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Parameter estimation methods for fault detection and isolation

LAAS-CNRS UPC

Teresa Escobet (UPC & LEA-SICA, Spain) Louise Travé-Massuyès (LAAS-CNRS & LEA-SICA, France)

March, 7-9 Via Lattea, Italian Alps DX 2001

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Parameter estimation methods for fault detection and isolation

Teresa Escobet (UPC & LEA-SICA, Spain) Louise Travé-Massuyès (LAAS-CNRS & LEA-SICA, France)

March, 7-9 Via Lattea, Italian Alps DX 2001

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March, 7-9 Via Lattea, Italian Alps DX 2001

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θ

Parameter Estimation

u e y

Calculation of process coefficients θ Theoretical modeling p=f -1(θ ) p Changes ∆p, ∆θ ∆p, ∆θ Fault decision

March, 7-9 Via Lattea, Italian Alps DX 2001

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Parameter estimation for fault detection

Recursive Least Squares algorithms

) ( ) ( ) ( ) 1 ( ) (

1 1 k b n k a n

n n t u b n t u b n t y a t y a t y

b a

− − + + − + − + + − = L L

Input-output model: Recursive Least Squares algorithm: ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( 1 ) ( ) 1 ( ) ( t t k t t t t t y t t P t t k t P t P t t P t t t P t k

T T T

ε θ θ θ ϕ ε ϕ ϕ ϕ ϕ + − = − − = − − − = − + − = ) ) )

[ ]

[ ]

T n n T b a

b a

b b a a nk n t u nk t u n t y t y t K K K K

1 1

) ( ) ( ) ( ) 1 ( ) ( = − − − − − − − = θ ϕ

where:

March, 7-9 Via Lattea, Italian Alps DX 2001

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Parameter estimation for fault detection

Recursive Least Squares algorithms characteristics

• The algorithm needs initial values: θ (0) and P(0)
• The result minimises the expression:
• LS estimate is consistent when:

E{ϕ(t) ϕT(t)} is non singular & E{ϕ(t) ε(t)} = 0

• The convergence speed is generally slow.
• Approaches for modifying the RLS algorithm to make it suitable as a

real-time fault detection method: Use of a forgetting factor Use of a virtual Kalman filter Use of sliding window data

• =

=

N t N

t V

1 2

) ( ) ( ε θ

March, 7-9 Via Lattea, Italian Alps DX 2001

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Parameter estimation for fault detection

Forgetting factor

The loss function to be minimized: The RLS method with forgetting factor is: Characteristics:

• =

−

=

N s s t N

s V

1 2

) ( ) ( ε λ θ

( )

) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( 1 ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( t t k t t t t t y t t P t t k t P t P t t P t t t P t k

T T T

ε θ θ θ ϕ ε ϕ λ ϕ ϕ λ ϕ + − = − − = − − − = − + − = ) ) )

• The parameter estimates converge to their true value quicker, thus decreasing the fault

alarm delay time

• But at the expense of increased sensitivity to noise. If λ is much less than 1 the estimates

may even oscillates around their true value

March, 7-9 Via Lattea, Italian Alps DX 2001

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Parameter estimation for fault detection

Forgetting factor- Approaches

• Time-varying forgetting factor [Fortescue et al. 1981]

( ) ( )

) 1 ( ) ( ) ( ) 1 ( ) ( 1 ) ( Covariance . 5 them if : note / ) ( ) ( ) 1 ( 1 1 ) ( Forgetting . 4 ) ( ) 1 ( ) ( 1 ) ( ) 1 ( ) ( Gain . 3 ) ( ˆ ) ( ) ( Error . 2 ) 1 ( ) 1 ( ) ( ˆ Prediction . 1

2

− − − = = < − − − = − + − = − = − − = t P t t k t P t t P ë (t) ë (t) t t k t t t t P t t t P t k t y t y t t t t y

T min min T T T

ϕ λ λ λ σ ε ϕ λ ϕ ϕ ϕ ε θ ϕ )

• The value of the constant σ0 is the expected measurement noise variance

which must be chosen based on the knowledge of the system.

• The minimum value for λ (t) is also to be chosen by the user.

March, 7-9 Via Lattea, Italian Alps DX 2001

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Parameter estimation for fault detection

Forgetting factor- Approaches

• Kalman filter

State equation: Recursive algorithm:

• The covariance matrix R1 has a similar role as the forgetting factor λ.
• These design variable should be chosen by trade-off between fast detection

(which requires λ “small” or R1 “large”) and reliability (which requires λ close to 1 or R1 “small”)

( )

) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( 1 ) ( ) 1 ( ) (

1

t t k t t t t t y t R t P t t k t P t P t t P t t t P t k

T T T

ε θ θ θ ϕ ε ϕ ϕ ϕ ϕ + − = − − = + − − − = − + − = ) ) )

{ }

s t T

R s v t v E t v t x t x

, 1

) ( ) ( ); ( ) ( ) 1 ( δ = + = +

March, 7-9 Via Lattea, Italian Alps DX 2001

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Benchmark process fault detection

The system can be modeled by:

n n

S z a z b h z a z b z h S z a z b Q z a z b z h ∆ − − ∆ − = ∆ ∆ − − ∆ − = ∆

− − − − − − − − 1 2 1 22 1 1 2 1 21 3 1 1 1 12 1 1 1 11 1

1 1 ) ( 1 1 ) (

March, 7-9 Via Lattea, Italian Alps DX 2001

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Benchmark process fault detection

Scenario I :

valve V1 blocked closed from time 1000

a1 b11 b12 a2 b21 b22

March, 7-9 Via Lattea, Italian Alps DX 2001

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Benchmark process fault detection

Scenario II :

V1 blocked opened

a1 b11 b12 a2 b21 b22

March, 7-9 Via Lattea, Italian Alps DX 2001

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Benchmark process fault detection

Scenario III :

leak in tank 1, the fault occurs at time 800 seconds

a1 b11 b12 a2 b21 b22

March, 7-9 Via Lattea, Italian Alps DX 2001

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Conclusions

• This benchmark is not ideal to test identification methods

because two out of three of the proposed faults change the model structure

• An advantage of these methods is to approach simultaneously

the fault detection and fault isolation problems

• A problem that must be reported about estimation methods is

their high sensitivity to the parametrization, i.e. in our case, the values of λ and R1

• Our experiments show that the convergence rate of the

algorithms is a critical issue