Fault Detection: A Quadratic Optimisation Approach

Fault Detection: A Quadratic Optimisation Approach Thomas S Brinsmead ? , J S Gibson , Graham C Goodwin , George H Lee , D Lewis Mingori y

z

yElec. Eng., Univ. of Newcastle, NSW 2308, Australia ? email [email protected] zMech. & Aero. Engr., UCLA, Calif., CA 90095-1597 USA

Abstract A fault detection lter is designed by minimising the response to confounding noises whilst keeping the fault response constant. A linear quadratic optimisation problem, related to H2 and H1 optimal control is used. The \Q" matrix of the associated Riccati equation is inde nite in contrast to the inde nite \R" matrix of H1 control. We extend linear optimal control theory to solve the fault detection problem in this framework. Key Phrases- Fault Detection, Linear Quadratic, Optimal Filtering, Riccati Equation, Inde nite Q.

1 Introduction

The area of fault detection and identi cation (FDI) has many potential practical applications. Fault detection (FD) studies whether a fault has occurred, whereas fault identi cation (FI) examines which fault has been detected. For sensor or actuator failure, a fault can be modelled as an unknown additive signal to an otherwise known system. In [6], a diagonalising post-compensator is used for FDI and it is suggested that optimisation might achieve a suitable trade-o between sensitivity to failures, and sensitivity to perturbations and measurement noise. However, constraining an FDI lter to be diagonal may compromise the performance with respect to other measures [4]. In [2], subspace techniques are used to isolate the various possible faults. Enhancement of fault signal transmission is not in the design criteria of the above methods and the eect of noisy measurements is not emphasised. In this paper, an FD lter is designed so that the fault response is large, but the response to non-fault signals including measurement noise, is small. Identi cation may be achieved by designing such a lter for each possible fault. An important issue is what assumptions about the various signals are permissible. In contrast to [2] we assume that the spectral characteristics of the fault and noise signals are known. Also in contrast to [6] and [2], fault input separability is not required. The design method results in the best achievable lter according to the design criteria.

y

z

z

We view the measured output y of a linear system as comprising of three components: yf , due to the fault; yd, due to disturbances; and ym due to measurement noise. For a good lter F , kFyf k is large but kFydk and kFyn k are small, for some norm. We maximise the ratio of the magnitude of the lter fault response to the magnitude of the lter response to other inputs, that is kFyf k : (kFydk + kFym k). This is equivalent to minimising kFydk+kFymk, with the constraint kFyf k = 1. We extend the result in [7] to a continuous time framework and investigate the lter properties as optimality is approached. Both nite and in nite time horizons are treated. Consistent results for the discrete time case are also shown. Both for clarity and for didactic reasons, we present the nite horizon case before the in nite horizon case.

2 Continuous Time 2.1 Problem De nition

The measurement signal y is generated by a stable multi-input, multi-output (m input and p output) linear dynamic system driven by (white) noise processes u and v and corrupted by (white) measurement noise w. The presence or absence of a fault is characterised by the presence or absence of the signal u. A minimal realisation of the system generating y is x_ = Ax + Bu + Ev; y = Cx + w; where x is the n  1 state vector, A is a stability matrix, A; B; E and C are time-varying, is a scalar boolean variable which equals 1 under fault conditions, and 0 otherwise, and w; u; v are white noise processes of appropriate dimension. Coloured noise can be modelled as being generated by white noise through an appropriate lter. The white noise process u has autocorrelation E fu(t)u(s) g = Ru (t ? s) with Rv and Rw de ned for v and w analogously, and Rw non-singular. The autocorrelation of the initial condition, E fx(0)x(0) g = W , is non-singular. It is assumed, without loss of generality, that u; v; w and x(0) are mutually uncorrelated.

2.2 Finite Time Filter

Consider nite time Z tf linear lters with scalar output, h(tf ?  )y( )d + l y(tf ): z (tf ) = 0

We aim to nd the impulse response which is optimal over the time [0; tf ] in an expected energy sense. Here,

h( ) is the strictly proper component and l is a feedthrough term. So that our lter responds to u (ie, when = 1), but not to v and w, we aim to maximise 1  E z 2j = 1 : (1) E fz 2j = 0g A convenient expression for (1), is aided by use of the the co-state variable , where _ = ?A  + C  g; (2) (tf ) = tf ;

= h(tf ?  ). and tf is an arbitrary constant and g( ) : def Without loss of generality, choose tf = ?lC . By considering the derivative dtd  (t)x(t) we see

z (tf ) = ?(0) x(0)+

tf

Z

0

For continuous time, we set l = 0, thus avoiding the

w from corrupting the output directly. The expectation of z 2 is found Zto be tf  E z (tf )2 =  (Q+ G)+g Rgdt+(0) W(0): 0

G  : def = BRu Bdef is the term due to u, Q : = ERv E , and R : = Rw . We see that maximising

defHere

(1) is equivalent to maximising Z Z

0

tf

0

 Gdt

 Q + g Rgdt + (0) W(0)

:

(3)

We call (3) informally, \the energy ratio" (ER), referring to the ratio of the extra expected energy in z due to u, to the expected energy in z due to [v w ] only.

2.2.1 Linear Quadratic Optimisation: We do not maximise (3) over g directly but use an approach standard in H1 theory by considering a certain class of cost functions de ned below. Z tf  (Q? ?2G)+g Rgdt+(0)W(0): J (g; tf ) = 0 (4) The g which maximises (3) corresponds to the largest

such that tf

Z

0

 Q + g Rgdt + (0) W(0)  ?2

tf

Z

0

 Gdt;

(5) holds for some non-zero denominator of (3). Hence, we consider nding the in mum, if it exists, of J (g; tf ) subject to the constraints (2), resulting in a class of optimisation problems, parametrised by , namely inf (6) g J (g; tf ): 1

E z2 = 1 is \the expectation of z2 , given = 1 " j

2.2.2 The Riccati Dierential Equation: Proposition 2.1 (Existence of Bounded In mum)

The value of (6) is bounded if the associated RDE (7) with initial condition P (0) = W has a unique solution over the interval [0; tf ]. The in mal value of the functional is inf J (g; tf ) = tf P (tf )tf . g P_ = P A + AP ? P C  R?1 CP + Q ? ?2 G: (7)

?( Bu+Ev)+gwd +l w(tf ): The proof of this is standard and follows the completion

l w(tf ) term thereby preventing the measurement noise

tf

The linearity of (2) in g and the quadratic relationship between g and J (g; tf ) together imply, that if and only if (6) is bounded (and positive), then (3) is less than or equal to 2 for all g. We therefore seek values of for which (6) is unbounded, and signals g which result in (5) an ER greater than 2 .

of squares argument from [1].

Corollary 2.1 (Existence of a Conjugate Point)

The converse is that (6) being unbounded is sucient for (7) to have a conjugate point on [0; tf ].

The continuous dependence of the DE (7) on the parameter implies that over any nite time interval, a solution exists for suciently large and hence (6) is bounded and infg J (g; 0) is zero. Any non-zero g falsi es (5) and so no lter response h exists with (3) greater than or equal to 2 . Conversely it is easy to show that for suciently small, (6) is unbounded and hence (7) has a conjugate point on the interval [0; tf ]. The following set is thus non-empty and bounded above. ? = f : the RDE (7) has a conjugate point on [0; tf ]g: Also ^ : def = fsup : 2 ?g is nite and non-negative. As in [1], ? achieves its supremum and the RDE for P ^ has a conjugate point at t = tf .

2.2.3 (Sub)Optimal Impulse Responses:

The following Caratheodory canonical equations [1], with X (0) = I and Y (0) = W , X_ = ?A X + C  R?1 CY ; Y_ = (Q ? ?2 G)X + AY : (8)

are related to the RDE (7) and the Hamiltonian [5] of the optimisation problem (6). A possible (sub)optimal impulse response is as follows. For any  ^ there exists a t? 2 [0; tf ] such that t? is a conjugate point and X (t? ) is singular [1]. Suppose  is any non-zero vector in the kernel of X (t? ). De ning  ?1 (t) for t 2 [0; t ] g (t) = R CY (9) 0 for t 2 (t ; tf ] gives a g(t) which satis es Z

0

tf

 (Q ? ?2 G) + g  Rg dt + (0) W(0) =  X (tf )Y (tf ) = 0:

Since both Y (0) = W and R are positive de nite, g is non-zero and the ratio (3) is well-de ned. The condition tf = 0 and hence constraint (5) are both satis ed. This can be used to show that ? is a connected set and contains points arbitrarily close to zero. For satisfying ^ 6= 2 ? the impulse response h (t) derived from g(t) as above is a suboptimal lter response, yielding an ER of 2 . Further, the h^ corresponding to ^ is an optimal lter response with ER equal to ^2.

2.2.4 Summary- Finite Horizon: The set ? of for which the RDE (7) has a conjugate point, is bounded above, contains its supremum ^, is connected and contains points arbitrarily close to zero. For any

 2= ?, the RDE (7) is de ned over [0; tf ] and no lter yields an ER greater than or equal to 2. For  2 ?, equation (9) yields a lter response, such that (5) holds with equality, and the ER is 2. For ^, the supremum (maximum) of ?, the RDE (7) has a conjugate point at tf and equation (9) yields an optimal lter response such that the ER is equal to ^2. 2.3 The Steady State Filter

Usually, the lter will be implemented in steady state so here we assume a time-invariant system. In the nite time case, the RDE has a conjugate point for the optimal lter, so it is not obvious how to de ne the in nite horizon optimal lter by taking the limit of the nite time results. We sidestep the issue by rst investigating the properties of the following Algebraic Riccati Equation (ARE) and the associated Hamiltonian matrix H .

P A + AP ? P C  R?1 CP + Q ? ?2G = 0: (10) 

  ?1 H = ?(Q ?A ?2 G) C ?RA C



We derive a particular stable lter for a given , which we term \the central lter" in deference to its analogue in H2 control, and parametrise all stable lters in terms of this central lter.

2.3.1 The Central Steady State Filter: A

dual state space form of the argument of (6) is _ = (A ? P (t)C  R?1 C ) + P (t)C  R?1y;  (0) = 0; z (tf ) = tf  (tf ):

Under certain conditions, the RDE converges to a stabilising steady state value (ie, A ? PC  R?1 C is stabilising) as tf ! 1. For the nite  time case, tf = 0 in order to have a bounded E z 2 . However, this results in a zero lter for the in nite time case. To avoid this, de ne a (suboptimal) central steady state lter,

as a function of , by replacing tf with an arbitrary output matrix F as _ = A + P C  R?1(y ? C ); z = F; (11) We can show that for suciently large, the ARE (10) has a stabilising solution by using its relationship to the corresponding Hamiltonian matrix [5], and the continuous dependence of eigenvalues on the matrix entries. The following can also be easily shown: Lemma 2.1 (All stable lters) The set of all stable non-zero lters may be expressed as the star product interconnection of the following FM with any strictly proper stable non-zero linear transfer function M (s). 3 2 A j K 0 f FM (s) : def = 4 0 j 0 I 5: ?C j I 0 where Af : def = A ? KC and K : def = P C  R?1 .

Theorem 2.1 If the ARE (10) has a stabilising solution then the expected energy in the response due to u is strictly less than 2 times that due to v and w for any stable linear non-zero lter, ie   E z 2j = 1  (1 + 2 )E z 2j = 0 : (12) Proof: By lemma 2.1, we can express the system

and lter as the star product interconnection of M (s) with 2 3 A j B E 0 0 G(s) = 4 0 j 0 0 0 I 5 ; C j 0 0 I 0 3 2 Af j K 0 FM (s) = 4 0 j 0 I 5 : C j I 0 By Parseval's  theorem, and because P is stabilising, then E z 2 is given by

Z

1



M C (j!I ? Af )?1 (Q + G + KRK )(?j!I ? Af )?1 C 

?1



?C (j!I ? Af )?1 KR ? RK (?j!I ? Af )?1 C  + R M  d!: We de ne Zu and Zvw as the expected energy in z due to u, and [v w ] respectively. Z

1

MC (j!I ? Af )?1 G(?j!I ? Af )?1 C  M  d!: Zu = ?1 ZZ vw = 1  M C (j!I ? Af )?1 (Q + KRK )(?j!I ? Af )?1 C  ?1



?C (j!I ? Af )?1 KR ? RK  (?j!I ? Af )?1 C  + R M  d!: 



so that E z 2 j = 1 = Zvw + Zu ;  E z 2 j = 0 = Zvw :

(13)

By rewriting the ARE (10) it can be shown that

Zvw ? ?2Zu =

Z

1

?1

M (j!)RM (?j!)d!:

Since R is positive de nite and M (s) is non-zero, Zvw ?

?2 Zu > 0 and so (12) follows easily from (13).

where Zu () and Zvw () have interpretation analogous to (13). By the fundamental theorem of calculus [3] for  suciently small,

 : def = Re (1 ) = Re (2 )  Re (i ); i 6= 1; 2

So for appropriately sized matrix X , Thus for suciently large, no lter achieves an ER 8 Z 1 x12 for i = 1; j = 2; of 2. This con icts with [6] and [2], where faults are <  1 for i = 2; j = 1; completely isolated into various subspaces giving a zero eD t XeD t dt = ? 2 : x212  xij  0 otherwise. Zvw . The apparent inconsistency is because we have a i +j non zero w and positive de nite R. In [2], R ! 0 in orAs  ! 0+ in the limit xij  2+ ! 0, and so der to isolate the faults and in [6], zero noise is assumed. i j Any non-zero lter has non-zero Zvw so by taking suf2 Z Z 1 1 ciently small the ER is achieved. Hence the ARE t A A t  ?2 e Xe dt ! ?2 V eD t U XUeDt V dt (10) has no stabilising solution for small . The fol?1 ?1 lowing set is therefore both non-empty and bounded = 2 Re [ u X u1 (v1 v1 )] 1 above. Z 1 ? : def = f : the ARE (10) has no strictly stabilising solutiong :  Also for constant X , ?2 eAt XeA t dt ! 0 and 0 Z 1 and ^ : def = fsup : > 0; 2 ?g is nite and  0. ^t At A ?2 e Xe dt ! 0 as  ! 0+. Hence 0 2.3.2 Approaching the Optimal Filter: By changing the sequence of logic in the proof of theorem ? Zv () ! Re [u1Gu1 (Fv1 v1 F  )]; 13.7 in [5] we can show that if (10) has no stabilis? Zuw () ! Re [u1(Q + KRK)u1 (Fv1 v1 F  )]: ing solution, the corresponding H has imaginary axis eigenvalues. The continuous dependence of eigenvalues The ARE can be written as on the entries of a matrix [3], and the fact that the imaginary axis is closed implies that ^ 2= ?. Also, as Af ( )P ( )+P ( )Af ( ) +K ( )RK ( )+Q? ?2G = 0:

! ^ from above, some eigenvalues of H approach the j! axis and Af ( ) becomes marginally stable. We show that the ER of the corresponding central lters Noting that Af (^ )P (^ ) has a pair of left eigenvectors u1 (for almost all F ) approaches ^2 . and u1 with corresponding eigenvalues j! and ?j!, by pre- and post-multiplying the above ARE with = ^, Theorem 2.2 (Properties of the Optimal Filter) by u1 and u1 , or by u1 and u1 , and summing we get Let P = P (^ + ) and assume that a limit P^ : def = lim !0+ P exists, so A^ : def = Af (^ ) is marginally Re [u1 (K (^ )RK (^ ) + Q) u1 ] = ^?2 Re [u1 Gu1 ] ^ stable. Assume A is diagonalisable and has a sinRe [u1 Gu1 ] 2 gle eigenvalue at 0, or two conjugate ones on the  + Q)u1 ] = ^ Re [ u ( K (^

) RK (^

) 1 ^ j! axis, so A^ = V^ D^ U^ where  ma D is a diagonal ^ u ! ^ 2 as  ! 0+ . v v : : : v , and trix of eigenvalues  , V = 1 2 n i It is now easy to see that ZZvw ^U = V ?1 =  u1 u2 : : : un  . Assume that all A : def = A ? K (^ + )C in a positive neighbourhood of 2.3.3 Summary: In nite Horizon: The set  = 0+ may be diagonalised as V D U , where V ! V^ , ? of

for which the ARE (10) has no stabilising solu^ ^ D = diag 1 2 : : : n ! D and U ! U tion, is bounded above, does not contain its supremum + as  ! 0 . Then, for any F such that Fv1 6= 0, the

^ , is connected and contains points arbitrarily close to ER of the corresponding central lters, Zv : Zuw ! ^. zero. For any  2= ?, the ARE has a stabilising soluProof: We give the proof for the slightly more comtion and it is not possible to construct a lter response plicated case where A^ has complex roots. The proof such that the ER is greater than or equal to 2 . For any for one root at 0 is similar. Note that

 2 ?, with strictly  < ^, it is possible to construct a Z 1   
? At  lter, by using the central steady state lter (11), with e ? eA t G eA t ? eA t dtF  : Zu () = F almost any output matrix F , and a choice of with Z0 1  

>

^ >

, suciently close to ^ such that the ER is ? At
Zvw () = F e ? eA t Q eA t ? eA t greater than or equal to 2. For ^, the supremum of ?, 0 this is not true, so the ARE (10) has a no stabilising  + eA t KRKeA t dtF  : solution and no stable lter has an ER greater than or equal to ^2 .

3 Discrete Time Formulation 3.1 Finite Impulse Response Filter

The discrete system is xk+1 = Aq xk + Bq uk + Eq vk ; yk = Cxk + wk : For simplicity Aq is assumed non-singular. We de ne  W : def = E fx(0)x(0) g, and E uk uj =
1 Ru kj with Rv and Rw de ned similarly. The lter output zK of a K + 1 tap nite impulse response lter is

zK =

KX ?1 i=0


hK ?i yi +
h0 yK ;

where hi is the lter impulse response. Also, h0 is a biproper component giving a further degree of freedom over the continuous time case. Again our goal is to maximise the discrete analogue of (1). The co-state equation is given by ?  k+1 = A? q k + A C
gk ; K +1 known : (14) def De ne gi = g(t) : = hK ?i = h(tf ? t) and write

zK = ?0 x0 +

K X i=0

?i+1 [ Bq ui + Eq vi ] +
gi wi ;

has ^ : def = fsup : 2 ?g nite and non-negative. For

= ^ the matrix R +
CPKq C  is positive strictly semide nite. We now solve for a lter response g. Let k? be the rst k to give R +
CPk C  not positive de nite. There exists a non-zero  2 Rm such that  (R +
CPk? C  )  0 and let 8 0 for k > k? ; < gk = :  for k = k? ;  ? 1  (R +
CPk C ) CPk A k+1 for k < k? : so J q (g; 0) =
 (R +
CPk C  )  0, and the ER (15) is de ned and less than or equal to 2 . Remark 3.1 For ^ 2 ?, the DRDE (17)q results in a singular positive semide nite R +
CPK C  . The corresponding h^ is an optimal nite lter response and  is the feedthrough term h0 = gK .

3.2 Discrete Time- In nite Horizon

The central in nite horizon discrete time lter is k+1 = Aq k +
Aq P q C (R +
CP q C  )?1 (yk ? Ck ); zk = Fk + h0 yk ;

Here P q is the maximal solution to the steady state  ARE given in (18) below, h0 is the lter feedE zK2 = 0 W0 +
i (Q + G)i +
gi Rgi ; Discrete through term, and F is chosen xed but arbitrary. i=0 i=1 (18) P q =
(Q ? ?2 G) + Aq P q Aq In the above, G =
12 Bq Ru Bq etc. The energy ratio is q   ? 1  q  ?Aq
P C (R +
CP C ) C P Aq K X

K X




0 W0 +


K X i=1

K X

i Gi K

X  Qi +
g Rgi

: (15)

i i i=0 i=1 The class of functionals J q is K K X X J q (g; K ) = 0 W0 +
i (Q? ?2G)i +
gi Rgi ; i=0 i=1

(16)

and the in misation problem is g ;iinf J q (g; K +1 ). i =0:::K

subject to (14). The recursive Discrete Riccati Dierence Equation (DRDE) is Pk+1 =
(Q ? ?2 G) + Aq Pk Aq (17)   ? 1  ?Aq
Pk C (R +
CPk C ) C Pk Aq ; The in misation problem is bounded with value

K +1 PK +1 K +1 if the DRDE (17) with initial condition P (0) = W results in R +
CPk C  positive de nite for all k 2 [0; : : : ; K ]. For suciently large the in misation is bounded and infg J q (g; 0) = 0. For small

The DARE (18) has a stabilising solution for suciently large and none for suciently small. Lemma 3.1 (All Stable Discrete Filters) The star product interconnection of the following FM (z ) and any stable linear non-zero transfer function M (z )+ h0 , where M (z ) is strictly proper and h0 is the lter feedthrough term, parametrises all stable lters. 2 q 3 A j
K q 0 f FM : def = 4 0 j 0 I 5: ?C j I 0 def q where Af : = Aq ?
Kq C and Kq : def = Aq P q C  (R +
CPC  )?1 . Theorem 3.1 If the DARE (18) has a stabilising solution then the expected lter energy due to u is strictly less than 2 times that due to [v w ] for any stable linear lter. Proof: De ning Zu and Zvw analogously to (13) we can show that 2
(Zvw ? ?2 Zu ) = (19) Z      M (ej! ) + h0 (R +
CPC  ) M  (e?j! ) + h0 d!

the in misation is unbounded. The set ? ? = f : the DRDE (17) results in R +
C  Pk C is non-negative. The result follows. For the central not positive de nite for some k 2 f0; : : :; K g g : steady state lter M (z ) = (F + h0 C )(zI ? A)?1
Kq and for a xed feed-through term, to minimise (19) we must set F = ?h0C .

Gain ratio− Mag v:Mag u & w

100

Gain dB

40

50 20

0 0 0

1

10 Frequency (rad/sec)

10

Gain/dB

−50 −1 10

−20

360

−60

0 −1 10

0

−80 −1 10

1

10 Frequency (rad/sec)

10

Figure 1: Frequency Response of Fc, =5.13 4 Example

Consider the following system with 3 2 2 ?15:2 1 0 0 6 54 0 1 0 77 66 x_ = 64 ? ?25 0 0 1 5 x + 4 ?50 0 0 0   y = 1 0 0 0 x+w Ru = Rv = Rw = 1

1

10

Central Filter Energy Ratio v. Gamma 36 34

gamma^2 Energy Ratio of F_c

32

3

2

10 7 77 66 11 5 u + 4 5

3

0:3 5:5 77 v 30 5 50

By numerical search, the optimal value of is found to be approximately ^  5:13. We look at the central steady state lters as approaches its optimal value from above. 4.0.1 Choosing the Output Matrix: In the limit, the output matrix F is inconsequential, however a secondary optimisation involving a generalised eigenvalue problem [7] results in a speci c choice of F and an \F-optimised central lter". Provided > ^, the energy distributiondefin the lter state vector,  , is vw : = E f  j = 0g and u : def = E f  j = 1g ? vw : If  is the smallest possible number such that the matrix vw ? ( ?2 +  ?2 )u is positive semide nite, an F which optimises the lter ER is ha null vector of the i?1

2 2 < 2. matrix. The F-optimal ER is 1 + (  ) Figure 3 shows that the ER of the F-optimised central lter approaches ^2 as approaches ^ from above. 4.0.2 F-Optimised Central Filter: Figure 1 shows the Bode Plot of the F-optimised central steady state lter for  ^  5:13. The lters become notch pass with notch frequency at approximately 1 rads?1 . Figure 2 shows that the ratio of the measurement spectral density due to u to that due to [v w ] , that (j!)kRu2 ; is maximised also at approxiis kT kT(yu yv j! )kRv2 + Rw2 mately ! = 1. Heuristically, the lter design results in a notch lter at the frequency which maximises the ER of the measurement y.

5 Conclusion

0

10 Frequency (rad/sec)

Figure 2: Energy Ratio in y:- u compared to [v w2]

Fault detection lter design is achieved using H1 -like methods to optimise a lter energy ratio criterion. The solution to the in nite horizon problem as posed results in a heuristically reasonable answer. Speci cally, the question- \What linear lter maximises the ratio of the

Energy Ratio, gamma^2

Phase deg

−40

180

30 28 26 24 22 20 18 16 4.5

5

5.5

6

gamma

Figure 3: ER of F-optimised Central Filter v. signal response to the noise response?" is answered by \A notch pass lter at a (single) frequency which maximises the ER of the measurement". This solves the problem as posed but is sensitive to the existence of fault components at the notch frequency. Nonetheless, the mathematical tools used in the derivation may be of independent interest since they extend H1 in novel directions.

References

[1] Tamar Basar and Pierre Bernhard. H 1 Optimal Control and Related Minimax Design Problems. Birkhauser, Boston, 1995. [2] Walter H. Chung and Jason L. Speyer. A game theoretic fault detection lter. Technical report, UCLA, 1996. Mech., Aerospace and Nuclear Eng. Dept. [3] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985. [4] G.C.Goodwin J.H. Braslavsky, M.M. Seron and R.W. Grainger. Tradeos in mulitivariable lter design with applications to fault detection. Technical report, Univ. Newcastle. Dept of Elect. Eng. [5] John C. Doyle Kemin Zhou and Keith Glover. Robust and Optimal Control. Prentice Hall, New Jersey, 1996. [6] M. Kinnaert and Y. Peng. Residual generator for sensor and actuator fault detection and isolation: a frequency domain approach. International Journal of Control, 61:1423{1435, 1995. [7] George Haotsung Lee. Least-squares and minimax methods for ltering, identi cation and detection. Technical report, UCLA, 1995. PhD Dissertation.