H∞ Low-Order Design for Fault Detection and Identification Problems

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H OO LOW-ORDER DESIGN FOR FAULT DETECTION AND IDENTIFICATION PROBLEMS A.M. Stoica t and M.J. Grimble: tUniversity "Politehnica" ofBucharest, Faculty of Aerospace Engineering, Str. Splaiullndependentei, no.3J3, Bucharest, Romania, e-mail: [email protected] ; University of Strathclyde, Industrial Control Centre, Graham Hills Building, 50 Oeorge Street, Glasgow 01 IQE, Scotland e-mail: [email protected]

Abstract: The aim of this paper is to present a design method for low-order solutions to a range of fault detection and identification problems. The theoretical results are developed in an H aD framework, using results from linear algebra. The design procedure mainly comprises two stages. In the first stage, the parameterised set of all fixed order solutions of the fault detection problem is obtained. In the second stage an optimisation problem is solved to determine the free parameter such that the disturbance attenuation and the fault identification requirements are accomplished. A numerical example demonstrates the effectiveness of the proposed method and a comparison of results illustrates the dependence of the performances on the order of the solution. Copyright © 2003 [FAC Keywords: Fault detection, fault identification, linear systems, H -infinity optimisation, algebraic approaches.

1. INTRODUCTION

Ding, 1997), (patton and Bou, 1999», eigenstructure assignment approaches (Choi, 1998), (patton and Chen, 20(0) and methods using the properties of some structural invariant subspaces associated with the model of the monitored plant (White and Speyer, 1987), (Chen and Speyer, 1999), (Saberi et. al., 2000), (Xiong and Saif, 2000). The H2 and HaD optimisation methods have been also used (see for instance, (Edelmayer et. ai, 1996), (Sauter et. al., 1997), (Douglas and Speyer, 1999), (patton and Bou, 1999) and their references). Necessary and sufficient solvability conditions have been derived in (White and Speyer, 1987), (patton and Cben, 2000), (Saberi el. al, 2000), in which fullorder solutions are determined. The main objective here is to present a design methodology to find low-order solutions to the FDI problem. Such a design is particularly useful in applications where the complexity of the plant monitored leads to high order solutions where implementation is difficult. The problem can ~ addressed using general order reduction techniques

The research on fault detection and identification (FDI) problem has prompted increasing attention during the last three decades. This interest is motivated by the continuous increase of plants complexity, which requires new capabilities to detect possible malfunctions and to initiate appropriate actions in emergency situations. The problem mainly consists in designing a system, often called in the control literature a residual generator, able to detect any failure of actuators and sensors of the monitored plant. The influence of disturbances must be considered in order to prevent false alarms. Additionally, the residual generator can satisfy fault identification requirements, to indicate which failure occurred. From the pioneering papers of (Beard, 1971), (Jones, 1973) (see also (Wilsky, 1976», a great variety of methods have been proposed to solve this problem. The following research areas can be highlighted: model-based methods (e.g. (patton and Chen, 1991), (Xiong and Saif, 1997), (Frank and

507

of the full-order solution but in this case a severe deterioration of the residual generator performances is possible. An alternative method to obtain loworder residual generators is described here. A general class of linear residual generators is considered, where only the order, and the number of outputs are fixed. It mainly uses the H'" framework but it is shown that in contrast with the H'" control problems the fault detection condition is not expressed as an H'" norm of a lower linear fractional transformation. The paper is organised as follows: in Section 2 the FDI considered problem is stated and equivalent optimisation conditions are derived. Section 3 includes the main result providing necessary and sufficient solvability conditions. Section 4 describes a numerical algorithm used in the design methodology of the low-order residual generator. An illustrative numerical example is presented in Section 5 and final conclusions are given in Section 6. 2. PROBLEM STATEMENT AND EQUIVALENT OPTIMISATION CONDITIONS Consider the linear time invariant system: = Ax+B.J +B2 d

x

where x E R It is the state variable, fER fault vector, d

E Rm~

m, is

the

denotes a disturbance signal

r =Ckxk +Dky

will be called H'" residual generator if the following conditions are fulfilled: • Stability: Ak is Hurwitz;

P > 0,

Gn( := KGyd'

f

and

b/';I Jis transformed to an optimisation problem on the infinite interval with D, left invertible (see also (Rambeaux et. al., 2000)). Using the assumption discussed above concerning the invertibility of DkDI , condition (3) can be rewritten as: 0'

.,

/l}

=

] [G -I U Cl) )] sUP_RO',/

>a ,

or equivalently, 1 ~" -1(;a>~'" <-. a

(5)

Then the design problem consists in determining a stable residual generator of form (2) with specified order such that conditions (4) and (5) are satisfied.

3. NECESSARY AND SUFFICIENT CONDITIONS

T

nA

+ XI AI

A; X;

d to r, respectively, ~O is the minimal singular

IOL denotes

the fault detection condition on the given interval

order residual generator of the

Bounded Real Lemma in a linear matrix inequality (LMI) form (e.g. (lwasaki and Skelton, 1994» it may be shown that (5) holds, if and only if there exists an (n + nk)x (n + nk) symmetric matrix XI > 0, such that

(4)

G" := KG yl are the transfer functions from value, and

lA)

form (2), with nk ~ I. Let (A;,B;,C;,D;) denote a

(3)

fGn((;a»L < P, a>

lA)

-I

Consider an

• Disturbance attenuation: given

(Oif{jlA)) =o (G>f{jlA)}). v WE[~/.;;;I] o(Oif (JIA)})~ inf S S;;;I !!.(G>f(Jw»). v IA)~[~/';;;I] . o

realization of the system G" -I . Then in virtue of the

• Fault detection: infGlER cio,,(;a»j> a ;

some

z(s) = Q(s )y(s) satisfy the conditions:

....R - ~(j)] 'J

vector and Y E RP represents the measured output vector. The system K with the state space model Xk = Akxk + Bky (2)

for

Remark 1. In applications in which D, is not left invertible, one can specify finite frequency ranges on which the fault detection condition is satisfied Introducing the filter Q(s), such that the output

inf

(1)

Y = Cx + D.J + D 2d

DiD, must be non-singular since otherwise, the fault detection condition (3) cannot be accomplished for high frequency fault signals.

T

B; XI

the H'" norm. A larger

ratio a / P > 1 provides higher sensitivity to failures in the presence of disturbances. As mentioned in the introduction, the attention will be focused here on the design of reduced order residual generators. Due to the very general structure (2), throughout the paper the monitored plant (1) is assumed stable. An additional design objective is the fault identification requirement. If the case when all failures can simultaneously occur at any time (failures of "type I"), then it is necessary to consider a residual vector r with the same dimension as the fault vector. This implies that DkD, is a square matrix. Moreover,

Cl

XIB; 1

--I

a "', DI

C/ <0.

D/ 1

-a I..,

Partitioning XI as: XI

R M] =[MT R > 0,

RE R

Itxlt

(6)

the inequality above can be written in the equivalent form:

Qt

ZI + J{ OQI + OT ~ < 0 , where the following notation has been used:

508

(7)

-CTDl+T ArM+ CTR RB,. +MB,Pl Ar R+JU +CTMT +MC T 0 M BI+RBkDI 0 MTA+ RC (8) T Ilffl ZI= BtTM +D/B /R -..!..(DtD.Y(DkDt) BtTR+ D/B/M a 0

-DI+C

a "'1

0

R

M'

[

I --I

1"'1

R 0 I M+D. TBk T-) - ~T

~ = -~/R+DtTB/MT)

-: 1 "'1

Remark 1. From condition (J 3) it follows that IDkDtI>a . A :=A-B. DI+C

Since R and S are the left upper corners of Xi and X,-I. respectively, the following additional conditions must be included :

r

C:= Bk~ -DIDI + - t := (DkD. )-1 Ck. C

[~ ~ ]~O

(14)

ran{[~ ~]) ~ n+nt.

(IS)

(-)+

denoting the pseudo-inverse of (.) . Without loosing the generality of the problem one can choose:

l[ P

Bk =

I 0

]

llll

if

nk

0] if

~ P. or nk

and

(9)

< p.

According to Iwasaki and Skelton, (1994), if ~~T >0 and QIWp1Wp/ QIT > 0, then the set of all solutions Cl of (7) has the parameterisation:

Indeed. if Bk is full rank then there exists a nonsingular transformation T (given by the singular value decomposition) such that TBk has the form (9). If Bt is not full rank, a small enough perturbation can be found to make it full rank and at the same time preserve the condition (7). Based on the socalled Projection Lemma (e.g. (Iwasaki and Skelton, 1994», it follows that there exists a matrix Cl verifying (7) if and only if

w",TZIW",
(10)

W01TZIW01 < 0

(11)

(16) wherecI>I,cI>2,cI>3 depend on PI,QI and Z, (see formulae (23) in (Iwasaki and Skelton, 1994» and < 1. L e R (Ill +",I ~'"I is a free parameter with

ILl

The condition PIPt > 0 can be checked directly. The second condition is satisfied for full column rank matrix solutions M (which always exist). Further, the disturbance attenuation condition (4) is fulfilled, if and only if there exists an {n + nk)x (n + nt) symmetric matrix X > 0, such that:

where W", and W01 are any bases of the null spaces of ~ and QI' respectively. Considering the partition : Xi-I

=[;T

;]>0, SeR"

K ",

direct algebraic computations show that conditions (10) and (11) are equivalent with: T (12) AS+SA -rIBIB. T <0 and ATR+JU +CTM T +MC

RB) +MBj;D)

BTR+D TB TMT

-; (ol DI (olD))

I

I

j;

r

[

-CTD +T I

<0

I

"'/

XBR

A./X;X AR BR X

-fR",~

CR

DR

CR~ DR

1

- fR"'l

where (A.R,BR.CR,D R) denotes a realisation of the system Grd = KGyd. With Cl defined by (8) and with its parameterisation (16), one can show that condition (17) can be rewritten in the equivalent form as: "N 12 "N13 T [ "NNil "N "N23 n 12 <0 (18) T "N(X,L)= "N / "N23 I - fJI "'~ T "N 44 "NI.' "N 24 T "N 34

N"] ~:

-~l a "'/

(13)

respectively.

with 509

<0 (17)

:Nil (X) = AT XI +XIA+CTB/ X/ +X2B.. C :N 12 (X,L)= AT X 2 +CTB/ X3 +X2(ll + 2I L3) :N 13 (X)= X I B2 +X2BkD2 :NI. = C TDI +T :N 22 (X,L)= X 3(1I + 2I L 3) + (ll + 2I LS' X3 :N n (X) = X/ B2 + X3B.. D2 :N 24 (L)= <1>1/ +<1>/ LT22T

In the following the attention will be focused on solving the system (18), (20)-(22) which includes non-linearities in the components :N'12(X,L) and :N'22 (X, L) defined by (19). These elements are multi-linear functions of the variables X and L, that is, if one of these variables is fixed then an affine dependence with respect to the other one is obtained.

:N34 = D/ DI+T :N44 = _/JIl-I, (19)

(20) where XI ER""", /j,i,j =1,2 is given by the

To accomplish the fault identification objective, the following two additional assumptions have been considered: the first is DkDI = 9"'1 which implies

partition:

[~ ~] =[<1>11 <1>12 ~I

~2

~

<1>21]

~' ~1I

ER'"

K",

,

Dk =ODI+for 0>0 such that p>0IDI+D211 (see

'6"22

and

Remark 3). The second isl&"'l -KGyfL

l·

I~DeDI Y(vI +D2

Theorem 1. Given a > P > O. assertions are equivalent:

where i is an (n+nk)x(n+nk) symmetric and positive definite matrix. The expression of , (x, L, will be not given here but it may be noted

p)

the follOwing

that its elements are also multi-linear functions of and L.

(i) It exists an n.. order stable residual generator K satisfying the conditions (3) and (4); (ii) The system (6), (/2)-(15) is feasible and there the

X 3 ER'"

symmetric

K", , 6. ER" 1 "". f

matrices and the

matrices X 2 E R ")(", ,L E R~' +'"f conditions (/8),(20), (21) and

).'"

In order to solve the system of matrix inequalities (18), (24) with respect to X> 0, x> 0 and L subject to the constraint (22), the following algorithm based on the analytic centre has been used:

XI ER""", , rectangular verifying the

Step I Find an initial feasible point Xo > 0 for (J 8). Step 2 Solve the LMI (18) with respect to L. Set

L ]>0. (22) L I", If the conditions above are satisfied then the matrices in the realisation of K are given by:

~} ~I + ~2L~3 )

1.0 =L. Step 3 Solve the LMI

(23)

Dk =UDI +, where U is defined by the Cholesky factorisation A

=UTU

X

4. A NUMERICAL ALGORITHM

[1"'~"'1

[ ~: ] =[I~.

for

(2 ,

The above developments are summarised in the following theorem:

exist


p > 0, small enough. This last condition has been treated in a similar manner to condition (4) and it leads to a matrix inequality of form !P L,p )< 0 (24)

Remark 3. The right lower 2 x 2 block in inequality (18) gives the necessary solvability condition as: p>

(Iwasaki and Skelton, (1994). Then the system (18), (20)-(22) is solved with respect to XI' X 2 , X 3 , A and L. A realisation of K is obtained introducing L into (23). Although the system (6), (12)-(15) includes a rank constraint, it can be treated using either one of the methods developed for rank minimisation problems (e.g. (Skelton et. al., 1998) and its references) or easier, exploiting the particular structure of conditions (12) and (13).

X for an xo=X.

arbitrarily Po > 0 large enough and set

Step

Make

'110

and B.. is given by (9).0

4

the

initialisations

k

'¥k+1

= A.IL/ Lel + (1- A.f¥k .

Step 6 Determine the analytic center (see e.g. (Skelton et. al., 1998)} L of:

510

=0 and

> ~Lo T Lo~ and choose a parameter A. E (0,1).

Step 5 Compute

The above result suggests the following procedure to find an nl -order residual generator. First solve the system (6), (12)-(15) whose solution allows cJ>1,cJ>2,cJ>3 to be computed of the parameterisation (16), using the formulae (23), in

'(i,Lo,p)< 0 with respect to

[

'Phi I., ••,

L

o

LT

I..

0

o

0

o

0

-X(X1 .L) 0

above state-space representation are defined as follows (Douglas and Spe)U, 1999): -0.0674

0.0430

0.0205

-1.4666

16.5800

0

0

Set Lt+1 =L. Step 7 Update Pt by Pt+1 = -Vi + (1- A.).ut , where fit denotes the smallest value of P for which

A=

:P(Xt ,Lt +1,p)s O. Step 8 Solve the LMI systemX(Xk+1,Ll + I )< 0,

0 0 H.= 0 0

:P(.Xt +1,Lt +I ,Pk+1)< 0

0.0430

-0.0299 -1.4666 -1.6788 , -1.6788 -0.6819 0 0 0 0

0.1377 0 0

0

-1.1948

-0.1672 -1.5172 • H, = -9.7842, 0 1.57 0

with respect to Xt+1 and

Xk+1 • respectively.

Step 9 If IL1+111 < I and Phi - Pt < e , where E> 0 is small enough, then STOP. Otherwise set k +-- k+ I and return to Step 5.

c- [ -

The initial feasible point X 0 is determined as follows: the condition (18) is written in the equivalent form Z + p1'LQ + Q1' L1' P < 0 and then, applying the Projection Lemma, necessary and sufficient conditions f
0

0 0.0591

0 0 1.0517 0.1495 -0.0299 -0.0677 0.0431 0.0171 0

0 0.0139

O.O~911

and, corresponding to representation (I),

~ o~l~ =B.,~ =[~ f} ~ =[~l =[B,

IILI

Note that ifDI above is not full rank the mult detection conditioo (3) caunot be accomplished for high freqtJ((llcy signals since lim_$!{G,,(jOJ})==O. In this situation GJ(must

be scaled according to Remark 1. Thus one obtains a scaled system G~ whose singular values satisfy the

Theorem 2. If then? exists a" nt order solutio" of lhe FD! problem satisfying (4), (5) and (24), tlren the computations peT/ormed at each step of the above centring algorithm are feasible at each step k ~ O. MOn?over, tire sequences {If't }t~O and {ut }l~O are strictly decreasing.

conditioos of Remark 1 for

~I

= 0.2 rad / sec and

DJI =IOOrad /sec . FllI'Ihff, based 00 Ibeorem 1, the FDI problem has been solved for different imposed orders of the r~dual gener"clWr. Some of the numerical results are preswted in Table 1.

5. SIMULATION RESULTS In order to illustrate the the
-0.8886 -0.5587

Tablet Q>mnal'l!!ive nlDIlerical results fer full::(!'4er generators n~ G ( I OJ id.~fl-·

._an
n.

6

5 3

7 6

A

x= Ax+B.OJ+B~ y=Cx,

U(.II"'/- "v

t

~.(T[G",(jdl)l

Cf."680"1 -·-' _.. _. - '-' 6:8845- ._.- ..

_ _9;...;.·~1!.7_._._.. _..___ ~_.1_87...;5"--_

The time responses of the residuals correspooding to individual and simultaru:oos wlures are plotted in Fig. I-Fig. 6 (1j -solid, r2 -dashed). In these simulations tmit mult signals, occurring at t =3 sec have been considered. These time responses reveal that the mult detection requirements are accomplished in both cases nt =5 and =3 .

where the :,1ate va;tor x E R 5 includes the longitudinal body axis velocity u , the nmnal body axis velocity w , the pitch rate q, the pitch angle (J and the first-order Dryden wind gust wg • The control variable is the eleven deflection angle 6 and OJ is the white noise input with tmit spectral density. The elements of the measured outpu18 wctor y are: the pitch rate q , the angle of attack a, the nmnal acceleratim az and the lmgitudinal accelerarim az . The two faults considered are the elevon and the normal acceleration accelerometer fiUlur~, respectivel y, therefore mI = 2 . Ai altitude

"t

050

1

2)'~6'e9"l

Fig.I. Residuals respowes:

h = 10,000 ft and Mach 0.9, the matrices in the

511

fi. occurs, nt = 5 .

:

,..-!'----;--:

i

Proceedings of the 3lf CDC, Phoenix, Arizona. USA. Oloi I.W. (1998). A simultaneous assigtUIJeIlt methodology of right/left eigenstructures. IEEE Transactions on Aerospace and Electronic Systems, 34(2), 625-634. Douglas R.K.. and I.L. Speyer (1999). H .BoWlded Fault Detection Filter. Journal of GuIdance, Control and Dyruzmics,ll(l), 129-139. Edelmayer A, I. Bokor and L. Keviczky (1996). H. detection filter for linear systems: comparison of two approaches. Proceedings of the lfA IFAC World Congress, Vol. N, 37-42. Frank P.M and X. Oing (1997). Survey of robust residual generation and evaluation methods in observer-based fault detection systems. Journal ofProcess Control, 7,403-424. Iwasaki, T. and RE. Skelton (1994). All controllers for the general H _ control problem: LMl existence conditions and state space formulas . Automattca. 30, 1307-1317. Jones H.L. (1973). Failure dettx.1ion in linear systems. Ph.D.thests. Department ofA.eronautlcs and Astronaulics, M1T, Cambridge, MA. Patten RI. and 1. Cllen (1991). Parity space approach to model-based fault diagnosis-A tutorial survey and some new results. Proceedings of IFAClIMACS Symposium of SAFEPROCESS'91, Baden-Baden. Patten RJ. and M Hou (1999). H.estimation and robust fault detection. Proceedings of European Control Coriference. Karlsruhe. Germany. Patten RI. and 1. Olen (2000). On eigenstructure assignment for robust fault diagnosis. International Journal of Robust and Nomlnear Conlrol,10,1193-1208. Rambeaux F., F. Hame1in and D. Sauter (2000). Optimal thresholding foc robust fault detection of uncertain systems. International Journal of Robust and Nonlmear Control. 10, 1155-1173. Saberi A, A. Stoarvogel, P. Sannuti P. and H. Niemann (2000). Ftmdamental problems in fault detection and identification. International Journal of Robust and Nonlmear Control, 10, 1209-1236. SauttJ' D., F. Rambaux and F. Hamelin (1997). Robust fault diagnosis in an H. setting. Proceedings ofSAFAPROCESS'97, 879-884. Skelton RE., T. Iwasaki and K.. cmgcriadis (1998). A Unified Algebraic Approoch to LInear Control Design. Tayloc & Fraocis. White lE. and lL. 8pe)U' (1987). Detection Filter Design: SpectralThe«y and Algtrithms. IEEETransactions on Autom. Control, 31(7), 583-603. Willsky AS. (1976). A survey of design methods for failure detection in dynamic ~tems. Automatlca, 11.601-611. Zad S.H and MA Massoumnia (1999). Generic IIOlvclbility of the f.Wun: detection and identification problem. Automatica, 35, 887893.

- !-- ;-~-

,• :•.·..' r Fl····:··Lr].·.~ .20~~'-+,~)-----',~-!-,---!-.---+7--,;'--9'-----.110

Fig.2. Residuals responses: f2 occurs ,nk: = 5 . 'OOI r--,--.,---,----.---r--.----~~

~ · · J· ! rJ;~H~+T:-

·:o~o---!-,~2~3--'~~5~6~'---'S'--9.l.-J,O

Fig.3. Residuals responses:

fi and fi occur, nk =5 .

.·:::r::::r:::·T.J.·. -j- . ·C:+ . . .l. . +. . ,t---;---;--J.·~·t·~·t·····l· ~·+~+-····l·-:·· .2'O~~'~2~3~.L-~5~i~,--.L-L9~'D

Fig.4. Residuals responses:

fi occurs, nk =3.

,·····;···;· ···:····f····· I······;·····L.) ... ) .....

: :::"[::::::l:::::: -~o

1

: ~t=t:I:::E: El:: :

]'5618"0

Fig.5. Residuals responses:

.

i(j;4:j~EH~ +. .+. .. L...

6 .. .

'1---";-'-"--' .... -2 0

fi occurs ,nk = 3.

t

-j-.....;.......

--!-....

2'''6&18''(l

Fig.6. Residuals responses: fi and f2 occur, nk = 3. The fault identification perfoonances are better in the full erdel' case which fact is also confirmed by the numerical results in the third column of Table 1. 6. CONCLUSIONS

The paper Jresented a tractable method for generating low-order solutims of FDI problems. A major benefit of the method is the possibility of deriving new results, corresponding to additional design objectives, often handled via H· tedmiques. Other versions, based on the ideas presented above, can be achieved from discrete-time and stochastic modelling of the monitored plant, for \\bleb Bounded Real Lemma type results are available. REFERENCES Beard, R.u. (1971). Failure accommodatien in

linear-systems through selfreorganizatien. Ph.D. Thesis, Department of Aeronautics and Astronaulics, MlT, Cambridge, MA. Chm, R.H. and J.L. Speyer (1999). Optimal Stochastic Multiple-Fault Detection Filter.

512