(i,i) element of T• .
x;
Clearly, is free of measurement faults. Define the estimation error of kalman filter and state propagator respectively
(18)
i.e .• A•.1 has
degree of freedom. we can perform the Chi-Square Test on individual A•.1 • The SCST result of every
(i2)
state forms a pattern which is mainly detennined by the fault location and can be classified using Fuzzy AR1MAP, so that the fault can be isolated and identified.
and define (13)
The covariance of
Pi
is
3. FUZZY AR1MAP NEURAL NETWORK AND ITS LEARNING ALGORITIIM
Fuzzy AR1MAP learns to classify inputs by a Fuzzy set of features. Fig. 1. shows the architecture of Fuzzy ARTMAP network. When there is no fault,
P. is a zero mean Gauss-
ian stochastic vector. When fault happens.
map field Fah
x; is still
x.
unbiased but is not, therefore P. is not zero mean any more. Similar to the residual Chi-Square Test, we can construct a detection function
A. = PiT.-1p.
(15)
It can be proven that when the initial values Po = poiU, p. = p.iU . We have
The detection procedure is the same as that of residual Chi-Square Test. There is shortcoming of
Fig. 1 Fuzzy ARTMAP architecture
245
When the Jth category is chosen, Y J = 1; and During supervised learning , ART"
receives a
Yj =
stream {a (P)} of input patterns, and ART;, receives
°
for j"* J. The ·
F; active vector x obeys
the equation
a stream {b (p) } of input pattern. where {b (p)} is modules are linked by a
FIIb
map field
X ={ lAWJ
whi~h
match tracking signal if {a (P)} is not
outputs a
matched to {b (p)}. In field
Fa" ,
a vector A is
fonned
coding
A = (a,a c ),
by
a~ = 1- a j
•
complement
a; and a: are component of a and a
if F2 is inactive (23) if the Jth F2 node is chosen
1
the correct prediction given {a(P)}. These two
3.2
Resonance or reset:
resonance occurs if
11AWJI , (pO or ph) -Ill-~P
C
(24)
respectively. Complement coding nonnalizes the input vector while preserves amplitude infonnation. The same as for B. For ART" and ART, ' Field
called a vigilance parameter pe (0,1). The larger
F;
the
otherwise mismatch reset occurs, where p
, receives both bottom-up input from Fo and
p, the closer the WJ to I. When
is
resonance
input from a field, F;, that represents
occurs, learning of WJ takes place to move WJ
the active category. The Fo active vector is denoted
closer to I. WJ is updated according to the equation
top-down
1 = (J I ,I2'···,I M) with each component 1; in the interval [0,1], i=I, ... ,M. The F; active vector is denoted
X = (XI' x 2' ••• , XM) and the F2 active
17 E (0,1) . Fast learning corresponds to where setting 17 = 1. When mismatch reset occurs, a new index J is chosen, by (22). The search process continues until the chosen J satisfies (24).
vector Y = (YI ' Y2 ' ... , Y M) . The number of nodes in each field is arbitrary.
Associated with each F2
category node j( j= 1, 2, ... ,N) is an adaptive weigh~ vector Wj =[w j" wj2 , ••• ,wjM] . Formapfield FaIJ, the output vector is X aIJ =[x~,xt,···,x:], apd
Ujab = [wt;, w~, ... , w~] represents
the
3.3 Map field activation: The map field is activated whenever one of the ART" or ART;, categories is active. The output vector x llb of
weight
vector from the jth node of F2" to Fe.
Fe obeys
3.1
I AW! if the JIh F; rrxk is a:tive cnd l{ is a:tive x" _ 11f if the JIh F; rrxk is a:tive cnd l{ is inative - {I if the JIh F; rrxk is iTllXtiw! cnd l{ is cx:tive
Category choice:
For each input I and F2
node j, the choice function, Tj
,
is defined by
11AWI
T .(J) = - - ' J a+IWjl
°
(19)
if the JIh F; rrxk is iTllXtiw! cnd l{ is iTllXtiw! If 1x lib 1< Plib 1yb 1, Plib is the
Match tracking: where the fuzzy "and" operator A
vigilance I?arameter of match field, it means ART"
is defined by
and ART, are mismatched, then p" is increased until the resonance condition (24) no longer holds. ART" restart to search and activate
(20)
and the nonn 1-1 is defined by
another node of F;"
condition and match (IXllbl=ly'AWJllbl~PllbIY'I>. w;(O)
M
Ipl=Llp;1
which satisfies the resonance
(21)
;=1
when Jth node of ART"
for any M-dimensional vectors p and q, a is a small positive number. The chosen category J satisfies
category
k,
W~
condition is set to 1,
predicts the
ART;,
= 1 for all time.
4. SIMULATION EXAMPLE (22) An integrated navigation system which consists of a strap-down inertial navigation system and a
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z(t) = H(t)X(t) +n(t)
Global Positioning Systems(SINS/GPS) is used in this simulation. The state variable vector X ER 18 where
z(t)=[ZI,Z2 ,Z3f,
n(t)=[nE,nN,nu]T,
H=[J3x3,03xI2,-J3x3] ' . Assume that the flight trajectory is level from south to north starting from 40 degrees latitude with velocity v = 300m/sec. where OrE ,l;'N' Oru - poslUon errors,
(NE'
/NN,
/Nu -
velocity errors' ;E ' ;N ' ;U - platfonn . angular errors, V V y ' V z - accelerometer biases, Jt ,
& :r '
& y'
&z -
Markov process parameters of gyro
errors, t5E , t5N , t5u - Markov process parameters of GPS errors. (E,N,U) represent east/northlup navigation coordinates. (x,y,z) represent body coordinates. System equation is described as X(t) = F(t)X(t) + GOJ(t)
Some parameters for simulation can be referred to Cheng(l994). The sampling period is one second. In the simulation, step-function faults of different components are introduced at 700th time step. First we use Chi-Square Test to detect the faults ,then we use Fuzzy ARTMAPl network to isolate faults and use Fuzzy ARTMAP2 network to identify the fault magnitude and fault occurring time. In the simulation, the following faults and the corresponding modes of Fuzzy AMfRAP are used as shown in table 1.
F and G can be found in Cheng(1994), or BarItzhack, et al.(l986) where a slightly different navigation coordinate system is adopted. Measurement equation is Table 1 Faults locations, magnitudes and corresponding modes of Fuzzy ARTMAP
...~!!~~..2U~.~~.L I......................~~~.~~~~.~f..f.~~.~~............__...~....B:!~..~~}...~.~~.~.. ..B:!~..~~.~ ..~.~~.. x accelerometer yaccelerometer
800,900,1000,1100, 1200, 1300(,ug) 800,900,1000, 1100,1200,1300(,ug)
1000 0100
1000 Cb 0100 Cb
x gyro
0.3,0.6,0.9,1.2, 1.5,1.8 (O/h)
0010
0010 Cb
y gyro
0.3,0.6,0.9, 1.2, 1.5,1.8
C /h)
0001
0001 Cb
Cb
=001 or 010 or 011 or 100 or 10 1 or 111 (corresponding to different fault magnitude)
Take the faults of x-gyro as an example, first we calculate the detection function values Al.; (SCST
By using Fuzzy ARTMAP 1 neural network, this pattern (denoted as s) is compared to all p trained patterns s· (k=1, .. .,p). If
results) using (18) for 18 states. As shown in Fig. 2 we have a pattern fonned by the SCST results (for clarity we have connected the points of Al.; by lines and different type line corresponds to different fault magnitude). Also we can see that different fault magnitudes only change a little of the pattern. We can take this effect as noise.
s
then is classified into category R and x-gyro fault is detected and isolated, where TR is a threshold Once a fault of sensor J is isolated, we can use Fuzzy ARTMAP2 neural network to identify fault magnitude and its occurring time. Suppose a sensor J has m different operational processes(each with different fault magnitudes and occurring times) and assume the faults are detected at time step k b •
0.8 0.6
Therefore we have m patterns as shown in Fig.3(m=3). Fuzzy ARTMAP2 stores p trained patterns in the data base through simulation, every pattern represents a certain fault magnitude and occurring time. We compare the SCST result AJ Fig. 2
with these stored patterns. If we have
SCST results for 18 states when x-gyro has different magnitude faults
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Table 2 Then it means the fault of x-gyro is in category ~ The magnitude and occurring time of fault are then identified from category R. In Table 2 and 3 we list the simulation results . As ,can be seen from these tables, the results are quite accurate.
Fault identification of x-gyro, fault occurring time=700sec
hC/h) kt(sec.) ,..·..··..·ftC/h) ..·..0·:4'..·..·......·· ·............ ·o·j..·....·......· ....·........·68·5·............. 0.8 1.15 1.4 1.7
0.6 1.2 1.5 1.8
690 705 705 703
Table 3 Fault identification of x-accelerometer, fault occurring time=700sec (pg)
h(pg)
k/ (seC)
810 ' 930 1020 1210 1290
800 900 1000 1200 1300
694 696 698 700 701
Fig.3: SCST modes under different fault magnitudes and occurring times where
ft
is the true fault magnitude,
h is
self-organizing neural network, Neural Networks, Vol.4, pp565-588. Carpenter, G. A., S. Grossberg, N. Markuzon. J. H. Reynolds, and D. B. Rosen(1992), Fuzzy ARTMAP: A Neural Network Architecture for Incremental Supervised Learning of Analog Multidimensional Maps, IEEE Trans. on Neural Networks, Vol. 3, No. 5, September. Cheng H. L.(l994), Doctoral Dissertation, Department of Automatic Control, Beijing University of Aeronautics and Astronautics, May. Da R.(1994), Failure Detection of Dynamic Systems with the State Chi-Square Test, Journal of Guidance, Control and Dynamics, Vo1.l7, No. 2, March-April. Da R. and.Ching-Fang Lin(1995), Failure Diagnosis System using ARTMAP Neural Networks, Journal of Guidance, Control and Dynamics, Vol. 18, No. 4, July-August. Kerr, T. H.(l987), Decentralized Filtering and Redundancy Management for Multisensor Navigation. IEEE Trans. on Aerospace and Electronic Systems, Vol. AES-23, No. 1, January, pp83-119.
the
identified fault magnitude, k/ is the identified faUlt occurring time.
5. CONCLUSION In this paper we have applied State Chi-Square Test and Fuzzy ARTMAP neural network to fault diagnosis of integrated navigation systems. The former one can be used to detect faults while the latter one can be used to isolate and identify faults. Further research is directed to compensate the fault effect on navigation accuracy by using the identified fault magnitude and occurring time. The influence of flight trajectories on the SCST resulted patterns should be investigated in the future. Acknowledgment The work was supported by the Research Grants Council of Hong Kong.
REFERENCES Bar-Itzhack, Itzhack Y. and Y. Medan(l986), GPS aided low cost strapdown INS for attitude determination. AIAA Guidance, Navigation and Control Conference, AIAA No.2149, pp516-521. Carpenter, G. A., S. Grossberg and J. H. Reynolds(1991), ARTMAP: Supervised real time learning and classification of nonstationary data by a
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