TP 2: P ARITY S PACE APPROACH

ESISAR 5A

Department : AUTOMATIQUE Damien KOENIG

2018 : Fault Diagnosis

TP 2: P ^ARITY S PACE APPROACH

Objective: Build a bank of residuals in order to detect and isolate the fault sensor

1 S

ENSOR FAULT DETECTION AND ISOLATION FOR A

BOILER SYSTEM EXCHANGER:

P

ARITY SPACE APPROACH

Theoretical study

Consider the following boiler system exchanger

The variables are:

 Tc : output water temperature at the boiler °C

 Tp : output water temperature of the primary circuit of the exchanger °C

 Ts : output water temperature of the second circuit of the exchanger °C

 Qg : gas flow m³/h

 Qp : water flow of the primary circuit of the exchanger l/h

 Qs: water flow of the second circuit of the exchanger l/h

The state of the system is desbribed by the following relation :

 

k



Tc

     

k Tp k Ts k



^T

x 

the control inputs by :

 

^{k }



^Q



^k2

 

^{Q k1}

 

^{Q k5}

 

^{Q k1}

 

u _{g p p s}

And the output by :

  k  T

_c

  k T

_p

  k T

_s

  k 

^T

y 

TpO

Chaudière

Circuit primaire échangeur

Circuit secondaire échangeur TpO

Qg

Qp

Qs

TcO TcO

TpO

TsO

AC-514

2 The matrices A, B and C are given by :



















33 31

22 21

12 11

0 0 0 a a

a a

a a

A ,



















34 32

24 22

13 11

0 0

0 0

0 0

b b

b b

b b

B , C = I3

1.1 Parity space

 Without increasing the time k, how many parity equations can be deduced?

 Gives the parity equations with a minimum horizon of observation, to limit the detection delays.

 Gives the fault location decision table, each sensor fault alarm with the logic decision.

2 S

TATIC PARITY SPACE WITH NON PERFECT DECOUPLING Consider the static output relation:

 

k Cx

   

k k Fd

 

k

y   

where xRⁿ, yR^m anddR^p .

y(k) is the output measurement, x(k) the state variable, d(k) the fault vector to be detected and (k) the noise measurement. Matrices C and F are known with appropriate dimension.

Assumption: m > n for a redundancy information existence.

We will find a parity vector sensitive to p-1 defaults and insensitive to di which represent the ith component of d.

This leads to explain the measurement vector in the form:

y

 

k Cx

   

k k F^d^

 

k F^d^

 

k

where d⁺ are d^- are respectively the sensitive and insensitive faults. F ⁺ et F ^– are the associated matrices.

2.1 Numerical Application:

Consider the following system:





















 2 0 2

1 0 1

1 1 1

2 0 1

1 2 1

C ,





















 1 0

5 2

0 0

2 1

2 1

F ,





















 1 1 3 0 1 F

 Find the kernel w with the optimization problem described bellow. Give w^TF^- and w^TF⁺.

 Check the products w^TC, w^TF^- and w^TF⁺ and conclude.

Solution:

The problem is reduced to find a matrices W such that

 

^{C F 0}

W (1)

where the parity vector is given by : p

 

k W

 

k WF^d^

 

k

AC-514

3 The exact solution W of (1) holds if and if the line rank of (C F^-) is deficient. If not the following optimization problem can be used:









 



 2 2

0

F w

F min w

C w

T T w

T

remark: max(f(x)) = - min(-f(x)) example f(x) = x²-2

Explanation

Consider 

 





2 1

C

C C where C1 is of full rank. The constraints w^TC = 0 is true by using the following changement :

Pw2

w

where

 











 ^ I

C

P C ^T

T 1 2

1 .

Proof :

 

1 2 2 1¹

2 1 2

1

0

0     

^



 



 

 w w C C

C w C w C

w

^{T T T T T} thus

 

⁰

2 1 1 1 2

2 



 



 ^  C I C C C

w^T .

With identification:

   

2 2 1 2

1 1 1 2

2 w w Pw

I C w C

I C C w

w ^T

T T

T   















 ^{ } .

Using the new variable w2, the problem ₂

2

0 





F w

F min w

; C

w T

T w

T becomes :

     

 

2 ^{22 22} 2

2 2

2 2

2

2 w B~w

w A~ minw Pw F F P w

Pw F F P minw w F F w

w F F minw F

w F

min w _T

T T w

T T

T T T T w

T T T T w

T

w   





















where ^{A~ PTF}

 

^{F TP and B~PTF}

 

^{F TP.}

Since the small eigenvalue  of the pair (A~,B~

) is the minimal of the criteria, the corresponding eigenvector w2 is the solution of the optimization problem. Find , such that



B~



w2 ⁰

A~

. From w2 and P deduce w and the parity relation:

 

_







 





 



d F d F w p

y w p

T T

That conclude the proof.