Residual Generator Design for Singular Bilinear Systems Subjected to Unmeasurable Disturbances: An LMI Approach

Copyright © IFAC Fault Detection, Supervision and Safety for Technical Processes, Kingston Upon Hull, UK, 1997

RESIDUAL GENERATOR DESIGN FOR SINGULAR BILINEAR SYSTEMS SUBJECTED TO UNMEASURABLE DISTURBANCES: AN LMI APPROACH E. Magarotto, M. Zasadzinski, H. Rafaralahy and M. Darouach

CRAN - CNRS URA 821, 1. U. T de Longwy - UHP - Nancy I 186, rue de Lorraine, 54400 Cosnes et Romain, FRANCE ~ : (0033) 3 82 25 91 13, e-mail: [email protected]

Abstract: In this paper, a method to design a bank of full-order observer-based fault detection and isolation is investigated. The plant model is assumed to be a bounded control inputs singular bilinear systems subjected to unknown disturbances. The design of the kth residual observer of the bank is divided into two parts. The first one consists in solving some algebraic (equality) constraints such that the kth residual is decoupled from the unknown inputs and the kth failure component. The second part consists in transforming the residual generator design into a dual robust stabilisation problem . Then the solution of an inequality constraint to ensure the exponential stability of the residual is obtained using a Linear Matrix Inequality (LMI) approach. Copyright © 1998 IFAC

Keywords: Singular bilinear system, Residual generator, Exponential stability, Unknown inputs, Linear matrix inequality.

1. INTRODUCTION

by (Yu and Shields, 1995) . Considerable attention has been focused on the state observation problems of singular linear systems. Various approaches, as singular value decomposition or extensions of Luenberger-like observers to singular systems have been proposed in the literature. Observer design for singular non-linear systems have been studied by (Kaprelian and Turi, 1992) using an extended linearisation technique. (Boutayeb et al., 1994) generalised those results to the case where the matrix EO is rectangular. Unfortunately, few efforts have been made to develop a theory of observers for singular bilinear systems. (Rafaralahy et al., 1996) proposed, for singular bilinear systems, an observer without using an extended linearisation technique and the observation error stability is obtained by satisfying a constraint on the pole placement of the observer linear part.

Bilinear systems are extensively used for the modeling of industrial processes such as chemical, biomedical or thermal plants and are used to approximate accurately a large class of nonlinear systems when linear ones are inadequate. The theory of observer-based fault detection for nonlinear systems has received considerable attention during the last decade. In survey papers, (Frank, 1993) and (Patton, 1994) considered the residual observer for nonlinear systems using a first order approximation, then the stability of the observation error is local. For bilinear systems, (Yu and Shields, 1995) extended the observer proposed by (Hara and Furuta, 1976) to the residual observer of bilinear systems. The parity space approach for bilinear systems is also considered by (Yu and Shields, 1995). (Kinnaert et al., 1995) treated the failure detection and isolation problem for bilinear systems using the regularly persistent observers proposed by (Bornard et al., 1988); this approach is less restrictive than those proposed

This paper is devoted to the design of a bank of residual generators for bounded control inputs singular bilinear systems subjected to unmeasurable

789

disturbances, without condition on matrix E' . The measurements and the algebraic part of the generalised state equation are bilinear in the control signals . This algebraic part and the measurement equation are decomposed according to the failures, the unknown inputs and the bilinearities. The kth exponential full-order residual generator of the bank is designed in two steps. First, equality constraints are solved to decouple the residual from the unknown disturbances and the kth failure mode. Second, this residual observer problem is converted into a robust stabilisation problem with structured uncertainties , then an LMI approach is used to solve a bilinear Lyapunov inequality. This approach avoids the conservatism due to the use of a pole placement constraint like in (Rafaralahy et al. , 1996) . The design procedure of the residual generator is given . As in (Kinnaert et al. , 1995) , bilinearities in the failure modes can be included in this approach .

Then system (1) is restricted system equivalent to m

i=l

m

(3.2) i=l

where

(1.2)

xn

and rank(E)

+L

uiC~x

+ Dn w

(5.2)

with VTy=[~~], VTCi=[~i] and VTD2 =[g~~]. Second , Y2 is decomposed in two parts : Y21 is affected by the unknown inputs while yn is free of unknown inputs. There exists nonsingular matrices VI and V2 such that vt D22 V2 = [b g] and vyt =!. Then equation (5 .2) becomes Tn

Y21 =cg1x

+L

uiC~IX + WI

(6 .1)

i=1

m

Y22=Cg2X+ LuiC~2X

(6 .2)

i=1

with VtY2=[~;;J, VTC2=[~t] and [:::~]=V;-I . Since yn contains a bilinear part, we need a new row compression on matrices C:22 to obtain a linear measurement Y221 . There exists a non-singular matrix V3 such that \t3 vl = I and V;T [ e ~2 .,. e 2'2] = C~2] where [e~n ' e2'22] is of full row rank. Then equation (6 .2) becomes (7.1 ) Y221 =C~2IX

[et,':

[1:] ,

,

(4 .2)

i:l

Now, since rank( E") = rI , there exists a nonsingular matrix P used to extract the algebraic part of equation (1.1) which can be reported into the measurement equation (1.2), i.e. PE" = [~] , PA'" = PE" = [~], PDj = [~:] and r

m

m

Y2=Cgx

The scalars fk (k = 1, ... , s) are arbitrary unknown functions of time. When no failure is present, these functions are all equal to zero. \Vhen the kth failure mode occurs , fk becomes non-zero . fk can represent both actuator failures or components failures.

E IR

an d

i=l

In this paper , we consider bilinear systems with bounded control inputs , i.e. u E n c IR m , where n = {UjU;"in ~ U' ~ u;" .. x for i = 1, .. . , m} (2) This assumption is generally satisfied for physical processes.

[~:] with E

1 ]

Now the measurement vector y is decomposed into four parts. The first one called YI is affected by failures , unknown inputs and bilinearities. The second one called Y21 is affected by unknown inputs and bilinearities , but is free of failure . The third one called ynl is free of failure , unknown input and bilinearity. The last one called Y222 is affected by bilinearities , but is free of failure and unknown input. First , in order to obtain a free of failure measurement equation , we make a row compression on matrix F2 : there exists a non-singular matrix V such that VVT = I and VT F2 = [F5 ' ] where F21 is of full row rank. Then equation (3.2) becomes m i YI=C?X+ Lu Cix+D2IW+F2tf (5.1)

where the state vector x E IRn , the control input vector tI E IR m with UT = [u 1 (tl ' um(tl], the unknown input vector w E IRq , the fault vector f E IR' with fT = [f1(t l' f'(tl] and the measured output y' E IW . E",A",E",C" , D;,D; , FI' and F2' are constant matrices where E" E IR rxn and rank(EO) = rl ~ min (r, n). Without loss of generality, matrices [~~] and [~~] are of full column rank .

=

= [iJD;

where z E IR and Bk E IR n (Bk is the kth residual vector) .

i;::;l

PFt

D2

;=1 n

m 2

'

m

+ BOu + D;w + Ft! (1.1)

·C·· =Co. X + '" LU X + DO2W + F'f

[Ai] C l.

Bk=Moz+ LuiMiz+Hy

i= 1

y°

=

m

m

u'A"x

C.

'

For the system (3) (or (1)) , the kth residual generator is described as follows

We consider the following time-invariant singular bilinear system described by

+L

= [-BU] y.

F2=[:;] .

2. PROBLEM FORMULATION

E·:i:=Ao·x

y

m

Y222=cg22 X +

= rl·

L U'C~22X i=1

790

(7 .2)

Wl'thVT 3 Y22

= [!I"') y;;~ an dVTCO 3 22 =

[C~",] Cg~2

(iii) for all w , Xo and zo , if there exists a failure mode Ii -:P 0 for j = 1, ... , s with j -:P k, then (h depends on

.

n.

Defining Ai = A' - Dll C21 with D, V2 = [Dll D'2) and inserting equation (6 .1) into (3.1), the system (3) is equivalen,f;, to Ex = A~x + L u'Aix + Bu + D ll Y21

Before to describe the residual generator design , note that the kth residual generator (4) (with k = 1, . . . ,5) can be expressed in terms of notations using in system (8) as follows i = G(u)z + Lg 1Y21 + Lg 21 Y221 + Lg n Y222 + lu

;=1

m

Yl =C?x+Lu' C;x+D21 W +F2Ii

m

(8 .2)

+L

;=1 m

U'(L21Y21

+ L221 Y221 + L222 Ym) (10 .1)

;=1

(8.3)

Ok =M(u)z

;=1

+ H221 Y221 + H222 Y222

( 10.2)

m

(8.4) m

O Y222 = C222 X+ ~ ~ U 'C'222 X

;=1

(8.5)

The correspondence between the notations used in (4) and (10) is given by

;=1

where the full column rank matrix F'2 and the fault vector h are given by [0 F12) = F, U and [~~] = UT I with UU T = I . h is obtained from the column compression on matrix F, and is the part of the fault vector I which can be detected and isolated. Let 5 be the size of h (F'2 E JR r, with rank(F12 ) = 5 ::; s) . Let nand Ft2 be the ph component of vector h and the ph column of matrix F'2 respectively.

.

In this section , the proposed approach for the kth residual generator design is based on Lyapunov stability of the error dynamics associated to the generator. This residual generator has an exponential decay rate and sufficient conditions for its existence are given . 'When no specify, the superscript i stands for i = 0, . .. ,m.

(9)

;=1

where Nt2 and F~2 are a constant real matrix and vector respectively which are of appropriate dimensions . In the sequel , if Nt2 -:p 0 then the vector Ft2 is replaced by the matrix [N{2 F~2), and the scalar

15

is replaced by the vector

(11.2)

H

3. DESIGN OF THE J(TH RESIDUAL GENERATOR

vector h may be bilinear (Kinnaert et al., 1995) , i.e. the term F'2h in equation (8 .1) can be rewritten as

•

(ILl)

where Lh = [L;" L;n) and H22 = [H'2' Hn2).

Remark 1. Without loss of generality, the fault

= L(Nt2 X + F~2) n

. [1 0S'] ] [0 L~, L~2) H = V [~ v, [l S,]] [0 0 n )

L' = V 0 v, [b

x.

F'2h

;=1

In order to satisfy condition (ii) , the eh component of the vector h is considered as an unknown input in the kth residual generator design .

["'45] .

For the kth residual generator , define the error vector e as e=z-TEx (12) where T E JR nxr ,. If L~, = 0 and L222 = 0 (for i = 1, ... , m), then the error vector (12) has the following dynamics e = G(u)e+(CoTE - TA~ + Lg 21 C~21 + Lg22C~22)X

The problem of failure detection and isolation can be treated by using a bank of residual generators (Frank, 1993 ; Patton , 1994) , each of these residuals are sensitive to some failure modes and insensitive to the others. To take a decision in a diagnostic procedure, the minimal required number of residuals is equal to the size of the failure vector h , this can be done if each residual (h is insensitive to one failure mode I; and sensitive to others failure modes with j = 1, .. . ,s and j -:P k . Thus the minimal number of residual generators in the bank is equal to s. Then the problem of the generation of residual (h (for k = 1, .. . ,5) can be stated as : find matrices C', L', l, H and M' such that the three following conditions hold

m

+L

u' (G'T E - T Ai

+ L221 C~21 + Lgn

C~22)X

;=1

+ (l - TB)u

n

+ (Lg ,

-T[D12 Ft2]

- TDll )Y21

[wIZ] - tTFt2fi 2

(13)

)=1 )~k

Note that putting L~, = 0 and L222 = 0 (for i = 1, . . . ,m) avoids to have products uiu) in the dynamic equation (13) . Inserting equation (12) into equation (10.2) , the kth residual is given by

(i) for all tv , Xo and '::0 , if 15 = 0 with j = 1, . .. , 5, then Ok decays exponentially to zero, (ii) for all tv , Xo and '::0 . if If -:p 0 and fi = 0 for j = L . . . , 5 with j -:P k, then Ok decays exponentially to zero,

Ok = M(u)e

+ (MOTE + H221C~21 + H222C~22)X m

,=1

The following assumption is made.

791

in Theorem 2 using an approach based on a linear matrix inequality.

Assumption (A 1) For all admissible inputs u EO , the vector fields TFr2 are not (C(u), M(u))unobservable , with j = 1, .. . , sand j '# k . If Nf2 '# 0 , then TFr2 is replaced by T(Nf2x + Fi2) '

Using (13) and (14), the solution of the residual generator problem for system (8) (or (1» is given by the following theorem .

3.1 Unknown inputs and failure mode decoupling

Since matrix E is of full row rank , the equation [D12 Flk2] = ~ (25) has always a solution given by cp = Et [D12 Ft2] (26) where Et is any generalised inverse satisfying EEt E = E . There exists a non-singular matrix V such that (27) cpU = ['" where cp is a full column rank matrix. Then equation (16) is equivalent to TE", = 0 (28)

Theorem 2. Put L21 = 0 and L222 = 0 (for i = ,m). The residual generator (10) (or (4» verifies condi tions (i), (ii) and (iii) if there exist matrices C', Lg 1, L~21 ' Lg 22 , J , Mi, H221 , H222 , T and Q = QT > 0 satisfying the following constraints for all admissible inputs u E 0 C'TE - T.4~ + L;21 C~21 + Lg 22 C;22 = 0 (15) 1, ...

T[D12Ft2]=0 J

0]

(16)

= TB

(17)

Lg 1 = TD ll (18) MOTE + H221C~21 + H222C~22 = 0 (19) M'TE + H 222 C;22 = 0 i = 1, ... , m (20) CT(u )Q+QC(u) < 0 (21) and (AI) holds.

Let <.pc be the orthogonal complement of <.p . Since matrix [<.p <.pc] is non-singular, then equations (19) and (20) can be equivalently rewritten as

PROOF. If L21 = 0 and Lb = 0 (for i = 1, .. . , m) , and if constraints (15)-(18) are verified , the error dynamics (13) becomes C(u)e - LTFr2f£

o

=

. .. °

[

(22)

)=1

° ...

H222

.,.

. . ..

0 (29)

--0-] 0

. .

.

.'. 0

.. ·

H222

Inserting (28) into (29) yields

)~k

Let V(e) = eTQe be a candidate Lyapunov function with Q = QT > o. The time derivative of this Lyapunov function along the dynamics (22) with f£ == 0 (j '# k) , can be expressed as . ( V( e, u) = e T (C T (u)Q + QC(u))e 23 ) If constraint (21)· holds , then the error dynamics (22) is quadratically stable . This error dynamics is exponential since each u' is bounded (see (2». If the constraints (19) and (20) are satisfied , the kth residual (14) can be written as 8k = M(u)e (24) Then conditions (i) and (ii) are satisfied. Assumption (AI) ensures that the contribution of the non-zero failures (with j '# k) on the residual 8 k is different from zero. Then , the residual 8k obtained from (22) and (24) fulfills condition (iii) if Assumption (AI) holds. 0

[Hnl Hm]6=0 where -

C

= [C~21 ° <.p C222

(30.1)

0 ... 0] 1

(30 .2)

m

C222 · · · C222

The solution of equation (30 .1) is given by

= l'V(I - 66 t )

(31) where vir is an arbitrary matrix of appropriate dimension . [H221 H222]

Note that , by using (28), (29) and (31) , there exists a matrix W of appropriate dimension such that

n

11.

=

[.v"~CC" °

wI'

-~C'lj j ••... °

...

1

: ~r(l-cct)[~

1 (32)

Remark .'J. If Nt2 '# 0 in equation (8.1) (see Remark 1), then equation (16) becomes (16 .bis) T D12 N12 F12 = 0

[

TE[-., j+ 11 [ H221 Hn2l

with 11.

-;

e=

C~21]l C~20 [[ :b: [..,j ~

1 [J. ~~

where =T C

k -k]

= [ q21 T q22 TCI222T

m T] . . .. C222 . Smce

(I - C<.p (C<.p) t) is a projector, a solution of equation (32) is given by W= 11. (I - C<.p (c
The design of the residual generator is decomposed in two parts. First , in subsection 3.1, the equality constraints (15)-(20) in Theorem 2 are solved in order to decouple the error e from the unknown inputs wand the failure component H· Second , in subse(,tion 3.2, the error stability is obt.ained by solving the inequality constraint (21)

By using equation (32), relation (29) becomes MTE
792

+ wc (I - <.p (C",

IT = [ M °TM

rc)

. .. M m T] .


=0

(34)

Putting

TE=(I

-~)

- )t~=

where

(35 )

u' in equation (13) and (14) (or (22) and (24)) as a "structured uncertainty" . Note that the definition of the "uncertainty set" n in relation (2) can leads to some conservatism in the use of an LMI approach (Boyd et aL., 1994) since, in general case, we have IU;"inl -::f:. lu;"ul with IU;"inl -::f:. 1 and lu;"&X1 -::f:. 1. To overcome this conservatism, each u i is rewritten as follows u'(t) = a' + O"'E'(t) (49.1) where a i and O"i (i = 1, ... , m) are constant real numbers given by a' = (U;"in + u;"ax)/2, 0"' = (u;"&X - u;"in)/2 (49.2) with 0'0 = 1 and 0"0 = o. Using (49 .1)-(49.2), the new "uncertain" variable E is defined as c E 0 C IRm, where 0= {C/C:"in:::; c i :::; c:"u for i = 1, ... (50) with C;"in = -1, c;"ax = 1 for i = 1, ... ,m

(36)

then the solutions of equation (34) are given by M=-W G (37) T=(I - ~)Et (38) Using (36) and (38), the existence of a solution to equation (28) is given by the following theorem. Theorem only if

4. Relation

(28) has a solution if and

(39)

,m}

PROOF. Insert (35) into (28), one obtain TE<,? = 0 {::? (I - ~)

(40)

The error dynamics (13) becomes

(41)

m

e= L

As <,? is offull column rank (see (27)), then relation (41) is equivalent to ( c<,?

r

G
=I

TB)u

+ (Lg 1 -

(42)

a

i

(

CiTE - TA;

+ L~21 C~21 + Lg 22 C~22)X

i=O

i=l

+ f>i c 'e ie-T[D I2 Ft,]

(43)

[7n - tTF{2fi (51)

i=1

+ J(~C~22TE + TA~ ~ i 0 0 i = L221 C221 + L222C222

]=1

]#k

Then inserting (35) and (43) into (15), matrices L221 and Lg 22 satisfy the following equations l\;C~21 TE

TDII )Y21

m

+L

and G<,? has a left inverse as in (42) if and only (39) holds. 0 Write matrices e i as follows e i = TA~ - J(;C~21 -l\~C~22

+ (J -

a.ieie

i=O

Using (43), the matrix G can be defined as

G=

[Co ... cm] = A - KG

(52.1)

m

(44)

A=

with

L aiT A;

(52.2)

i=O

Equations (44) can be equivalently rewritten as [Lg 21 ... L;I Lg n ] r = III (45) where

J( eT

with

and III

= [",OC~"T

= T [A~~

...

A~~] +

[([ J{~ J{~ J[ ~~::] TE) .{[ J{;" J{n[ ~H~] TE)]

J(~

Ki ... J(~]

",OC~22T

",IC522

T

... ",mc

(52.3)

n2

T ]

(52.4)

The insertion of equations (15), (18) and (52) into (13) gives (53.1 ) e=(G + Htl.(c)Z)e

(46)

r=

= [J(?

H= tl.(c) =

[O"le l ... O"me m]

(53.2)

(~EiTi) ® In

(53.3)

(53.4) . Z=[In ... Inr where T' is the elementary matrix of order (m x m) which has unity in the (i, i)th position and all other elements are zero, ® is the Kronecker product. From (50), the matrix tl.(E) in (53.1) is bounded as (54)

(47)

A solution of equation (45) is given by [Lg 21 .. . L;I Lg 22 ]=llIr t (48) The existence of this solution is discussed in (Magarotto et aI., 1997) via a parametrization of the solution Q of inequality (21).

Let V(e) = eTQe be a Lyapunov function candidate with Q a symmetric positive definite matrix. The uncertain system (53)-(54) is robustly asymptotically stable for all c E 0 if the time derivative of V (e) along the trajectory of (53)- (54) satisfied

3.2 StabiLity anaLysIs via an LMI approach

The aim of this subsection is the use of a Linear Matrix Inequality (LMI) approach to determine matrices Q . Ki and Kg such that the inequality constraint (21) in Theorem 2 holds. Consider each

T V(e,e) =eT(CQ+QG + Htl.(e)ZQ + QZT tl.(E)ZHT)e

793

< 0 (55)

control inputs . No assumption is made on the size of the matrix E· and on the regularity of the matrix pencil sE· - AO •. Bilinearities in the failure modes are considered in this approach (see Remarks 1 and 3) . Sufficient condition is given in order to obtain disturbance decoupled residuals. Considering the bounded control inputs as "structured uncertainties", the exponential stability of the kth residual generator of the bank is guaranteed by solving an LMI associated to a robust stabilisation problem. To reduce the conservatism inherent to the robust control theory, a control inputs change of coordinates has been used . The LMI to be solved has an affine dependence on the variables, then the residual generator gams are obtained via convex optimisation .

Relation (55) can be converted into the following LMI

1< 0 [ ~:q.~.~~~~.~.~l!.~.j.9.~.~ ZQ ' -S

using the Schur complement and that (Boyd et al. , 1994) Si1(c)

S

= b.(c)S

(56)

= ST > 0 such (57)

(58 .2) (58 .3) and * is the transpose of the off-diagonal part .

6. REFERENCES

It is easy to see that if inequality (58.1) holds , then constraint (21) in Theorem 2 is verified and the error dynamics (22) is exponentially stable. Since the matrix inequality (58.1) has an affine dependence on variables Q and Y, then the solution and the feasibility of the LMI (58 .1) can be performed with convex optimisation methods (Boyd et al. , 1994) .

4. SYNTHESIS OF THE

[(TH

Bornard, G. , N. Couenne and F . Celle (1988) , Regularly persistent observers for bilinear systems. In : Lecture notes in Control and Information Sciences. pp. 130-140 . New Trends in Nonlinear Control Theory. Boutayeb, M., M. Darouach , H. Rafaralahy and G. Krzakala (1994). Asymptotic observers for a class of non linear singular systems. In: Am. Cont. Con! Baltimore , USA . pp. 1440-144l. Boyd , S., L. El Ghaoui, E . Feron and V , Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory. Chap . 15. Studies in Applied Mathematics . Frank, P.M. (1993) . Advances in observer-based fault diagnosis . In: Int. Con! on Fault Diagnosis TOOLDIAG'93. Toulouse, France. Hara, S. and K. Furuta (1976) . Minimal order state observers for bilinear systems. Int. J. of Cont. , 24 , 705-718. Kaprelian , S. and J. Turi (1992) . An observer for a non linear descriptor system , In : Con! on Dec. and Cont. Tucson, USA. Kinnaert, M., Y. Peng and H. Hammouri (1995) . The fundamental problem of residual generation for bilinear systems up to output injection. In: 3rd Eur. Cont. Con! Rome, Italy. Magarotto, E., M. Zasadzinski , H. Rafaralahy and M. Darouach (1997) . An LMI approach to FDI problem of singular bilinear systems subjected to unknown inputs . In : IAfACS ~Vorld Congress 97. Berlin , Germany. Patton, R.J . (1994) . Robust model-based fault diagnosis : the state of the art. In: IFAC SAFEPROCESS '94. Helsinki , Finland . Rafaralahy, H., M, Zasadzinski , M. Boutayeb and M. Darouach (1996) . State observer design for descriptor bilinear systems . In: lEE Control '96. Exeter, UK . pp. 843-848. Vu, D. and D.N. Shields (1995). Fault diagnosis in bilinear systems: A survey. In : 3rd Eur. Cont. Con! Rome , Italy.

GENERATOR

The design procedure of the kth residual generator (4) can be summarised as follows. 1 Find matrices P , V, 1l 1 , V2 , V3 and U to transform the bilinear system (1) into (8); 2 Compute 'is, [j and r.p using (26)-(27); 3 Compute and T using (36) and (38); 4 Compute scalars cri and u i using (49) ; 5 Compute A, C and C using (52) and (58 .3); 6 Choose S satisfying (57) , and compute Q and Y by solving the LMI (58 .1) ; 7 The gains [(i and [(2 are given by (52.3) where K is deduced from (58 .2) ; 8 Compute C i and C(u) using (43) and (10.3) ; 9 Choose Wand compute C, H221 and H222 using (30.2) and (31); 10 Compute 1i, C, W, M ' and M(u) using (32), (33), (37) and (10.3) ; 11 Check Assumption (AI); 12 Compute wand deduce L221 and Lg 22 using (4 i') and (48) , put L21 = 0 and L222 = 0; 13 Compute J and Lg 22 using (17) and (18) ; 14 Matrices L' and H are given by (ll) .

5. CONCLUSION The objective of this paper is the design a bank of full-order disturbance decoupled residual generators for singular bilinear systems with bounded 794