Assessment of observer based fault estimators for TS fuzzy models

(1)

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Assessment of Observer Based Fault Estimators for TS Fuzzy Models

Zahra Shams Elec. Eng. Dept.

Shahed University Tehran, Iran

Email: [email protected]

S. Seyedtabaii Elec. Eng. Dept.

Shahed University Tehran, Iran Email: [email protected]

Abstract- Fault detection of nonlinear systems become more feasible when it is conducted over Takagi-Sugeno (TS) approximated fuzzy models. Proportional plus integral observer (PIO) and robust observer (RO) have already been developed for the estimation of the system states and actuator/sensor faults. In this paper, the algorithms are implemented for the detection of valve and level sensor faults of a two-tank system. Our simulation results indicate that both algorithms run well in estimating states and sensor fault, however, there is obvious differences in how they detect actuator fault in the presence of noise. From viewpoint of estimation variance, RO renders cleaner estimate of the fault than PIO, whiles PIO has faster fault tracking speed than RO.

According to the achieving result, RO algorithm is recognized to be a more attractive in estimating actuator faults in noisy environments. The results are validated through simulations.

Keywords- observers; TS fuzzy models; actuator; sensor faults;

noise

I. INTRODUCTION

Any unacceptable deviation of system parameters from their nominal values, that may be a sign of near future complete failure, is called fault. Naturally, all processes are vulnerable to faults, therefore, maintaining long run system performance in the form of product quality and or behavior demands certain measures that can be accomplished by automatic fault diagnosis and isolation (FDI) to give early warning and to ensure safety and quality of operations.

Among a few fault diagnosis approaches, the quantitative model based techniques and in particular the observer based method continues to gain an increasing attention in the fundamental and applied research fields [1]. The methods embark on analytical, rather than, measurement redundancies.

In practice, due to model uncertainties, disturbances, perturbations and noises, accurate fault estimation, such as the fault size and shape, is a very challenging procedure [2-4].

Most nonlinear behavior systems can be appropriately approximated by Takagi-Sugeno (TS) fuzzy models [5].

Therefore, the problem of fault detection in nonlinear systems is transformed to the fault analysis of TS fuzzy system models.

In this respect, plenty of researches have been conducted. A proportional integral (PI) observers has been introduced in [6- 8], which detects and estimates the size of actuator faults. The algorithms performance degrades in case of measurement noise [4]. Another technique for actuator fault estimation for T–S fuzzy model has been reported in [9, 10]. Estimation of

actuator and sensor faults is the subject of study in [11], where the faults are introduced as auxiliary variables to be estimated.

A revised version of PI observer for the detection of sensor and actuator faults has been detailed in [12]. Design of a reduced order robust observer with unknown input for estimation of faults in continuous and discrete TS models has been investigated in [13] .

In this paper, the problem of state estimation and actuator/sensor fault detection in a noisy environment of a two- tank system is investigated. The algorithms are the PI observer (PIO) [14] and the robust observer (RO) [4]. It is exhibited that both algorithms operate well in estimating the state and sensor fault; however, contrasts’ are recorded in the detection of the actuator faults. PIO quickly tracks the actuator fault as soon as it occurs but the estimate is noisy, whiles low speed fault tracking RO yields lower variance estimate, which is demanded.

In section 2, the general TS fuzzy model and PIO and RO observer based fault detection algorithms are introduced. In section 3, the simulation results in detecting actuator and sensor faults in a two tank system is investigated and lastly conclusion comes in section 4.

II. TS FUZZY MODEL FAULT DETECTION A. TS Fuzzy model

Most nonlinear physical systems can be accurately modeled by Takagi-Sugeno (TS) fuzzy systems [5]. The TS fuzzy set approximates a nonlinear system by a combination of several linear local dynamics where each is represented by one of the fuzzy implications. Thus, the global behavior is obtained by the contribution of each of the linear sub-models. It is argued that if nonlinear system state variables are used as the fuzzy decision variables, the fuzzy model is capable of replicating a large class of nonlinear systems [15]. Thus, it is worthwhile to exploit observer based fault detection on these models which it would be applicable to many nonlinear systems [16].

A TS Fuzzy model subject to actuator fault, sensor faults, and unknown disturbance is expressed by,



















) ( ) ( ) (

) ( ) ( ) ( ) ( ( ) (

1

t Ff t Cx t y

t w H t f F t u B t x A t

x

s

i a i i i r

i

i

 (1)

where x(t)∈ⁿ is the state, u(t)∈^p is the input vector, w(t)∈^d

(3)

is the exogenous disturbance, y(t)∈^q is the measured output, fa(t) ∈^p1 is the actuator and fs(t) ∈^m is the sensor faults. Ai, Bi, Fi, C, F, D and Hi are constant real matrices of appropriate dimensions. The μi(x) is a membership function of the state variables, x. It is defined by,

 

1

( ) 1 , 0

0 ( ) 1 , 1,...,

r i i

i

x t

x i r





   



    



 (2)

Where "r" represents the local models index.

B. PI observer (PIO) design

To estimate the states and faults of (1), first, the output is filtered by a stable matrix

( ( ))

f f

y  T y y t (3)

and (1) is reformulated by introducing a new state variable, r(t) as follows [14],



q



i i i i i

i

i i s a f

i i i i r

i i

I H C

B H TF B

o F F

T TC

o A A

t f

t t f t f y

t t x r

t r C t q

t w H t f F t u B t r A t

r

0 0 , 0 ,

0 ,

) , (

) ) ( ( ) , (

) ) ( (

) ( ) (

)) ( ) ( ) ( ) ( ( ) (

1 1

1 1 1 1 1

 



 







 







 







 





 



 







 























(4)

Where the matrices F1i are full column rank. If it is assumed that the fault is in polynomial form of k-1 degree and their k^th derivatives are bounded, i.e. it obeys the following model,

1

1 0

( ) ( )

( ) ( ) ( )

k k

k

f t f t

f t f t f t f



 



 

 



(5)

A proportional integral observer is set up to estimate the states x, yf and the fault and its derivatives as given below,







































1 ...

1 : )

( ) ˆ ( )) ˆ( ) ( ( ˆ) ( ) ˆ(

) ( ) ˆ( )) ˆ( ) ( ( ˆ) ( ) ˆ(

) ˆ( ) ˆ(

) ( ))) ˆ( ) ( ( ) ˆ( ) ( ) ˆ( ˆ)(

( ) ˆ(

1 1

1

1 1 1 1

k j for t g t f t q t q K r t f

t g t f t q t q K r t f

t r C t q

t g t q t q K t f F t u B t r A r t r

fj j j

li r

i i j

f li

r

i i

r pi

i i i r

i i





 (6)

Where KPi, KIi and K^jIi are the proportional and integral gains, respectively. The variables gr (t) and gf(t), gfj(t) are introduced in order to compensate the influence of the immeasurable decision variables. For a matter of simplicity, the system (4) and the observer (6) is restructured to the following compact form,

































) ˆ( ) ˆ(

) ( ))) ˆ( ) ( ( ) ( ) ˆ( ( ) ˆ(

) ( ) (

) ( . )) ( )

( ) ( ( ) (

2

2 2

1 2

2 2

2 1

t r C t q

t g t q t q K t u B t r A t

r

t r C t q

t f J t w H t u B t r A t

r

i i i

r

i i

k i

i i

r

i i





(7)

Where

 0 0 0

,

0 0 0 , 0 0 0 0 0 0

0 0

0 0 0 0

0 0 0 ,

0 0 0

, ) (

) (

) ( ) ( , ) ˆ (

) ˆˆˆ((()) ) ˆ( , ) (

) (

) ( ) (

1 2

1 1

1

2 1

1

2 1

2

1 1





 



C C K

K K K K

B B I I I F A A H H

I J t g

t g

t g t g t f

t f

t r t r t f

t f

t r t r

k li li li pi

i

i n n n i i

i i

i

fk n f f r

k k

v v v

f





























































































































 



(8) Where nf = nfa + nfs and Inf is an identity matrix.

The observer estimation error,

) ˆ( ) ( ) ( , ) ˆ( ) ( )

(t r t r t e t q t q t

e   y   (9)

Is expressed by,

1

2 2 2 2

1 1

2 1 2

0 0

2 1 1

( ) ( ) . ( )ˆ . ( ) . ( ) ( )

ˆ

, , , ( ) ( )

, , 1

(1 ) 1

( ) ( )

( ) , ( )

( ) ( )

0 ˆ

r

i i

i

r r

i i i i i i i i i i

i i

i

k

y T

e t r A e t G t z t g t

A A K C A A B B r r

G J H A B

f t r t

t z t

w t u t

g if e

g r

 

    

  

  



 



 

    

       

 

          

   

   

   





 

1 2 1

2 2 2 2

ˆ

2 2

T

T T

y y y

T T

y y y y

r u u

P C e P C e if e

e e ^   e e ^ 



  



(10)

The estimation error e̅(t) asymptotically stable and L2 stability is guaranteed [14], if there exist a matrix P>0, matrices Ni >0 and the positive scalars  and 0 for each fuzzy implication that satisfies the following LMI condition under minimized μ,

T i i i

i i

C N PA C N PA

I I I I I

I I P PH PJ

) (

0

*

0

*

0 0 .

*

0 0 0 .

*

0 0 0 0 .

*

.

2 2 2 2

0 1 2









































(11)

Due to ϵ (t), z (t) and g (t) some resides remain on the estimation as (10) indicates. The gain of the PI observer for each fuzzy implication is given by,

(4)

i

P N

K 

^¹

.

(12)

C.

Robust state/fault observer (RO)

By introducing a new state variable x̅ from the series of x and the sensor fault fs, a new arrangement for TS fuzzy model (1) is derived which is expressed by [4],

C F

F C H E

B H F B

F

F A A

Q I t f t x t x

t x C t y

t f E t w H t f F t u B t x A t x Q

m n d q

i i p q

i i

i i m q T n

T s T

s i a i i i r

i i

 



 







 







 







 







 





 



 























1 1

1 1 1 1 1 1

0 , 0 ,

0 , , 0 0 0 , 0 )]

( ), ( [ ) (

) ( ) (

)) ( ) ( ) ( ) ( ) ( ( )

( 

 (13)

Adding K1iẏ to both sides of (13) yields,



 





 



 





 





















K F Q diag

K K F C F Q I

F Q F H Q H E Q E B Q B A Q A

t Ff t x C t x C t y

t y K G t f E t w H t f F t u B t x A t x

i i iq i i i i

i i i i i i i i i i i i i i

s

i i s i i a i i i r

i i

, 0 ) ,..., ( 0 ,

, ,

) ( ) ( ) ( ) (

)) ( ) ( ) ( ) ( ) ( ) ( ( ) (

1 1 1 12 1 12 1

1 1 2 1 1 2 1 1 2 1 1 2 1 1 2

0 1

1 1 2 2 2 2 2 1

*



 



(14)

It is shown [4] that under the defined conditions, there is a transformation matrix, T that transforms (14) to the following equation,



 



 

















 



 













 



T F N F T F K Q T N

H T H E T E B T B T A T A

t r I t

Tr C t y

t y N t f E t w H t f F t u B t r A t

r

t Tr t C x

T C

i i i i

i i i i i i i i

q q m n q

s i i a i i i r

i i T

, 0 ,

, ,

) ( ] 0

[ ) ( ) (

)) ( ) ( ) ( ) ( ) ( ) ( ( ) (

) ( ) ( ,

1 2

1 3 1 1 1

2 1 3 2 1 3 2 1 3 2 1 3

) ( 1

3 3 3 3 3 1

1

1 1

*



 

(15)

Now, system (15) is decomposed into two set of variables, variables dependent and independent of the measured y as follows,

i i i i

i i i i i i i i i i i

i i

T T T q

q m n

q q m n q

s i i a i i i i r

i i

s i i a i i i i r

i i

F F N F N H N H E H E B E B A B A A

A A

t r t r t r t r t

r

t t r r

t I r t

y

t y N t f E t w H t f F t u B t r A t r A t r

3 32 31 2 1 3 32 31 3 32 31 3 32 31 3 22 21

12 11

2 1 2

1

2 2 1 ) (

2 32 32 32 32 2 22 1 21 1 2

1 31 31 31 31 2 12 1 11 1 1

, , ,

, ,

] ) ( ) ( [ ) ( , ) ( , ) (

) ) ( (

) ] ( 0 [ ) (

)) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) (





 



 



 



 



 



 



 



 



 



 



 

































 



 



























(16)

r2 has already been measured, thus the unmeasured r1 is required to be estimated. By considering the following equations for fs,

11 12 1

21 1 22 2

21 22 2

s s

x T T r

x Tr f T r T r

f T T r

     

       

   

 

(17) a reduced order estimator is designed for r1 and fa as follows [4],







  

 









































































g

i g

j g

k k j i g

i g

j j i g

i i k

j i j i i

i i i i i i

i j k k i j

i j r

k j i

i j i j

i j i k k i j i

a i j i r

k j i

i j i i j i k

k k a

k

k k

denote

T F A A T F A A

t y A K N K K F K

t u B K t r A K t z F K t z

t y A K A N K K N A K A

t f F K F t u B K B t z A K A t z

t y N K K t z t f

t y N K K N t z t r

1 1 1

1 1

1 ,

, ,

22 31 22 22 21 31 21 21

22 31 12 12 21 31 11 11

22 3 2 3 3 32 3

32 3 ,

,

1 21 3 2 32 3 2

22 2 12 2 2 2 1 21 2 11

32 2 31 ,

,

32 2 31 1 21 2 11 1

2 3 3 2

2 2 2 1 1 1

, , ,

, ,

)]

( ) ) (

(

) ( ) ˆ( ) ( [ ) (

)]

( )) (

) )(

((

) ˆ( ) (

) ( ) (

) ( ) [(

) (

)]

( ) [(

) ( ) ˆ(

)]

( ) [(

) ( ) ˆ(





(18)

The observer estimation error is denoted by,

1 ˆ1 ˆ

( ) [( ( ) ( )) , (^T _a( ) _a( )) ]^{T T}

e t  r t r t f t f t (19)

It is shown [4] that If there exist a symmetric positive- definite matrix P, and matrices Mi >0 (for each fuzzy implication) and j=1, 2… g that satisfies the following linear matrix inequalities,

0, 2 0, 1

ii 1 ii ij ji

G G G G i j g

 g      

 (20)

Where

4 4 1 1 1 2

1 2

11 31 31

4 1 21 32

2

2 32 1 1

*

, [ ] , 0

0

0 0

[ 0],

T T T

i i j i i j i i j

ij

i i i

i j j j i

j j

A P PA J M M J M PH M J

G M

H

A F

A J A B H

I

J H



 

      

   

   

    

 

 

(21)

Then, the estimation error has H∞ performance level γ1 as expressed below,

1 12 2

0 0

lim ( ) 0, ( ) [( ( )) , ( )] 0

( ) ( ) ( ) ( ) , ( ) 0

T T T

t a

t t

T T

e t t w t f t

e s M e s ds s M s ds if t



   



   



  

 

(22) With the following observer gain:

2 1

4 3

i

i i

i

K K P M

K

  

 

 

(23) III. SIMULATION TESTS

In this section, fault estimation in a two tank hydraulic unit using the PIO and RO algorithms are examined. There are two interconnected tanks, two level sensors, two electric valves and a fluid supplying pump. The system state variables and control inputs are,

1 1 2 2 1 2

[ , , , ] ,^T [ , ]^T x h s h s u v v

Where hi, si and vi are the fluid level in a tank, valve opening and the valve controlling voltage of component i.