Design of Finite-Dimensional Observer-Based Fault Diagnosis for a Class of Distributed Parameter Systems

Copyright @ IFAC Fault Detection. Supervision and Safety for Technical Processes. Espoo. Finland. 1994

DESIGN OF FINITE-DIMENSIONAL OBSERVERBASED FAULT DIAGNOSIS FOR A CLASS OF DISTRIBUTED PARAMETER SYSTEMS H. RAFARALAHY, M. BOUTAYEB, M. DAROUACH and G. KRZAKALA C.R.A.N.-E.A.R.AL (C.N.R.S. UA 821). Universite de Nancy I. 186 rue de Lorraine. 54400 Cosnes-et-Romain. FRANCE A bstract. A new method for the design of finite-dimensional observer-based fault diagnosis for a class of distributed parameter systems is presented. The considered system is a set of parabolic equations with nonlinear boundary conditions which is reduced to a set of ordinary differential equations by the use of generalized orthogonal polynomials. The simultaneous state and sensors fault estimation of a class of distributed parameter systems is transformed to the state observation of finite dimensional nonlinear descriptor system without assumptions about the dynamics of fault components. The proposed observer is a finite dimensional non singular system. The exponential convergence in the original coordinates is stated and a numerical example is given to illustrate the efficiency of the method. Key Words. Distributed parameters systems; sensor failures ; nonlinear singular systems; finite dimensional observers

1. INTRODUCTION

must be approximated for numerical simulation or implementation.

In the last decade, considerable attention has been focused on the theory of observer-based fault diagnosis of finite dimensional linear systems. Diagnostic observers. instead of observers-based control, are output observers and residuals are given by the estimation errors. Two approaches based on observers are available in the literature, the dedicated observer scheme and the unknown input observer scheme. Recently, Seliger and Frank (1991), Frank (1993), Schneider and Frank (1993), have proposed observer-based fault diagnosis schemes for nonlinear uncertain systems. They have extended the theory of linear unknown input observers for residual generation of nonlinear uncertain systems. Since it's introduction by Thau (1973), the theory of nonlinear observers has been a topic of interest of many publications. Different design methods based on nonlinear state transformation and output injection have been developed. Unfortunately, little effort has been made for observers design of nonlinear descriptor systems. Their importance arises from the ability to represent nonlinear systems with unknown inputs, partial dynamics with nonlinear algebraic constraints and non causal systems. However, few effort has been made to develop the theory of observer-based fault diagnosis for infinite dimensional systems described by partial differential equations. Moreover, design of observers for distributed parameter systems requires to generate a strongly continuous semi-group operator which

In this paper, we consider the simultaneous state and sensors fault estimation for a class of distributed parameter systems by the use of a finite dimensional observer. The main idea of our work is the use of generalized orthogonal functions to transform the original distributed parameter system into a set of ordinary differential equations to reduce the complexity of the state and fault observation problem. The technique of generalized orthogonal functions has been used extensively for system analysis. parameter identification, model reduction and control of distributed parameter systems. The approximated system thus obtained is a finite dimensional nonlinear system which is transformed into a nonlinear descriptor system for simultaneous state and fault reconstruction.

2. PROBLEM STATEMENT Let us assume the process governed by the following set of partial differential equations SI: aXl(Z,t)

(I-a)

at aXl(ZlO,t)

dn

:::0

dX'(ZIl,t) dn -S,(U,Z'l>Z20,t)

(I-b) (l-c) (I-d)

41

T

S2 : dX2(Z,t) d2X2(Z,t) dt a2 dZ 2 ZE [Z20,Z21]. tE [O,oo[

X2(t)

d2(z2o)

dl

=

~/X}(t)2(Z20)-U(t))+Y2 (9 2(X}(t))-9 1(xT(t))) (5-c)

(I-e)

T

d2(z21)

dX2(Z20,t) dn S2(U,Zll>Z20,t)

(I f) -

X2(t)

dX2(Z2I,t) dn =0 X2(Z,t=O)=X20

(I-g)

where 9j(xT(t)) = xT(t)QjXj(t)xT(t)Qjxj(t) for j=I,2

(I-h)

and

dl

with

with al>a2>O, and zlO
4

4

4

SI(.) = ~1(XI(ZII,t) - u(t)) + YI(XI(Zll,t) - X2(Z20,t)) ~2(·) = ~2(X2(Z20,t) - u(t)) + Y2(X2(Z20,t) - XI (Zll,t)) These equations express the mathematical model of heating process where x/z,t) is the temperature distribution of Sj' u(t) is the surrounding temperature and the boundary conditions (I-c) and (l-f) express the simultaneous heat transfers by convection and radiation.

=0

(5-d)

QI = 1(ZII) T(Zll)

Q2 = 2(Z20) }(Z20)· Substituting (5-a)-(5-d) into (3) and (4) yields dXI(t) "dt = Al XI(t) + BI u(t) + Cl g(XI(t),X2(t))

(6)

dX2(t) "dt = A2 X2(t) + B2 u(t) + C 2 g(XI(t),X2(t))

(7)

-I

where

A

T

Al = al WI (1-'1 QI - VI) , -I

A

T

A2 = - a2 W2 (1-'2 Q2 + V2), -I

BI=-al~IWI 1(ZII),

B2 = a2

~2 Wil 2(Z20), -I

Cl = - al YI WI 1(ZII)' -I

Assume that variables Xj(z,t) are square integrable therefore there are expandable in orthogonal series Nj·1 Xj(z,t) '" ~ Xij(t) j(z) j=I,2 (2)

C 2 = - a2 Y2 W2 2(Z20), g(XI(t),X2(t)) = 9 1(x T(t)) - 9 2(x}(t)).

T

where

Xj (t) = [xo/t) ... x(Nj-I)}t)] /z) = [T(z) and
3. OBSERVER DESIGN A great number of methods for observer based residual generation have been proposed in the literature. The greater part of these methods are devoted to finite dimensional linear systems. Little effort has been developed for the fault diagnosis of non linear systems or for infinite dimensional systems. Based on recently proposed observers for a class of distributed parameter systems (Rafaralahy et aI., 1993) and for a class of nonlinear singular systems (Boutayeb et at., 1994), this paper is devoted to the simultaneous state and fault estimation.

~wI=alxl(t)

dI(Zll) .... T( ) dI(ZIO) .... T( ) V}(3) dz 'V I Zll dl 'V I ZIO I

{

T

dx2(t)

T

Assume the measurements taken at point zmij of each subsystem Yll(t) XI(Zmll,t)

~ W 2 = a2 x 2(t)

d2(z21) .... T( ) d2(z2o) .... T( ) V } (4) dl 'V 2 Z21 dl 'V 2 Z20 2

{

Zjl with

Wj =

f /z)T (z)dz,

ZjO dj(z)

dz

XI(Zmldl,t)

= Dj/z), Vj = DjWpj Y2d 2(t) T

Yij(t) '" i (Zmij) Xm + h ij dij(t)

The boundary conditions (I-b), (I-c), (1-f), and (1g) can be approximated as follows T dI(ZIO) x I(t) dl 0 (5-a)

The equivalent system under approximation is

=

d1 (ZII)

dl

X2(Zm2d 2,t)

which can be approximated as

over [zjo,zjd.

T

X2(Zm2I,t)

T

and Dj is the operational matrix of differentiation

XI(t)

=

yet) =

(l:)

=

~I (xT(t)1 (ZII)-U(t))+YI (9 1(XT(t))-92(xJ(t))) (5-b)

42

{

~ ch = A x(t) + B u(t) + f(x(t)) yet) = HI x(t) + H2 d(t)

Q/t) fI

x/t)

fI

11(t) =

•

a/t)

ap) where x(t) is the state vector, u(t) and yet) are the input and the output variables respectively. The signal d(t) represents the sensors fault to be detected or estimated . For the simultaneous state and sensors fault estimation, consider the augmented state 11(t) =

Proof Under assumption (iii), there exists matrices P, K, P I and S I of appropriate dimension which satisfy

equation (8). Let e(t) the estimation error e(t) = ~(t) - 11(t) = \)(t) + K yet) - 11(t)

[:~:~J

eet) = 1>(t) + (K H - I) ~(t) (16) which, by virtue of equation (8), can be written as

the system (L) becomes a singular nonlinear system (La) (La)

{

EdT]~t) =

eet) = 1>(t) - P E ~(t) eet) = N \)(t) + L yet) + M u(t) + g(\)(t),y(t))

A11(t) + E u(t) + [(11(t))

yet) = H 11(t) H = [HI H2]

+ ( M - PE) u(t) + g(\)(t),y(t))-P[(11(t))

-

f(.) = f(.) and B = B. Notice that no assumptions are made about the dynamics of d(t).

+ ( NPE + LH - P A) 11(t) (19) Using equations (9) to (IS), the following Sylvester equation holds, where N is a stability matrix for a suitable choice of 9

Theorem under the following assumptions: (i) the pair (A,H ) is observable, I f( .) is locally Lipschitz, (ii)

-

{

[~ is of full columns rank.

~ ~(t) = eT(t) (NT S - S N) e(t) ~(t) ~-geT(t)Se(t) + 2e(t)TSP( f(~(t))-f(11(t)))

(23)

+ 2 k "'max(S) IIPII)~(t) (24) ch "'min(S) where "'min(S) and "'max(S) are respectively the smallest and the largest eigenvalues of S. Therefore, there exists kl >0 such that the error dynamics is exponentially stable for 9 >

(11)

N=PA-GH

(22)

~ ~ (- 9

(9) (10)

-

+ 2 eT(t) S P (f(~(t)) - f(11(t)))

Under assumption (ii) and using Cauchy-Schwarz inequality one obtains

(8)

-

(21)

Let ~(t) = eT(t)Se(t) a Lyapunov function candidate where S satisfies the Lyapunov equation (14). The time derivative of ~(t) equals

11(t) = \)(t) + K yet) with

M=PB L=G+NK

-

eet) = N e(t) + P a(\)(t)+Ky(t)) - [(11(t)))

= N \)(t) + L yet) + M u(t) + g(\)(t),y(t))

g(\)(t),y(t)) = P f(\)(t) + K yet))

-

NPE+LH-PA=0 (20) Finally, e(t) is governed by the following differential equation

then the following system (Oc) (Oc) :

~(t)

(18)

eet) = N e(t)

whereE = [In 0], A = [A 0]

(in)

(17)

- P (A 11(t) + E u(t) + [(11(t)))

-

(15)

(12)

and G = S-I HT (13) where S is solution of the following Lyapunov equation for 9>0 large enough

(

2 k "'max(S)IIPII) "'min(S)

2-kJ:

ile(t)11 ~ klile(O)ilexp[ - 9 "'max(S) (S) IIPII ) t] (25) ( min i.e. ~ and <1 converge exponentially to x and d respectively.

- 9 S - (AT pT S + SPA) + HT H = 0 (14) is an exponential state observer of the system (La)

J(z)~/t) is an estimation of xjCz,t) and

43

Finally, the state estimation of the original system is given by fI T fI x/z,t) = j (z) x/l) j= 1,2 (25) The error estimate ~ in the original coordinates is given by ~/z,t) = £j(z,t) - Xj(z,t) (26) It is well known that (Szego, 1975) Nj-I lim IXj(z,t) - } xiJ(t) i]'(Z)I = 0 (27) N)--+00 t:"b

250 200

.J50 . .L : JJ_. ,.~_..~~~.• of :~ r'Y" .. ..~t " \ rwr.jlk", bo .~

:

'; 11 • • r ••.••I• ••• t ' ' "

Therefore 1~/z,t)1 :S; IxN/t)1

-50 0

e

"'mru,cS)

+ kiz) lIe(O)lIexp[(-2"+k"'min(S) IIPII and finally

~

100

. lim

N]-+~ . t-+~

5

10

15

20

25

30

35

t (mn)

) ] t

(28)

1~;(z,t)1 = 0

Fig. 2 Fault reconstruction (d l )

(29)

J

•

which completes the proof.

:;~k:····

Hence if the error dynamics of the approximated system is exponentially stable then the error dynamics is asymptotically stable in the original coordinates with a proper choice of e.

_ I

1280 ~ :.\ ... 1260 ., \ . , . . . .... : . . .. .. . 1240 1220 . "'r\k;,;.. :... :

. . . ...

'\\'V" . :....:.......... . '.

1200

1180 ' .

4. NUMERICAL EXAMPLE

. .'; *;~:; ~;'\"'" 'a:j~; "

1160 .

1120 0

.

:/: --:~J~ . V:'.'; ··· .: .. ~:·.il.'~·, ......... . . .

······· ···¥... t·,i.·,;., l ·:'\ ··

1140' .......... ..

Let us consider the following example of coupled parabolic system given by equations (l-a)-(l-h) with:

.: <.~~.

.. .... .. ..

10

.. ~.tr... ... :... 'f1~ ..... :.. .. ..:.... .

20

30

40

50

60

70

80

t (mn)

Fig. 3 System output (S2) al 3.2 e-5 a2 3.2 e-5

~I - .3738 ~2 .3738

YI -3 e-lO Y2 3 e-lO

ZIO 0 Z20 .9

Zll .1 Z21 1

XIO

l0000K X20 13000K

50 40 ........ . 1

" ,c , '" c,1 ,. ' •• •• •••• 1

30 .. 20 ..

The input u(t) which is the heat source term is assumed to be constant i.e. u = 1200oK. The state vectors of each subsystem are approximated by the shifted Legendre polynomials of fourth order. Simulation results are shown in figures 1 to 6. Figures 1 and 3 represent respectively the system outputs of SI and S2' In figures 2 and 4 are shown the fault estimation . Finally, figures 5 and 6 represent the actual value (solid) and the estimated value (dashed) of the state of the original distributed parameter system at z=O for the subsystem S I and at Z= 1 for S2'

20

25

30 35

40 45

50

55

60

I

· (y:tL~J~~4 -c - - "c-

'u_

l~f--- -- -- ' 9500

10

20

30

40

60

70

.. . ~ ...l

... --.~"-': . C.. .. : .... : .... :'

800 ~: ·

-: .

. .:

...... : .. .. . : . . . . . : .. . .. : -

- !I

600~""

"

...... .... .

" .!I """" 400~ -r:

1

'"

-50

••

70

Fig. 4 Fault reconstruction (d 2)

I~

::1-( • • •"• '• -"

65

t (mn)

2~i

... .. .. '..... . , ... . . '...

. .. -.. . . -.. .

1

-..........: :

OoL---~--~--~--~--~--~--~--J

10

80

20

30

40

50

60

t (mn)

t (mn)

Fig. 5 State estimation (z=O)

Fig. 1 System output (SI)

44

70

80

" ~~ '~~::-'-'-~-. . . .:.:~.

. . . .;. . . . . .:. . . . .

Hou, M. and P. C. MiilIer. (1992). Desing of observers for linear systems with unknown inputs. IEEE Trans. Aut. Control. Vo!. AC 37. No. 6. Kou. S. R.. D. L. Elliott. and T. J. Tarn. (1975). Exponential observers for nonlinear dynamic systems, Information and Control. 29. 204-216. Krener A. J. and A. Isidori. (1983). Linearisation by output injection and non linear observers. Systems & Control Letters 3. 47-52. Krener A. J . and W . Respondek, Nonlinear observers with linear error dynamics. SIAM J. Control Optim. 23. 1985, 197-216. Lahouaoula A . and M. Courdesses. (1989). Identifiability and identification of a class of parabolic distributed systems under approximation. Int. J. Systems Sciences. Vo!. 20. No. 4. 683-697. Rafaralahy H .• M. Boutayeb. M. Darouach and G. Krzakala, (1993). Exponential state observer for a class of distributed parameter systems. 32nd IEEE Conference on Decision and Control. San Antonio. Texas. Schneider H. and P. M . Frank. (1993). Observerbased supervision and fault detection for robots, TOOLDIAG'93 Internationnal Conference on Fault Diagnosis. April 5-7. France. Seliger R. and P. M. Frank. (1993). Robust residual evaluation by threshold selection and a performance index for nonlinear Observer-based Fault diagnosis, TOOLDIAG'93 Intemationnal Conference on Fault Diagnosis. April 5-7. France. Syrmos. V .• (1992). Observer desing for descriptor systems with unmeasurable disturbances. IEEE Conference on Decision and Control, Tucson. December. Szego. G . (1975) . Orthogonal polynomials, American Mathemetical Society. Thau. F . E. (1973). Observing the state of nonlinear systems. Int. J. of Control. Vo!. 17. No. 3. 471-479. Tomambe. A. (1992) . High-gain observers for nonlinear systems. Int. J. Systems Sciences. Vo!. 23, No. 9. 1475-1489. Viswanadham N. and R. Srichander, (1987). Fault detection using unknown input observers. Control-Theory and advanced Technology. Vo!. 3. No. 2. Wa\cott. B. L. and S. H. Zak. (1987). State observation of nonlinear uncertain dynamical systems. IEEE Transactions on Automatic Control. Vo!. AC-32. No 2. Yang, F. and R. W. Wilde. (1988). Observer for linear systems with unknown inputs. IEEE Trans. Aut. Control. Vo!. AC 33. Yoichi Uetake. (1989) . Pole assignment and observer desing for continuous descriptor systems. Int. 1. of Control, Vo!. 50. No. 1. 8996. Zeitz. M., (1987) . The extended Luenberger observer for nonlinear systems. Systems & Control Letters 9. 149-156.

~-~-I'I _. . . . . .. .,

.... .. ....····1 ······:- ··· · ·:-·· · · ·:-· ··· 1

" '1

····1 : :: : ...... ........ : : : :I 10

20

30

40

50

60

70

80

t (mn)

Fig. 6 State estimation (z= 1)

5. CONCLUSION In this paper. a thermal process with mixed heat transfers by conduction. convection and radiation is modelled as a distributed parameter system which is reduced to a set of ordinary differential equations by the use of generalized orthogonal polynomials. Then the system is transformed to a singular non linear system for the simultaneous state and fault reconstruction without assumptions about the dynamics of fault components. The observer thus obtained is a finite dimensional non-singular system. The exponential convergence of the observer in the original coordinates is stated. A numerical example is presented to illustrate the effectiveness of this approach.

6. REFERENCES Birk. J. and M. Zeitz. (1988). Extended Luenberger observer for nonlinear multi variable systems. Int. J. of Control. Vo!. 47. No. 6. 1823-1836. Boutayeb. M .• M. Darouach. H. Rafaralahy. and G. Krzakala. (1994). Asymptotic observers for a class of nonlinear singular systems, accepted for the 1994 American Control Conference, Baltimore. Maryland. Deza. F. E. Busvelle. J. P. Gauthier and D. Rakotopara. (1992). High gain estimation for nonlinear systems, Systems & Control Letters 18.295-299. El Jai A. and M. Amouroux. (1988). Sensors and observers on distributed parameter systems. Int. J. of Control. 1988. Vo!. 47. No. 1.333-347. Frank. P. M .• (1991). Enhancement of robustness in observer-based fault detection. IF A C SYMPOSIUM on Fault Detection Supervision and Safety for Technical Processes SAFEPROCESS'9I, Baden Baden. Frank. P. M .• (1993). Advances in observer-based fault diagnosis. TOOLDIAG'93 Internationnal Conference on Fault Diagnosis. April 5-7. France.

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