(c) 1998 W. Nuninger Centre de Recherche en Automatique de Nancy
Fault detection for time-delay Systems within robust residual based on finite memory observers.
By W. Nuninger, F. Kratz, J. Ragot C.R.A.N
IFAC, 3rd Workgroup on On line fault detection and supervision in the chemical process industries
1998, june 4-5, Solaize, France
W. NUNINGER, F. Kratz, J. Ragot, Fault detection for time-delay systems within robust residual generation based on finite memory observers., 3rd IFAC Workshop on On line fault detection and supervision in the chemical process industries, Vol. 1, Solaize, France, june 4-5, 1998.
PLAN
n
Introduction
n
Problem formulation
n
Finite memory observer
n
New residual
n
Sensitivity
n
Simulated example
n
Conclusion
Introduction n
Fault detection problem Ë
Diagnostic procedure (1) Residual generation (2) Residual evaluation
n
Residual generation Ë
Massive redundancy
Ë
Analytical redundancy - Parity space - Observers
Estimation error
- Parameter estimation n
Robustness Ë
Residual both l
sensitive to faults
l
insensitive to disturbancies performance degree
Drawbacks of observer based methods
n
Infinite memory (all process history) Ë
residual less sensitive to recent data i.e. incipient faults
n
Model uncertainties + infinite memory Ë
Accumulation in the estimation error Divergence phenomenon [Heffes, 1966] [Toda et al., 1978]
n
Solutions Ë
Fading memory filter
Ë
Finite memory observer
[Sorenson et al., 1985]
l
continuous time
[Medvedev, 1991]
l
discrete time
[Kratz et al., 1994]
More sensitive to recent measurement Finite memory => how many data ?
Diagnostic Problem n
Time-delay system xÝ(t) =
(1)
J
S
∑ A x(t − t ) + ∑ B u (t − t ) s
i
s
s =0
j
j =0
y( t) = Cx (t ) + v( t) Ë
Measurement noise v(t) gaussian, zero-mean, covariance matrix (V)
n
constant
known
Problem Ë
Estimation of y(t) of model (1) based l
on the simpler model (2) xÝ( t) = Ax( t) + Bu (t )
(2)
A = A0 et B = B0
y( t) = Cx (t ) + v( t) l
upon (k+1) data defined at instant t by y(t − τ 0 ) M Yk (t) = y(t − τ ) k
Ë
fixed size sliding window
Residual based on the previous estimation l
to detect output addive faults
l
despite the model uncertainties
Problem resolution n
System model xÝ( t) = Ax (t) + Bu (t )
y( t) = Cx (t ) + v( t) n
Stactic form Ë
y( t-τi ) is expressed in function of x(t) y( t − τ i ) = w i x( t) − u ( t − τ i ,t ) + v( t − τ i ) t
w i = Ce
− Aτ i
u(t − τ i , t) =
∫ Ce
A (t −τi −δ )
Bu(δ)dδ
t −τi
Ë
On the observation window Z k ( t) = Wk x(t) + N k (t )
with
Z k ( t) = Yk ( t) + U k (t ) u( t − τ 0 , t) U k (t) = M u (t − τ k , t)
Wk = [w i ] v(t − τ0 ) N k (t) = M v(t − τk )
Variance of Nk R k (t) = diag (V(t − τ 0 ),K,V(t − τ k )) = (Q k Q k ) T
−1
Finite Memory Observer
n
State estimation Ë
In term of maximum likelihood Minimisation of φk =
1 Z k (t) − Zˆ k (t) 2
2 R −k1
under the constraint ˆ (t) + U (t) = W xˆ (t ) Zˆ k (t) = Y k k k k
n
Solution in the least squares sense xˆ k ( t) = Ω −1k W kT R −1k Z k ( t) Ω k (t) = WkT R −1k (t )Wk
where
n
Existence if and only if Ωk is invertible if W k is full column rank Observability condition
Ë Ë
Finite Memory Observer
n
State estimation error Ë
zero mean (if no model uncertainties)
Ë
variance-covariance Var {x k (t ) − xˆ k (t) } = Ω −1 k
n
Observation window’s length l
Minimal size = observability index
l
Optimal size
[Murthy, 1980]
Ë
Ë
Recursive formulation : At the frozen instant t, consider one more data (zk+1)
Ωk solution of an Algebraic Riccati Equation the serie converges [Caines, 1988] Evolution of Ω −1k while k increases
Optimal k
New residual n
System output Z 0,* k ( t) = Y0*,k ( t ) − U 0 ,k ( t ) = W0 , kx (t ) + ∆Z 0,k(f ( x,u, t) ) S
J
∑ A x(t − t ) + ∑ B u(t − t )
f (x, u, t ) =
s
s
s =1
i
j
j =1
Z 0,* k ( t) = W0 , kx (t )
n
Model output
n
Output measurement
noise
Z0 , k ( t) = Z ( t ) + D 0, k ( t) + N0 , k ( t) * 0 ,k
additive faults n
Residual
e k/ r (t) = xˆ 0 / k (t) − xˆ r / k (t) x^0,k(t) time
t-τk
t-τr
t-τ0
t-τk
t-τr
t-τ0
^x
r,k (t)
Y0 , r−1 Y0 ,k = Yr,k
time
ek/r(t) time
t-τk
t-τr
t-τ0
t-τr-1
t
Detection window
Residual formulation n
Additive parameter variations Ë
Influence
Ë
Residual
∆ Z 0 ,k ( t ) = ∆W0 , k x( t ) − ∆U 0 , k ( t)
ek / r (t) = e k / r (t )+ ∆e k / r (t ) T −1 e k / r (t ) = Ω −1 0 , k W0,r −1 R 0 ,r−1 N 0 ,r−1 (t) +
with
(Ω
−1 0, k
T −1 − Ω −1 r, k )W r,k R r, k N r,k (t)
Detection window −1 0 ,k
∆ e k / r (t) = Ω W
(Ω
−1 0, k
T 0 ,r−1
R
[∆Z
0 , r −1
]
(t) + D 0 ,r−1 (t) +
W rT,k R −1 (t ) + D r, k (t) − Ω −1 r, k ) r, k ∆Z r , k
Not the same gains
[
]
overlapping data-sets
improve the sensitivity to incipient faults
Choice of (k,r)
n
−1 0 , r−1
Time delay ∆ e k/ r (t) = −Ω −1 Wr,Tk R −1 ∆Z r, k + D r, k }+ r, k r, k {
Ω 0, k W0 , r−1 R 0 ,r−1 {∆Z 0 ,k [0,r − 1] + D 0 , r−1 }+ −1
T
Ω W R −1 0, k
T r, k
−1
−1 r, k
{∆Z
0,k
[r,k ] + D r, k }
Sensitivity n
Residual sensitivity (Jacobian matrices)
Deterministic additive parameter variations l
data of the detection window
∂e k / r (t ) ∂e k / r (t ) T −1 = = Ω −1 0, k W 0, r−1 R 0, r−1 ∂D T0, r−1 (t) ∂∆Z T0 , r−1 (t) l
the overlapping data-sets
∂e k / r (t) ∂ e k / r (t ) −1 T −1 = = (Ω −1 0,k − Ω r, k )Wr, k R r,k ∂D Tr, k (t) ∂∆Z Tr, k (t ) n
Choice of (k,r) Ω −1k
Ë
k
evolution of
Ë
r < k,
evolution of the performance index ρD 0,r, kr−1 =
∂e k / r T ∂D 0,r −1
∂e k/ r ∂D Tr,k
2
>1 2
Time delay not taken into account Ëk still chosen the same way Ër if no faults, the 2d estimation equals the first one in the residual only one data in the detection window
Simulated Example n
Chemical process [William et Otto,1960]
Raw materials FP FA
valuable product
coolant
FB
∞
heat exchanger
Distillation column
decanter
reactor coolant Coolant
FW2
FW1
by-products n
System equations [Ross, 1971] xÝ( t) = A 0 x( t) + A 1x (t − 1) + Bu( t) y( t) = Cx (t ) + v( t)
n
The observer is based on the model xÝ( t) = Ax(t) + Bu (t )
y( t) = Cx (t ) + v( t) −4.93 −3.20 A0 = 6.40 0
A = A0 C = [0 0 1 −1]
−1.01 0 0 0 0 1 1.92 0 0 0 1.92 0 −5.30 −12.8 0 0 B= A1 = 0.387 −32.5 −1.04 0 0 0 1.87 0 0.833 11.0 −3.96 0 0 0.724 0 0
0 1 0 0
Simulated example
n
4 additive faults to detect ‘ ‘
‘
n
t = 0.8s, 4s ∆t = 1.6s, 0.5s
random value
t = 7s, 8s
Measurement noise ‘
n
20% of magnitude bias
5% of magnitude
Output ‘
not faulty : red plot
-0.04 -0.06 -0.08 -0.1 1
2
3
4 5 Time (s)
6
7
8
9
Simulated example n
Norm of the estimation error variance 8
6
4 5
10
15
20
k
n
Ë
k = 15
Ë
r = 2 : only 1 data in the detection window
Residual when no faults
ε k / r (t ) = yˆ 0 ,k − yˆ r , k = Ce k , r (t) 0.02 0.01 0 -0.01 -0.02 1
2
3
4
5 Time (s)
6
7
8
9
Simulated example n
Output estimation error
0.02
does not converge to zero (even while no faults) Ë adaptive threshold required
0.01
0 -0.01 -0.02
n
ˆ ˆ New residual (k,r) =(3,2) 1 2 3 4 5 ε k / r6(t ) = 7y 0 ,k −8y r, k =9 Ce k , r (t)
0.02 0.01 0 -0.01 -0.02 1
Ë
2
3
4
5
6
7
8
9
only a fixed threshold is necessary
Conclusion n
Finite memory observer Ë Ë
n
finite number of data required on line computation
New residual defined detection window, more influence to incipient faults Ë deterministic additive parameter variations practical way of choosing (k,r) -> small detection window in general Ë
n
Applied for time-delay system Ë
residual based on a simpler model only one data in the detection window
easier computation, simpler residual evalution Ë
n
performance index more complex to prove because of coupling terms
Extension [Nuninger, 1997] Ë
output and input simultaneous estimations residual proprieties remain to be proved
