A Fault Detection Isolation Observer Design by Hybrid Disturbance Decoupling Approach

Copyright C IFAC Intelligent Manu!r.cturing Systems, Scoul, Korea, 1997

A FAULT DETECTION ISOLATION OBSERVER DESIGN BY HYBRID DISTURBANCE DECOUPUNG APPROACH J
Tae--~n

Park

Department of Electrical Engineering Dankook University #8 Hannam.Dong, Yongsan-Gu, Seoul, 140-714, Korea Phone: 82-2-709-2575, Fax: 82-2-795-8771 e·mail: [email protected]

Abstract: A fault detection isolation observer developed so far rt:quj~s that the mlmber of unknown inputs (including the faults of no interest and di.stwbances to be algebraically rt:jected) must be less than that of outputs. TIle existence condition is hard to meet and restricts the practical use of a fault detection isolation observer. The development of the fault detection isolation observer with some mild existence condition has been an open problem. 1be pwpose of this paper is to propose a fault detection isolation observer which gives a viable solution to the problem. 1be basic corrept of this proposition is the use of the unknown distwbanc:e modeUing approach together with thc algebraic decoupling approach. A numerical example shows the applicability of the proposed fault detection isolation observer. Keywords: Fau1t detection isolation; Fault detection isolation observer, Unknown input observer, Algebraic decoupling approach; Di.stwba.oc:e roodelling approach

faults, distwbances and modelling errors so that the fau1ts can be isolated by the simple decision logic with the prescribed confidence: level. Hereafter. an unknown input observer (UIO) will be called by a fault detection isolation observer (FOIO) when it is sensitive to some particular fau1ts and invariant to some other faults and unknown disturbances, while an VIO is invariant to all the unknown inputs (disturbances) to the process. No intentional classification of distw'banc:es is made in the design mo. There have been a number of studies for the robust residual gereration based on the UIO theory . Viswanadham and Srichander (1987) developed FDIS by directly utilizing the UIO of Kudva, et al. (1980). FI1lIlk and Wilnncnberg (1989) systematically described a unified approach to the design of robust observer schemes for fault detection isolation (FOO. More recently, an approach for detecting and identifying sensor and actuator faults in uncertain systems was presented by Saif and Guan (1993). A design procedure and the necessaty and sufficient condition for the

1. INlRODUCTlON

Increasing complexity and large sca1e property of today's industrial process not oruy make the achievement of the desired level of system reliability difficu1t but also increase the ~hutdown cost, and various theories and techniques have been developed for enhancing system reliability. Among them, the model-based fau1t detection and isolation system (FUIS) has been an important research
73

existence of FDI0 were given by Hou and MUiler (1994) through a new fonnulation of the FOIO problem (Massoumnia, et a/., 1989; Koenig, et al., 1996; Koenig, et 0/., 1997). The reduced order UIO and its application to the FDI problem in a class of stale delayed dynamical systerm were addressed by Yang and Saif (1996). It is well known that almost all the existing UIOs and IDIOs take the algebraic decoupling approach to guarantee the invariant property to the faults of 00 interest and distwbances, and they have the existence condition that is hard to meet That is; the number of distwbances must be less than that of outputs. This condition restricts the practical application of the UIO to the FDI problem in which multiple faults are concerned (Saif and Gum, 1993). This problem has been well rec:ognized and is still an open problem to be solved.

set of the fault mode to which an FDIO is sensitive by the fault of interest; otherwise. the fault of no interest. Rewriting the system (I) with the two subsets of the fault mode gives

where

i=A.r+Bu+Fd+Fu:

(2a)

y~C%

(2b)

we R ",

de R -,

(F)~f,. rank(C)~p

rank (F) = a,

and

a+p~

rank

• . In (2a),

Fa and Fu are defined as Fd= . ~

..1.4-1 Tjm,

Fw=

t.

;./. i _ I

Tim,

(3a) (3b)

where J is the index set of the faults of interest, and J denotes the index set of the faults of no interest, u:. Obviously, 1/\1= 0 and fVf ~ U.2 ... ·.kl. The matrix F consists of the signature matrix of the faults of no interest and therefore may be regarded as the coefficient matrix of some unknown inputs. F denotes the coefficient matrix of the faults of interest. IDID design problem is to construct an observer which is insensitive to the faults of no interest, u: in (2a). As it was noticed before, almost all the e>cisting FDIOs take the algebraic decoupling approach, so they have the existence condition;

a,

The purpose of this paper is to suggest the FOIO that yields a viable solution to the problem. The basic c:oncept of this proposition is the use of the unknown distUlbance modelling approach (Lee, et a/., 1996) together with the algebraic decoupling approach. The paper is organized as follows. In Section 2, the system including the faults is fonnulated and the structwal constraints imposed upon the previous studies of FOIO (or mO) is pointed out Such a crucial limitation is removed through a new design c:oncept of IDIO and the feature of the proposed FDIO is illustrated in Section 3. SectiOD 4 shows the applicability of the IDIO through an example.

rank (C F ) - rank ( F)

This condition cannot be satisfied whenever the number of outputs is less than that of the faults of no interest. lbe condition restricts the practical application of the FDIO and must be replaced by some mild condition to achieve reliable fault diagoosis.

2. PROBLEM DESCRD'nON Assume that our system with the faults can be described by the linear time invariant model (Koenig, et al., 1996);

3. A NEW DESIGN CONCEPT OF FOIO

x= Ax+Bu+ ,~ ~ Tjm, y~

C%

(la)

In this section a new FDIO design concept is proposed. Remember that the problem is to design an FDIO such that the fault vector u; in (2a) should not affect the quality of the state estimates, or equivalently, the penormance of the observer. 1ne main idea of our proposition is the combined use of two useful but different techniques: the algebraic decoupling approach and the unknown disturbance modelling approach.

(tb)

where xe R· is the state vector, ue R" is the input vector, A: is the number of possible faults, miE R If are the ith fault mode, and TjE R .. are the fault signature matrix. Matrices A , B and C an: (n. n), (n ...) and (P. n) matrices., respectively. In multiple obsen'er based FDI schemes, a number of IDIOs are employed to isolate the fault mode(s). Each IDID is a special kind of UIO which is sensitive to a subset of the fault modes and insensitive to the remaining fault modes. How to partition the fault modes into two subsets depends on the design fearures : simplicity of isolation logic. ability of multiple fault isolation. and existence COnditiOD of each FDIO. etc.. Hereafter we will call the

3. / Partitioning the faults of no interest The flI'Sl step of the FDIO design is to partJllon the faults of no interest, u in (la), ilUo two

sub·vectors:

w=

74

TJ T

T· [ WI : W2

(4)

2~= Eu 2z,.

where the effect of the fault vector WI E R Q (q(p) on the state estimates can be removed by an algebraic way, and all remaining faults fonn another fault vector W:2E R I-q. Then, the system (2) becomes x=Ax+Bu+Fd+(F I F 2 y~

)(:j

Cx

UJ:2,. =

Hu

(Sa) (5b)

Eu= [0 (' , -lld

l,···,P-Q

2z,.

° °

1 (&,_0

I "I

(6a) (6b)

Because the effect of the fault vector W2 cannot be algebraically removed, they should be tIea1ed in some other way SO that the resultant FDIO provides the correct state estimates in the face of the faults of DO interest. The unknown disturbance modelling approach is taken in this w\ by the study. Hereafter, let us call algebraically rejectable fault vector and W:2 by the modelled fault vector. Theorem 1 gives the explicit partitioning conditions.

1ER I{I-q),,6(I-q)

1"(',- 1)

0 I ,, (I,-ll1 e R (I-oil ,,6(I -tl .

Hu= (II

where 8, is the order of the polynomial:

The dynamical equation (6) provides an effective model not on1y for unknown disnllbances but also for process faults and sensor faults with the assumption that they are of the wavefonn structures. The (p-Q)-dimensional fault vector is modelled by arranging the model of each fault vector to diagonal fonn as

Theorem 1. A partitioning of (4) is prop." if (a) omk (CF,)- omk (F,) - Q; and (b) the pair (C : PAl is completely observable

22= E z 22

when:

W2=

(1.- F,(CF,)+C) eR"·

(CF,)+ ~

=

If the sufficient information of the faults are available. the parameter matrices can be identified. Otherwise, the faults may be modelled as the solution of (6) with the matrices:

where FleR ""Q and F2ER "" (I- q).

P~

i

H z 22·

(7a)

(7b)

3.3 Augmented system and FDJO design

« CF,)T( CF,»-'( CF,)T.

TIle augmented system is obtained by combining the system (S) with the fault model (7):

Condition (a) is necessary for the algebraic rejection of the fault vector Wt. It is easily proved by showing that there exists a transfonna1!on matrix. f, for (5) that makes PFI=O. Since rank(CFI)=Q, we have (CF J )+( CFI) = Iq . Therefore,

:Ca = AaX,.+ Bau+F,.d+ FaW I y = CaXa

(Sa) (8b)

when:

PF, ~ (I. - F,(CF,)+C)F, ~

F, - F,(CF,)+(C F,) ~ O.

Condition (b) is to guarantee the existence of an FDIO for the transformed system. It should be noticed tha1 the transformation matrix is almost always singular, and the pair (C: PAl may not be observable even if the pair (C : A) is proved to be observable. In such cases the number of the rejectable fault vector WI that will be algebraically rejected shou1d be reduced and reananged so that the observability condition is satisfied.

Aa ER <-+6(I-q).-+«.I-q), B,. eR <,,+6(I-tl . • ), F", eR ("+6(I-q).,,,), Fa eR (.+
al., 1997). E(WI) =

3.2 Modelling Ihe faull vector The effect of the fault vector W2 on the state estimates cannot be algebraically rejected. In many cases, however, the faults thaJ: may occur in physical processes have some wavefonn struc:tures such as jwnp(step), ramp and exponential function etc.. So, a fault can be modelled by the differential equation:

I

y-

C",Aaxa - C",B"u - C"F",d- C",F",WI12

(9)

where y - C"A",x", - C",B",u - C"F",o. is a fixed vector and I· I 2 denotes the Euclidean nonn The solution is given by w\ =

75

(Cs F",)+ (y

- C",A",x.- C.B.u - C,.F,.d) +UQ-(C.F",)+(C",F,.»'W; (10)

with rank (C.. F .. ) '" rank (F.. ) -4, where can be considered as new diS1UIbances and

(C,F,)' ~

« C.F,)T

(C,F,»-I (C .. F..)T eR

WI

e= Ne- PF.. a.

The error dynamic equation (18) shows that the fuIl-order FOIO (14) with the coefficient matrices (16) is invariant to the faults of ro interest and sensitive to the faults of interest. Theorem 2 summarizes the existence condition of the asymptotic FOlD (14) for the augmented system (8).

'IX'

With the rank condition above, P F .. = 0 where

P= (I.+4l.P-q) - F .. (C.F.. )+C.) eR (,,+4l.P-.» x(_+4l.P-.)}.

(11)

Theorem 2: If (a) The pair (C.. : PA.) is observable; and

Therefore, substituting (10) into (Sa) gives

(b) rank (C,F,) = rank (F,) =

then there exists an FOIO coefficient matrices (16).

(12a)

(12b)

by

Defining a new state z=;';- F.(C.F.)+y in (13) removes the time derivative of the measurement output vector and gives the weIl·known expression:

f,;=z+Ly

(l4a) (l4b)

where zER (,,+M.P-q», X:;eR (-+ «P-q», and NeR (. + 4l.P-q» x(. +4l.P-q», (;eR (.+4l.P-q» x,. HeR (.+4l.P-fl) lI"', and LeR (.. +4l.P-.» 1(, must

be determined such that x:; will asymptotically converge to Xa if no fault of interest occurs. With the new state z= x .. - -F.(C.. F .. ) > = ;;; - Ly, the FOlD (13) becomes

-

z=

(PA .. -KC.)z+PB"u

+ [K(I.

- C,L)

+ PA,LJy.

-. (15)

are obtained from (15):

G~

PA, - KC, K(I, - C,L)

+ PA,L

H~PB,

L ~ F,(C, F,)'

and FI = F. then all the faults of no interest can be rejected by the algebraic method and the observer design procedure is therefore reduced to the pme algebraic approach, and (n)th order (lowest) observer may be constructed. (ii) if U7 = It: and F2 = F. then all the faults of no interest must be modelled and the resultant observer with highest order of (n + df!J may be constructed by the use of the unknown fault vector modeUing approach. It is noteworthy that case (ii) is the worst case in view points of

(l6a) (16b) (16c) (l6d)

Defme the state estimation error vector by

e= x.-x..

the

RemOl"k: As a special case, (i) if we can partition the fault vector of no interest and comsponding transmission matrix as wl = It:

Then, the coefficient matrices of the FOIO (14) N~

(P)Q), (14) with Q

It is straightforward to show that the coooitions (a) and (b) in Theorem 2 are equivalent to two conditions in Theorem 1 because the model (7) of the fault vector W2 is an observable canonical fonn The proposed new methodology yields the following characteristics, namely · it resolves one of the major practical difficulties of all the observer based FDISs previously reponed. that the number of unknown inputs must be less than or equal to that of outputs and is capable of detecting and isolating simultaneous faults more than the number of outputs. · in conventional FDIS based on UIOs or FOIOs, the eigenspectrum of the UlO (or FOlD) cannot be arbitrarily assigned if the number of outputs is equal to that of unknown inputs (Saif and Guan, 1993). This problem was removed in the proposed scheme. • the residuals for detecting and isolating faults are indepeooent of the modeUed fault vector. Therefore the proposed FOIO has the invariance property to unknown inputs (including the algebraically rejectable fault vector). • the FOIO offers additional degree of freedom in the selection of the fawts to be rejected by the algebraic decoupling approach. · as a byproduct, the FOIO regenerates the unknown fault modes so that they can be used to acconunodate the faull

where the algebraica1ly rejectable fault vector was disappeared. If the pair (C.: PA .. ) is completely observable, a full-order FOIO is given

z= Nz+ Gy+Hu

(18)

(17)

then, the error dynamics become

76

dimensionality observer.

and

estimation

errors

of

the

ml = m2 = m3-2+O.3sin(O.1t) and the system input, u = 1 + O.2sin(t) , Fig, 1 shows the FDI results for three cases that two simultaneous faults occur: (a) ml and m2' (b) ""2 and PnJ. and (c) m3 and. ml ' It is assumed that the faults 0CCUJTed at 5 second, From those figures it is easily recognizable tbat non-zero residual values due to the faults of no interest disappear after a soon transienL By proper choice of the threshold values for each residual . the detection delay can be minimized,

4. NUMERICAL EXAMPLE

A numerical example is taken to the system with matrices:

A =

[-Io 0 1] B=m·

c= [~

0 -2 1. 1-3

nl (19)

T1- (I 0 0]'. T 2 - [01 Or. T1 - [00 11'.

P=2

and a-I. and defme /=0). 1= (2.3) for FDI0 1 and /=(2).1= (3.1) for FDI0 2 and /=(3).1= 0.2) for FOIO 3. In this c:ase, FOIO t should be sensitive to only fault ; and robust to all other two faults. Choose matrices F. F= [F I : F 2 ] and vectors d. w= [ WI: Wz ] with q= 1 as in Table 1, where P = /J. Matrices E z and H z for modelling the: fault vector Wz are given with 8 = 2 in Table 2. In Table 3, the coefficient matrices N, G, H . L of each FOIO with eigenvalues of N= (-1.-2.-3.-5.-5) '"" represenled. The residual generation process simply consists of subuac:ting the measured state vector from the state vector estimated by the observer I: Choose

r/= I .%'-xl.

i = 1. ···.3.

F

dr

1

[T,]

2

[T, J

3

i= 1.2 ..... 3.

i f r;) E;. £1

=

E2

=

E3

[m, J

[F,; F,J [T,; T,J

[ WIT ·: WzTJ ['"2 : PnJ)

[m, J

[T,;T,]

{m3 : md

[T, J (m, J

[T,; T, J

[ml :mz ]

FDIO

E,

H,

[~ ~l [~ ~l [~ M

1

2 3

fl OJ fl OJ fl OJ

(20) FDIO

1

r/
where the thresholds are

FDIO

d..u

and vectors

Table 2 Matrices E2 and H z

where i' ER· is the: state vector estimated by the observer I. The detection logic: is if

E.. E.

Table I ' Matrices

Assume the total number of possible process faults is k = 3. Possible faults are described by the: fault signatures T i • i = 1. ... • 3 with the: indices corresponding to the: fault number:

(21)

= 0.7.

2

All zero v~' mean that the faults of interest has not occurred. while if at least one among v,s' is not equal to zero. it indic:ates the OCCWTeIlce of the: fauJt(s) of interest.. In related to Table I , the isolation logic of Table 4 is chosen to identify the faults where S,' = 1 means that fault i has occurred. 1bere are two zeros in each column, which implies that FOIO i should be sensitive to the fault ; and invariant to the other two faults for correct isolation. Two faults that occur simultaneously are taken into acc:ount In order to show the effectiveness of the proposed FDIOs and unknown fault modelling, in FOI simulatioDS are performed for the fault modes,

3

FDIO

-7 -0.5 0.5 o 0 13 -4 .5 -0.5 -10 -13 -0.5 -4.5 1 0 -61 0 o1 0 -30 0 0 o0 -8 -0.5 0.5 I 0 0 -4 -1 o 0 0 -I -4 o 0 -17 0 0 o1 -10 0 0 o0 -5 0 0 o0 0 - 6.6 -3 .6 1 0 0 -0.4 -4.4 o 0 o -13.4 -13.40 1 0 -7.5 -7 .5 o 0 1

o 1.4 o 13.4 o 7.5 3

H7 (I 0 0 0 OJ (I 1 - 1 0 OJ

(0 1 0 0 OJ

L7 [0 0000]

[1 0000] 00000

01000

77

2

G 6 0 -13 -1 13 1 61 0 30 0 7 1 0 1 0 -1 17 0 10 0 0 0 o 4.6

[00000] 00100

of the unknown disturbance modelling approach together with the algebraic decoupling approach. An important contribution of this paper is to resolve one of the major practical difficulties concerning about the existence condition of all the observer based FDIS. The IDIO may also regenerate the fauJt modes that can be used to accommodate the fauJt. The FOIS with the FDIO offers the designer additional degree of freedom such as the number of faults to be rejected by an algebraic approach and. can be constructed to detect and isolate simultaneous faults more than the number of outputs. A mnnerical example shows the applicability of the proposed FDIO for FDIS.

Table 4 Isolation Logic Fault

v,

v,

v,

""

I

0

0

SI -

0

I

0

S2

0

0

I

S3 -

""

'"3

Decision Logic

0-

1

(V I

=

V2 V3

-"

, ...-"

00

Vl

.- '.

TO

REFERENCES

30

Frank, P.M. and J. Wi,innenberg (1989). Robust fauJl diagnosis using unknown input observer schemes. In: Fault Diagnasis in

=I, s2 = 1)

DynamiC(JI Systems: Theory and App/iC(Jtions (RJ. Patton, P.M. Frank and R.N. Clarl<, (IS! Ed)), pp. 47-98.

15 20 tim.[nc)

=I,v2=1 , Va =0

,SI

25

Prentice Hall. New York. M. and P . C. MUller (1994). Fault detection and isolation observers. Int. J. Contr.. 60, pp. 827~846 . Koenig, D., S. Nowakowski and A Bowjij (1996). New design of robust observers for fault detection am isolatioIL Proc 35th IEEE Con] Decision and Control, Kobe, Japan, pp. 1464-1467.

Hou,

•!

•I •

1.6 1.' 1.' 1

....... '.

0.' 0.' 0.' 0.2 00

., ,

"

10

'-."

15 time

20

25

Koenig, D., S. NowakoMki and T. C=hin (1997). An original approach for acruator arv:l

30

component fault detection aId isolatiOIL To

[.~[

appear in Proc

of the IFAC Synposium

"Sajeprocess9r. (b) faults

,• •• ~ •

mz

Kudva, P., N. Viswanadham and A. Ramakrislma (1980). Observers for linear systems with wUrnown inputs. IEEE Trans. Aulomal. Contr., 25, pp. 113~1l5 . Lee, K.S., S.W. Bae and J. Vagners (1996). On the fault detection isolation systems based on GOS using functional observers. ?roe. 351h IEEE Con] Decision and Control, KOOe, Japan" pp. 1181-1183. Massoumnia, M.A., G.C. Verghese and A. S . Willsky (1989). Failure detection and identificatioIL IEEE Trans. Automat. Contr., 34, pp. 316~321. Saif, M am Y. Guan (1993). A new approach to robust fault detection and identificatiOIL IEEE Trans. Aero. and £lec. Sys., 29, pp. 685-695. Viswanadham, N. and R Sricha.nder (1987). Fault detection using unknown input observers. Control Theory and AdQ'llanced Technology, 3, pp. 91·101 , 1987. Yang, H . and M. Saif (1996). Observer design and fault diagnosis for retarded dynamical systems. ?roe. 35th IEEE Con] Decision and Control, Kobe. Japan, pp. 1149-1154.

and m l

('.."

'-..,

3

2 00

, .-- '. 10

/" 15

20

25

30

time!ncl ( VI

=I , v2 =0 ,V3 =1.s3 =1,sl = 1)

Fig. 1. FDI results for two simultaneous faults

5. CONCLUSIONS

This paper proposed an FDIO that accentuates the effect of the preselected fau1l on the residual. The basic concept of this proposition is the use

78