Publication soumise à European Journal of Control, 1998 Submitted paper to European Journal of Control, 1998 Paper finally refused and not published. (c) 1997-1998, W. Nuninger

LINKS BETWEEN BOTH DIRECT AND INDIRECT RESIDUALS BASED REDUNDANCY WITHIN PARITY SPACE AND UNKNOWN INPUT OBSERVERS.

ON

ANALYTICAL

W. NUNINGER, 1

Centre de Recherche en Automatique de Nancy (CNRS UPRESA-A 7039), I.N.P.L., 2, Avenue de la Forêt de Haye, F 54 516 Vandoeuvre-lès-Nancy Cedex, France, Tel. (33) 3 83 59 59 59 - Fax (33) 3 83 59 56 84 E-mail : [email protected]

To solve the Fault Detection and Isolation problem, a diagnostic procedure is used. It is usually composed by two steps : the residual generation this paper deals with and their further evaluation within decision function. This paper considers two methods based on analytical redundancy: parity space approach and unknown input observer based approach that respectively generate so called direct and indirect residuals. These methods are known to be linked under certain circumstances so that one might wonder which method to use in order to design a robust detection system in some sense. Considering unknown input observers, a link had already been proved by the authors between each residuals. Therefore the relevance of this paper with respect to the previous results is the more precise hypothesis for the theorems so that one residual can be generated from the other one. Therefore, the design of the corresponding parity space or observer is given. Finally, the conclusion is obvious that parity space is the more efficient method; this was not so clear in the previous results by Nuninger, et al. [1]. In addition, the sensitivity of each residual with respect to additive faults on the outputs is studied and some results about the robustness with respect to parameter uncertainties are given too. Finally, a simulated example is given.

Key words: Redundancy, fault detection, fault isolation, observers, unknown input, parity space, robustness. 1. Introduction Fault diagnosis is actually a problem of great interest because of the increasing complexity of industrial processes. Many surveys on design methods for the diagnostic procedure exist in the literature. Among them, the following works can be quoted by Iserman [2], Frank [3, 4], Gertler [5] and Patton [6]. The aim of the detection system is to detect and isolate the faults in order to further maintain the required performances of the process. Considering the residual based detection systems (that use analytical redundancy) several methods were develop such as parity space approach, observer and filter based approach and the parameter identification method. All theses methods required a process model which can not be known with certainty. Therefore such methods have to be robust to system uncertainties, i.e. the generated residuals (and further the residuals evaluation procedure) have to be sensitive to faults (and especially incipient ones) but still remain insensitive to other disturbances as parameter uncertainties and measurement noises; on the contrary, false alarm and no detection occur which are drawbacks for detection systems and degrade the

robustness degree [4]. All the quoted methods are known to be linked and to generate so called "dual" residuals, i.e. each residual can be deduced from another one under some hypothesis. Such results were proved by Patton and Chen [7] that recall parity space leads to reliable FDI schemes. They stress on the fact that even though their potential robustness was studied, less work has been done to show the powerful correspondence existing between both techniques. In addition, Frank and Wünnenberg [8] and Frank [4] pointed out that under certain conditions both residuals were equivalent within a choice of the observer gain to get the same proprieties on the estimation error. This result was proved afterwards by Magni and Mouyon [9]. Several authors, as Frank and Wünnenberg [8], Wünnenberg [10], Gertler [5], Staroswiesky [11] and Frank [4] evidenced that parity space approach leads to certain types of observer structures and is therefore structurally equivalent. The recent work by Delmaire et al. [12] also proved the result for residuals generated by parity space and parameter identification methods. With respect to unknown input observer and parity space, the results were first proved by Nuninger et al. [13] for one way (i.e. direct residual can be generated from the indirect residual) and further completed by Nuninger et al. [1] for the reciprocal part. Anyway, no results have been given with regard to the robustness aspect. Indeed, unknown input observers are known to be helpful within Generalised Observer Scheme to improve the robustness degree with respect to system uncertainties. It is therefore interesting to evaluate the sensitivity of both residuals with respect to assumed uncertainties and then to conclude thanks to the equivalent relations. Are these relations still true when faults or parameter uncertainties occur? The relevance of this paper is to give more precise hypothesis for the already proved theorems. Then, the conclusion is more obvious and parity space can be (in theory) seen has the more efficient technique (not taking into account computational problems to find the projection matrix). As a consequence of these theorems, one knows how to construct one residual from the other one and how to design the corresponding observer or parity space ; the solution is not always unique. As a consequence, the robustness of each residual is not always improved and it can be useless for fault detection. This is based on the sensitivity study of each residual with respect to faults and parameter variations. This paper is organised as follows. First the problem formulation is quickly summarised and the results about direct and indirect residual generations are recalled. Second, the theorems are given with more precise assumptions. In addition, an additive assumption of theorem 2 is presented and also the new theorems 5 to 7 which are relevant to this paper. So the following works by Nuninger et al. [13, 1] should be checked when necessary. Then, the

special case of strictly equal residuals is given. Further, the sensitivity of the residuals to additive faults on the output measurement and to parameter uncertainties are studied too. Finally, a conclusion is given after a simulated example.

2. Problem Formulation Let us consider the linear time-invariant system subject to disturbances:

xÝ(t) = Ax(t) + Bu(t ) + Hv(t ) y(t) = Cx(t) + Du(t) + Fv(t )

(1.a) (1.b)

where x ∈R is the state vector, u ∈R r the control p input, y ∈R the measured output vector and v ∈Rq the disturbance vector which components are unknown functions of the time. All the matrices are of appropriate dimensions and of real constant components. n

Lemma 1. For system (1) that satisfies: Hypothesis 1: C is a full row rank matrix. Hypothesis 2: there is more measurements than unknown inputs (p > q). the following reduced equations can be written:

xÝ(t) = A 1x(t) + B1u(t ) + H 1y1 (t ) + H2 v 2 (t) y1 (t) = C 1x(t) + D1u(t) + v1 (t) y 2 (t) = C2 x(t) + D2 u(t)

(2.a) (2.b) (2.c)

with where v1 ∈R , v 2 ∈R , y1 ∈R , y 2 ∈R m = rank(R) ≤ inf(p, q) = q where R is a diagonal matrix defined by the following singular value decomposition of matrix F: m

R F = U 0

0 T V 0

q− m

m

p− m

(3)

where U and V are square matrix of dimension pxp and qxq respectively. The matrices and vectors are given by:

R Σ= 0

0 v  y  , VT v =  1 , Σ − 1U Ty =  1  , I v2  y2  C D     Σ − 1U TC =  1  , Σ − 1U TD =  1  , A 1 = A − H 1C1 ,  C2   D2  B 1 = B − H1 D1 and HV = (H1 H 2 ) ♦ Proof. The proof is based on the singular value decomposition (3) of matrix F. Indeed, with the above definitions, equations (2) are obtained by pre−1 T multiplying equations (1) by the quantity Σ U . ♦ Remark 1. Note that assumption 1 (which is not restrictive as any matrix built with some rows of matrix C is also a full row rank matrix) implies that there is no massive redundancy in equation (1.b). Indeed, if it is not the case, a singular value

decomposition of C can always been performed so that massive and analytical redundancies are driven apart. As a consequence, assumption 2 is sufficient to provide analytical redundancy. ♦

if and only if hypothesis 3 is true as (9.a) and (9.c) can be rewritten under the form:

2. Residuals Generations

Under hypothesis 4, H2 does not appear anymore in the equations as the singular value decomposition of F leads to:

2.1. Direct residual generation Theorem 1. Consider system (1) of reduced form (2) that satisfies hypothesis 1 and 2. Then a parity space, defined by the projection matrix :

[

Ω(s) = Ω 3 (s)C 2 (sI n − A1 )

−1

]

Ω 3 (s)

0 (n +q ),m

(4)

of dimension (n+q)x(n+p) exists if and only if : (i)

Ω 3 (s)C 2 (sI n − A 1 )− 1 H2 = [0](n +q ),( q− m )

(5)

⇔ (ii) Hypothesis 3:  sI − A1 −H 2  matrix  n  is full column rank 0   C2 or under the additive hypothesis: Hypothesis 4: F is full row rank if and only if : (iii) the pair (A1,C2) is observable and Ω3(s) can therefore be chosen with no constraint except the dimension. For both cases, the direct residual is given by: −1 r(s) = −Ω3 (s) C2 (sI n − A1 ) B1 + D 2 u(s)

{[

]

+C 2 (sI n − A 1) H 1y1 (s) − y 2 (s)} −1

(6)♦

Proof. Redundancy equations (8) are simply obtained by elimination of the unknown variables, x and v, in equation (2) within the use of the Laplace transform and a given projection matrix Ω(s) = [Ω1 (s) Ω 2 (s) Ω 3 (s) ] so that relation (7) is satisfied.

(Ω

1

 sI − A1 −H 2  (s) Ω 3 (s))  = (0 0 ) 0   C2

(10)

 R  T V F = U  0 (n −q ),q  with m=rank(R)=q. This implies that the dimension T of v2 is null. So, v1 is defined by V v = v1 and v2 does not appear in the equations (2) anymore. As a consequence, the matrix H2 does not appear either. Indeed, HV=H1. A simpler form is given so that constraint (5) has not to be satisfied anymore. Then, the necessary and sufficient condition for the parity space is reduced to (iii). In this case any matrix Ω3(s) of dimension (n+q)x(p-m) can be chosen. In both cases, within relations (9), the direct residual expression is based on the expansion and simplification of the redundancy equation (8) rewritten like (11). So, residual r(s) is equal to zero (or close to in presence of noise) when no failure occurs on the measured input u or/and output y1 and y2 and differs from this value otherwise.

Ω 1 (s){H 1 y1 + B1 u}+ Ω 3 (s){y2 − D 2 u}= 0

(11)♦

Remark 2. Thanks to the reduced form (10) of the constraint on Ω1(s) and Ω3(s), it is proved in [13] that the search of Ω(s) is reduced to the calculation of the kernel of a binomial matrix (see Annexe). The projection matrix Ω(s) can therefore be chosen of minimal degree.

2.2. Indirect residual generation

 sI − A 1 0 −H 2  Ω(s) C1 I 0  = 0    C2 0 0 

(7)

On the other hand, implicit redundancy equations can be generated as the difference between the measured output and the estimated output based on an unknown input observer of the state defined by:

 B1 H 1 0  u (s)  Ω(s) −D1 I 0  y1 (s)  = 0     −D 2 0 I   y2 (s)

(8)

zÝ(t ) = Nz(t) + L 1y1 (t ) + L2 y 2 (t) + Gu(t) xˆ (t) = z(t ) − V1y1 (t ) − V2 y2 (t) + Mu(t)

The simplification of the expanded system (7) leads to constraints (9) on the matrices defining Ω(s):

Ω 1 (s) = −Ω 3 (s)C 2 (sI − A1 ) Ω 2 (s) = 0 −1 Ω 3 (s)C 2 (sI − A1 ) H 2 = 0

−1

(9.a) (9.b) (9.c)

Theses constraints are reduced to constraint (5) on matrix Ω3(s) (i.e. (9.c)) within the definition (4) of matrix Ω(s). It is therefore clear that a solution exists

(12.a) (12.b)

where z(t) is of dimension n and xˆ (t) denotes the state estimation of size n. Lemma 2. Consider system (1) of reduced form (2) that satisfies hypothesis 1 and 2. Then the estimation error, ε(t) = xˆ (t) − x(t ) , based on the unknown input observer (12) tends asymptotically to zero if and only if the following relations are satisfied: N stable

V1 = [0]n ,m

(13.a) (13.b)

PH 2 = [0]n ,q −m

(13.c) (13.d) (13.e) (13.f) (13.g) (13.h)♦

M = V2D 2 L 1 = PH1 P = I n + V2C2 L 2 C2 = PA1 − NP G = PB1 − L 2D 2

Proof. The matrices of the unknown input observer are designed so that the state estimation error expressed by (14) converges to zero despite of the unknown input. The derivative of the state estimation error is given by (15) using equations (2) and (12.a).

ε(t) = xˆ (t) − x(t) = z(t) − V1 y1 (t) − V2 y2 (t) + Mu(t) − x(t) εÝ(t) = zÝ(t) − Px(t) − Q Ý u(t ) − V1 vÝ1 (t ) P = I + V1C1 + V2 C2 Q = V1 D 1 + V2 D 2 − M

(14) (15.a) (15.b) (15.c)

First, the derivative (15.a) is rewritten taking into account the expression of the derivative (12.a) of z(t) and introducing (2.b) and (2.c). Second, it is assumed that matrix Q is the null matrix so that M satisfies (16). Then the expression of z(t) is deduced from (14) and replaced so that the derivative of the state estimation error is given by (17). As a conclusion, the state estimation error tends asymptotically to zero if relations (13) are satisfied.

M = V1 D 1 + V2 D 2 eÝ(t ) = Ne(t) − V1 vÝ1 (t ) − PH2 v2 − QuÝ(t) + [NP + L 1 C1 + L 2 C2 − PA1 − PH1 C1 ]x(t)

[L D + L D + G − PB [L − PH + NV ]v (t) 1 1

1

2

1

2

1

1

− PH1 D 1 ]u(t)

x   In x =  x 1  ∈Ker C  2  2

H  2 0 p−m,q−m 

x = −H x  then x is defined so that:  C x 1 = 0 2 2  2 1 p−m,1 x ∈Ker(C ) ∩ Im(H ), ∀ x  2 . ⇔ or x1 ∈Ker (H2 ), x = 02  2 2 1 From (b), the kernel is reduced to the singleton and x1 = x 2 = 0 . This implies that Ker ( H 2 ) = {0}

and Ker (C 2 ) ∩ Im( H 2 ) = {0} . From the definition

(16)

of the rank of a matrix which is equal to the difference between the number of column and the dimension of the kernel, the following relation is given:

(17)

rank (C H ) = rank(H ) + 2 2 2

1

♦

Theorem 2. Consider system (1) of reduced form (2) that satisfies hypothesis 1 and 2. Then: (a) for a square matrix P of rank n-(q-m), and Hypothesis 5: rank (C 2 H 2 ) = rank ( H 2 ) = q − m ,

−H  In  2 ⇔ (b) rank C  = n +q− m 0 p−m,q−m   2 ∀s ∈C, Re(s) ≥ 0 the unknown input observer (12) exists if and only if: (i)

Proof. The relevance of this paper is the equivalence between hypothesis 5 and condition (b). The necessary and sufficient conditions for the unknown input observer have already been proved by several authors as Kudva [14], Hou and Muller [15]. The equivalence between the three propositions (i), (ii) and (iii) was proved by Darouach et al. [16] under the given assumptions 1, 2 and 5 and the condition (a). The result is simply applied to our case and the complete proof is therefore omitted because of the number of pages required for this paper [1]. Let us prove that hypothesis 5 implies (b). Under hypothesis 2, consider vector x so that:

Hypothesis 6: the pair (A1,C2) is detectable

sP − PA1 ⇔ (ii) rank   = n ∀s ∈ C, Re(s) > 0  C2   sI − A1 −H 2  ⇔ (iii) matrix  n  is full column rank 0   C2 sI − A H ⇐ (iv) rank  n  = n + q ∀s ∈C, Re(s) ≥ 0 0  C2 ♦

{

} {

}.

dim Ker (H ) − dim Ker(C H ) 2 2 2 As x ∈Ker(C2 H 2 ) ⇔ C2 H 2 x = C 2 (H 2 x) = 0 x ∈Ker(H )  ⇒ or x ∉Ker(H 2 ), H x ∈Im(H )∩ Ker(C )  2 2 2 2 it is deduced:

{

}

rank (C H ) = rank(H ) − dim Im(H ) ∩ Ker(H ) 2 2 2 2 2 that proves rank (C2 H 2 ) = rank(H 2 ) . As a consequence, (b) implies Hypothesis 5. The reciprocal is obvious as Hypothesis 5 implies that Ker (C 2 ) ∩ Im( H 2 ) and the kernel of H2 are reduced to the singleton. Therefore, for any vector x so that:

 x   x ∈Ker(C ) ∩ Im(H ) = {0}, ∀x 2 2 2 x =  1  /or 1  x 2   x2 ∈Ker(H 2 ) = {0}, x1 = 0 it can be said that x is the singleton to which one is reduced the kernel of the considered matrix. Condition (b) is proved. ♦ Remark 3. Note that the N stability problem (13.a) is reduced to an eigenvalue assignment problem for the detectable pair (PA1,C2) within the use of a gain. Indeed, replacing the expression (13.f) of matrix P in

equation (13.g), the choice of K with respect to (18.a) stands for the pole placement of the pair (PA1,C2) and implies L2 (18.b).

N = PA1 − KC 2 L 2 = K − NV2

(18.a) (18.b)

Remark 4. Note that the matrix V2 has to be chosen so that (19), deduced from (13.c) and (13.f ), if true. Then V2 is defined by (20) where V20 is a constant matrix that can be chosen as a null matrix as this choice does not modify matrix K. Note that (.)+ = [(.)(.)T]-1(.)T stands for the right inverse of the given matrix.

V2 (C 2 H 2 ) = −H 2 + + V2 = −H 2 (C2 H 2 ) + V20 I − (C 2 H 2 ) (C2 H 2 )

[

]

(19) (20)

The expression (21) of P is further deduced within equation (13.f). Note that rank(P) = n-(q-m). Finally, L2 is given by (18.a) and the other matrices M, L1 and G from the definitions (13.d), (13.e) and (13.h).

P = I − H 2 (C 2 H 2 ) C 2 (21) Lemma 3. The indirect residual generated for the defined unknown input observer (12), under hypothesis 1, 2, 5 and conditions (a) and (iii) of theorem 2, is given by:

theorem 1. Therefore the proof is reduced to prove that for the given choice of the projection matrix (25), the constraint (5) is also satisfied and that the relation (24) is true. Using relations (13.f), (13.h) and (13.c), expression (26) of Ω3(s)C2 is obtained that leads to (27), i.e. constraint (5). Finally, basing the demonstration on relations (25) and (26), the direct residual (6) is easily rewritten as (28). Thanks to relation (13.d), it is clear that (28) is equivalent to relation (24).

Ω 3 (s)C2 = T(s)C 2 [I + V2 C 2 − (sI − N) −1 L 2 C 2 ]

= T(s)C 2 (sI − N) [(sI − N)P + L 2 C 2 ] (26) = T(s)C 2 (sI − N) −1 P(sI − A 1 ) −1

Ω 3 (s)C 2 (sI − A 1 ) −1 H2 = T(s)C 2 (sI − N)−1 PH2 (27) =0 r(s) = T(s) [C2 (sI − N) −1 G + D2 + C 2 V2 D2 ]u (s)

{

− C2 (sI − N) L 1 y1 (s) −1

}

+ [I + C2 V2 − C 2 (sI − N) L 2 ]y 2 (s) −1

(28) ♦

+

[

]

e(s) = − C2 (sI n − N) −1 G + C2 M + D2 u(s) −1

−C2 (sI n − N) L1y1 (s)

[

−1

]

(22)♦

+ I p −m + C2 V2 − C 2 (sI n − N) L 2 y2 (s)

Theorem 4. Consider system (1) of reduced form (2) that satisfies hypothesis 1 and 2. Consider the direct residual (6) based on the parity space (4) where matrix Ω3(s) is given of dimension (n+q)x(n+p). Choose two matrices : - a square stable matrix N of dimension nxn, - a constant matrix V20 of dimension nx(q-m).

Proof. The indirect residual is defined by the output reconstruction error (23) within the model (2.c) and can be therefore be expressed such as (22).

e(t ) = y2 (t) − yˆ 2 (t) = y 2 (t ) − C 2xˆ (t) − D 2 u(t) (23)♦

Then an unknown input observer of form (12) defined by V2 (20), P (13.f), L2 (18.b) where K satisfies (18.a) (i.e. the pole placement of the pair (PA1,C2)) and the defined matrices M (13.d), G (13.h), L1 (13.e) and V1 (13.b) exists if and only if hypothesis 5 is true. Besides, the signal e(t) generated by:

3. Structural Equivalence

e(s) = ϕ(s)r(s)

3.1. Indirect residual to direct residual

[

]

ϕ(s) = C 2 (sI n − N ) P(sI − A1 )C T(s)

Theorem 3. Consider system (1) of reduced form (2) that satisfies hypothesis 1, 2 and 5. Consider the indirect residual (22) based on the unknown input observer (12). Then, for any stable matrix T(s), the signal r(t) based on the expression:

r(s) = T(s)e(s)

−1

− 2

(29.a) (29.b)

is the indirect residual based on the previously defined unknown input observer if and only if T(s) is a stable matrix such that:

T(s)Ω 3 (s) = I n + p

(30)♦

(24)

is the direct residual based on the parity space defined by the projection matrix Ω(s) (4) where Ω3(s) is:

[

3.2. Reciprocal: direct residual to indirect residual

−1

Ω 3 (s) = T(s) I p −m + C2 V2 − C 2 (sI n − N) L 2

] (25)♦

Proof. Both the existence of the unknown input observer and the parity space are guaranteed thanks to the necessary and sufficient condition (iii) of theorem 2 which is equivalent to the hypothesis 3 of

Proof. Note that (.)+ = (.)T[(.)(.)T]-1 stands for the right inverse of the given matrix and (.)- = [(.)T(.)]-1(.)T for the left inverse. First, the existence of the parity space is guaranteed from hypothesis 3 of theorem 1. As this assumption is the condition (iii) of theorem 3, the existence of the unknown input observer is guaranteed if and only if hypothesis 5 is also true. Note that if the parity space exists under hypothesis 1, 2 and 4 then hypothesis 5

is also satisfied because H2 does not appear in the equations anymore. Then, only relation (29) remains to be proved under constraint (30) on the choice of T(s). Second, the indirect residual e(s) (22) is now rewritten to make appear the expression (6) of the direct residual. Matrices M, L1 and G are replaced in (22) by their expression to get:

e(s) = −α(s)u(s) − β(s)y1 (s) + γ(s)y2 (s)

(31.a)

α(s) = C2 (sI n − N) −1 [PB1 − L 2 D 2 ] + C 2 V2 D 2 + D 2

(31.b)

−1

β(s) = C 2 (sI n − N) PH1

(31.c)

[

γ (s) = I p− m + C 2 V2 − C 2 (sI n − N) L 2 −1

]

(31.d)

The previous theorems prove that for a given unknown input observer, several parity spaces can be designed so that the performances of the indirect residual can be changed within a good choice of matrix T(s). On the contrary, the reciprocal is not always true. Indeed hypothesis 3, that guaranties the existence of the parity space, can be satisfied whereas hypothesis 5 is not, i.e. the unknown input observer does not exist. This is proved thanks to the following counter example, it is obvious that hypothesis 3 is true but not hypothesis 5:  0  0 1 0  1 0 0   A1 = 0 0 1 , C2 =   and H 2 =  0      0 1 0  1  0 0 0

 0 rang C 2 H 2 = rang  ≠ 1 = q − m = rang H 2 .  0

(

)

( )

Replace Ip-m in (31.d) by its expression introducing − I p− m = C2 C2 the left inverse of C2 (defined by

C 2− = C T2 (C 2 C T2 ) that exists) and further replace V2C2 by its expression (13.f) in (31.b) and (31.d). −1 Then, within factorisation of the term C 2 (sI n − N) , α(s) and γ(s) are expressed by: −1

N, V20 et T(s)Ω3(s) = I Theorem 4

Hyp. 1,2,3

No solution

Hyp. 1,2,3,5

α(s) = C2 (sI n − N) −1 [PB1 − L 2 D 2 + (sI n − N)PC2−D 2 ] γ (s) = C 2 (sI n − N) −1[(sI n − N)PC −2 − L 2 ]

Hyp. 1,2,4,6 Hyp. 1,2,5,3

Theorem 3

The product NP is then replaced by (13.g). After factorisation by P(sI n − A 1 ) the following expressions of α(s), β(s) and γ(s) are obtained: α(s) = C 2 (sI n − N) −1 P(sI n − A 1 )[(sI n − A 1 )−1 B1 + C−2 D 2 ]

β(s) = C2 (sI n − N) −1 P(sI n − A 1 )[(sI n − A 1 ) −1 H1 ] γ (s) = C 2 (sI n − N) −1 P(sI n − A 1 )[C−2 ]

Finally, define ζ(s) so that the following relation is true:

ζ(s)C 2 = C 2 (sI n − N)− 1 P(sI n − A1 )

(32)

Then, α(s), β(s) and γ(s) are given by (33) :

α(s) = ζ(s)[C2 (sI n − A1 )−1 B1 + D 2 ]

[

−1

β(s) = ζ(s) C 2 (sI n − A 1) H 1

]

γ (s) = ζ(s)

(33.a) (33.b) (33.c)

Finally, the comparison of the direct residual (6) and the indirect one (31), taking into account (33), leads to the relation (34) which is true if and only if the stable matrix T(s) satisfies the condition (30). In addition, definition (32) of ζ(s) implies (35). From (34), (29.a) is true with ϕ(s) = ζ(s)T(s) .

e(s) = ζ(s)T(s)r(s) ζ(s) = C2 (sI n − N) −1 P (sI n − A 1 )C−2

(34) (35) ♦

3.3. Conclusion

T(s) Parity space based solutions

Unknown input observer based solutions

Fig. 1. Get one residual from the other one and reciprocal. The consequence of such theorems is illustrated in Figure 1. Note that when hypothesis 4 is true, hypothesis 5 is also satisfied. Besides, under hypothesis 6, hypothesis 3 is true if hypothesis 4 is satisfied. In Figure 1, ‘x’ stands for one solution that belongs to one set of solutions (circle). The ways to get one residual from the other one is given by the arrows. Finally the case when both residual (e and r) are equal is a special case of the previous theorems [17]. Although from theorem 3, the indirect residual is directly obtained within T(s) chosen as the identity matrix (theorem 3) the reciprocal is more complex. Indeed, the solution exists if and only if hypothesis 5 is true and if N and V20 can be found so that (theorem 4):

[

]

ϕ( s) = C 2 (sI n − N ) P(sI − A 1 )C −2 T( s) −1

Up to now, no proof has been given for the existence of such matrices under the previous constraint.

where the dynamic of x depends on ∆A and ∆B. A Hyp. 1,2,3

S opt

Hyp. 1, 2,4,6

D

B Hyp. 1, 2,5,3

C

S opt

no solution

S opt

Hyp. 1, 2

Hyp. 1,2,5

no solution

E

Fig. 2. Sets of solutions for both approaches. As it was already stressed by Gertler [5] for Luenberger observer based residuals, both approaches are dual but of distinct complexity. In addition, it seems more convenient to choose the parity space approach to design an « optimal » diagnostic procedure. Indeed, if one « optimal » solution (Sopt) can be found within unknown input observers (set B) it can always been found within parity space approach (set A ∪ C that contains B) as it is illustrated in Figure 2. On the contrary, if Sopt belongs to A \ (B ∪ C ) , only the parity space approach will solve the problem. Considering the numerical point of view, this is not so simple as the exact projection matrix can not always be found. As a consequence, less-optimal solutions are given and parity space is not so efficient. Anyway, it is simpler to take into account structured constraints on the direct residual (choice of the projection matrix Ω3(s)) than on the indirect one except by using Generalised Observer Scheme. This remark introduce the robustness performance criteria of the detection systems.

4. Residual Sensitivity to Faults and to Parameter Unvertainties 4.1. Definitions and notations

Therefore, the previous results on the parity space and the unknown input observer designs are still correct (the model (1) is still the same) if y(t) is replaced by y(t) − ∆y(t) − d(t) as the output measurement is now y(t) (36). As a consequence, it is clear, by applying the same method used for lemma 1, that the reduced form (2) of the system is now obtained by replacing y1 and y2 by y1 (t) − ∆Y1 (t) − d 1 (t) and y2 (t) − ∆Y2 (t) − d 2 (t ) respectively so that the reduced form of the real system is rewritten as:

xÝ(t) = A 1 x(t) + B 1 u(t) + H 1 y 1 (t) − H 1 d 1 (t) + (37.a) ∆A 1 x(t) + ∆B 1 u(t) + H 2 v 2 (t ) y1 (t) = C1 x(t) + D 1 u(t) + v 1 (t) + ∆Y1 (t) + d 1 (t) (37.b) y2 (t) = C 2 x(t) + D 2 u(t ) + ∆Y2 (t ) + d 2 (t) (37.c) with the following new definitions of vectors:

 ∆C 1  Σ −1 U T ∆C =  ,  ∆C 2 

 ∆D 1  Σ −1 U T ∆D =  ,  ∆D 2 

∆A1 = ∆A − H 1 ∆C 1 , ∆Y1 (t ) = ∆C1 x(t ) + ∆D 1 u(t) , ∆B1 = ∆B − H1 ∆D1 ,

∆Y2 (t) = ∆C 2 x(t) + ∆D 2 u(t) .

4.2. Residuals modifications due to faults and parameter variations Lemma 4. Assume that the direct and indirect residuals are generated by parity space and unknown input observer designed on the model (A, B, C, D, H, F) of the real system which is normally described by (A+∆A, B+∆B, C+∆C, D+∆D, H, F). Then, these residuals are re-written in the following way:

r (s) = r(s) + ∆r(s)

(38.a)

e(s) = e(s) + ∆e(s)

(38.b)

In the following, the real system is submitted to parameter uncertainties. This uncertainties are deterministic additive and constant parameter variations on matrices A, B, C and D modelled by ∆A, ∆B, ∆C, ∆D respectively. The given formalism means that the matrices of the real system are (A+∆A, B+∆B, C+∆C, D+∆D, H, F) whereas the model of the system is still described by the time representation (1) with (A, B, C, D, H, F). The assumed model is therefore not exact and this will put bias on the residuals. In addition, additive faults d(t) are assumed on the output measurement which is therefore given by:

where r(s) (6) and e(s) (22) stand for the values of the residuals when there is no fault (d=0) and no parameter variations (∆A=∆B=∆C=∆D=0) and where the variations ∆r and ∆e from these values are given by:

y(t) = y(t) + ∆y(t) + d(t)

Proof. Because of the proprieties that give the reduced form of the system model, it is easily proved that the defined direct and indirect residuals differ from their original values by ∆e(s) and ∆r(s). The proof is easy if one consider the definitions of the indirect residual (23) and the direct residual (6) with

(36.a)

where y(t) stands for the true value of the output given by equation (1.b) and ∆y(t) for the bias due to parameter variations:

∆y(t ) = Cx(t ) + ∆Du(t)

(36.b)

{

∆r(s) = −Ω 3 (s) −C2 (sI n − A 1 ) H 1 [∆Y1 (s) + d1 (s) ] −1

− ∆Y2 (s) − d 2 (s)}

∆e(s) = C 2 (sI n − N) L 1 {∆Y1 (s) + d 1 (s)}− −1

[I

p−m

]

+ C 2 V2 − C 2 (sI n − N) −1 L 2 {∆Y2 (s) + d 2 (s)} ♦

the right notations measurements.

y1 and

y2

of the output ♦

4.3. New results on the residuals submitted to faults and parameter variations

Theorem 7. Under the assumption of theorem 5 both sensitivities of the designed residuals are linked by: ∂r(t) ∂e (t) = T(s) T with i = 1, 2 (45) T ∂∆ i (t) ∂∆ i (t) whereas under the assumption of theorem 6, theses sensitivities are linked by:

Theorem 5. Consider the assumptions of theorem 3. Then, for any previously chosen stable matrix T(s) so that (24) is true, relation (39) is also satisfied:

∂e (t) ∂r(t) = ϕ(s) T with i = 1, 2 T ∂∆ i (t) ∂∆ i (t )

∆r(s) = T(s)∆e(s)

Proof. These relations are directly proved from the relations (39) and (41) as the choice of T(s) and the definition of ϕ(s) do not rely on the additive fault di(t), nor on the parameter variations coming from ∆Yi(t) (i =1, 2) ♦

(39) ♦

Proof. Basing the proof on the transformation that gives the reduced form of the system taking into account the new measurement (36), the result is obvious. Indeed, under the assumptions and choice of T(s) given by theorem 3, it is proved that Ω3(s)C2 satisfies relation (26) that leads to (40) thanks to equation (13.e). −1

−1

Ω 3 (s)C 2 (sI − A 1 ) H 1 = T(s)C 2 (sI n − N) L1

(40)

Then, the constraint (25) on matrix Ω3(s), leads to the relation (39) between ∆e(s) and ∆r(s). ♦ Theorem 6. Consider the assumptions of theorem 4. Then, for any previously chosen matrices N, V20 and T(s) so that (30) is true, relation (41) is also satisfied where ϕ(s) is given by (29.b).

∆e(s) = ϕ(s)∆r(s)

(41) ♦

Proof. The result is based on the proof of (29) given for theorem 4 in this paper. Indeed, note that ∆e(s)can be written as:

∆e(s) = β(s)∆Y1 (s) − γ (s)∆Y2 (s)

(42)

where β(s) and γ(s) are defined by (31.c) and (31.d) respectively. As it was previously proved, β(s) and γ(s) are also defined by (33.b) and (33.c) respectively under the definition (35) of ζ(s). So expression (42) is rewritten as (43) and it is obvious that relation (41) is true under constraint (30) on T(s) and the definition (i.e. (29.b)) of ϕ(s) = ζ(s)T(s) .

{

∆e(s) = ζ(s) C 2 (sI n − A 1 ) H1 [∆Y1 (s) + d 1 (s)] −1

− ∆Y2 (s) − d 2 (s)}

(43) ♦

4.4. Sensitivity of the residuals to the faults and parameter variations The sensitivity matrix is defined as the Jacobean of the residual vector with respect to the vector:

∆ i (t ) = ∆Yi (t) + d i (t) with i=1,2.

(44)

that introduced both influences of faults and parameter variations on the measurements.

(46)♦

This theorem is important as it proves that the sensitivity of a given indirect residual can be improved (or not) within the only choice of a correct matrix T(s). For instance, first the matrix allows one to structure the residual with respect to d1(t) and d2(t) and second, to simply increase the sensitivity to the faults. On the contrary, the reciprocal is less easy as (N, V20, T(s)) has to be chosen correctly in order to satisfy constraint (30) whose solution is not proved (theorem 4). Of course, because of the structure of matrices, such a choice is not always simple. Note that whatever the choice of T(s) is, there is no way to separate the influence of faults di(t) from the influence of ∆yi(t) (fixed i equal to 1 or 2).

4.5. Optimal choice of T(s) Consider the sensitivity performance index, Sij(e), of the indirect residual with respect to the jth component of the vector ∆i (44), ∆i[j] defined by the Euclidean norm of the jth column vector of the sensitivity matrix (Jacobean). The same definition of Sij(r) is applied for the direct residual. Then, assuming that T(s) is chosen as a symmetric matrix of dimension equals to the one of both residuals, the theorem of CourantFisher proves the following inequality:

{

}

{

}

Min λ k {T(s)} ≤ ρi , j (r,e ) ≤ Max λ k {T(s)} k k

(47)

where λk{T} stand for the eigenvalues of matrix T and with the sensitivity performance index rate definition:

ρ i ,j (r,e ) = S ij (r) S ij (e) =

∂r ∂∆Ti [j]

2

∂e ∂∆Ti [ j]

(48) 2

As a consequence, in order to improve the sensitivity of the direct residual in comparison to the sensitivity of the indirect residual, T(s) should be chosen as a symmetric matrix (for instance a diagonal matrix) of the same dimension than the indirect residual with a minimal eigenvalue so that λ min {T(s)} > 1 . Note that

In the following, the simulated example is a fourthorder system [13] described by the following matrices appearing in equations (1):

−0.75 −1.18 −0.60 −0.22

 1 0 0 0 C =  0 0 1 0 ,    0 0 0 1

0.41 −0.81 2.32 1.71 −2.46 0.93  , B=  1.40 −1.32 0.05    1.59 −0.51 0.50 0.20 0.32   1  0.99 −0.21  0.5 , F = H=  0.16 −0.01    0  0.01 0.86 

-7

-8

-9

-10

-11

-12

-13

-14 20

30

40

50 60 Time (s)

70

80

90

Fig. 3. Not faulty (dash) and faulty (continuous) output y2 which is close to the estimation.

5. Simulated Example

 −0.46  2.80 A = 1.24   −0.14

Note that when a failure occurs on the first output of system (1) a corresponding failure occurs on only two outputs of system (2): y1(1) and y2. Only the output y2 is plotted in Figure 3 as the indirect residual is based on.

y2

λmin{T(s)} relies on s. Therefore, the diagonal components of T(s) should be chosen as filters (first order filters for instance) which leads to a direct residual less sensitive to measurement noises in comparison with the filtered signal e(s). Then for some frequencies (fixed s), Sij(r) can be greater than Sij(e). Anyway, this does not mean that the detection of fault will be better than with the indirect residual as the influence of parameter uncertainties is also greater. Nevertheless, a specific evaluation procedure can be designed so that a good robustness degree is achieved anyway. Note that the same result should be given to get the indirect residual from the direct one within the choice of a constant symmetric matrix ϕ (s) but is still not proved that such a choice is possible because of the more complex constraint (30) on the choice of (N, V20, T(s)) given by theorem 4.

2.37 2.52 3.70  1.59

Under the given assumptions, theorems 3, 5 and 7 can be applied. Therefore, a direct residual is designed with T(s) chosen as a first order filter of time constant τ = 20s and static gain equal to 200. 2

x 10-3

1 0

0 0  1

-1 -2 20

30

40

50

60

70

80

90

F = U (R T 0 T ) V T , D = [0]3 x 2 and x(0)=0. T

Note that rank(R)=2 so H2 and v2 do not appear in the reduced form (2) anymore and 2 2 1 v 1 ∈ R , y1 ∈R , y 2 ∈R . First the unknown input observer design is given with K defined by the pole placement of the detectable pair (PA1,C2). The four chosen poles are stable and quicker than the eigenvalues of PA1 so that the error converges rapidly. The control inputs are a succession of steps of different lengths and magnitudes. The first unknown input is a random number function of the time whereas the second one is a step of value equal to one beginning at time t=25 s. Note that an additive white noise (5%) of variance 0.01 is simulated on each output measurements. In the following, no parameter uncertainties are simulated. The test will indeed be useless as it is proved that both influences of fault and parameter variations can not be driven apart whatever the residual is. Then, only additive faults are assumed on the output measurements of system (1). These faults are composed by gradients (from instant 300 to 500 and from instant 700 to 800) leading to a step of unit value (from 500 to 700).

Fig. 4. Residuals when faults occur (no noise) 0.02 0.01 0 -0.01 -0.02 20

30

40

50

60

70

80

90

Fig. 5. Residuals while measurement noise and fault. Both residuals are plotted in Figure 4 when there is no measurement noise on the output. The direct residual is a filtrate indirect residual. As a consequence, the sensitivity of the direct residual to

faults is greater than the sensitivity of the indirect residual. The plotted residuals in Figure 5 in presence of measurement noise confirms that conclusion. Despite of the noise, r(s) is more useful for fault detection than e(s) but the chosen filter T(s) should be optimised. 2

1.5

that structured residuals are obtained. Future works should now focus on the influence of parameter uncertainties as the robustness criteria is a compromise between a high sensitivity to faults and a low sensitivity to disturbances. In that context, a residual defined as the difference between two finite memory estimations based on overlapping sliding observation horizons could be of great interest as it is shown in [17] and [18] in both discrete and continuous cases.

1

References

0.5

0

-0.5

-1 20

30

40

50

60

70

80

90

Fig. 6. Step unknown input (dash) and estimation (continuous) while there is no faults. The estimator gives a pretty good estimation of y2 which can not be distinguished from the faulty measurement (continuous plot in Figure 3). The unknown input v1(t) can be estimated too within model (2.b): vˆ (t) = ˆv1 (t) = y1 (t ) − C 1 xˆ (t ) . This estimation is plotted in Figure 6. Note that the observer gain could be optimised to be less sensitive to measurement noise. Such consideration can also be taken into account for the redundancy equation generation but parity space do not allow the output and unknown input reconstruction.

6. Conclusion This paper considers the structural link between direct residual generated by parity space approaches and indirect residual based on unknown input observers. Some previous results by Nuninger et al. [13, 1] and Nuninger [17] are recalled but more precise and correct assumptions for the theorems are given. The consequence is that it is now obvious that in theory within the parity space approach any solution can be designed (not taking into account numerical problem to find the projection matrix). In addition, the results are also extended when additives faults occurs on the output measurement but also when parameter variations occur. It is shown that in that case, the equivalence is still true for the given assumptions and choices and that the sensitivities of dual residuals only rely on the choice of the transfer matrix. It is therefore easier to improve the performance (i.e. the sensitivity to faults) of the dual direct residual of a given indirect residual than the contrary. The simulated example shows that the transfer matrix between both residual should be chosen so that a filtration of some modes is made and

1. Nuninger, W., F. Kratz and J. Ragot. Structural equivalence between direct residuals based on parity space and indirect residuals based on unknown input observers. SAFEPROCESS’97, 1, 1997, pp. 462-467. 2. Iserman, R. Process fault detection based on modelling and estimation methods, Automatica, 20 (4), 1984, pp. 387-404. 3. Frank, P.M. Fault diagnosis in dynamic systems using analytical and knowledge base redundancy A survey and some new results. Automatica, 26 (3), 1990, pp. 459-474. 4. Frank, P.M. Advances in observer-based fault diagnosis. Tooldiag'93, Inter. Conf. on Fault Diagnosis, 1993, pp 817-836. 5. Gertler, G. Analytical redundancy in fault detection and isolation. SAFEPROCESS'91, 1991 pp. 9-22. 6. Patton, R.J. Robust model-based fault diagnosis: the state of art. SAFEPROCESS'94, 1994, pp. 1-24. 7. Patton, R.J. and J. Chen. A parity space approach to robust fault detection using eigenstruture assignment, European Control Conf., 1991, pp. 1419-1424. 8. Frank, P.M. and I. Wünnenberg. Robust fault diagnosis using unknown input observer schemes. In: Fault Diagnosis in Dynamic Systems (R. Patton, P.M. Frank and R. Clark, Ed.), 1989, pp. 47-97. Prentice Hall. 9. Magni, J.F. and Mouyon Ph. A generalized approach to observers for fault diagnosis, 30th IEEE Conf. on Decision and Control, 3, 1991, pp. 2236-2241. 10. Wunnenberg, I. Observer-based fault detection in dynamic systems, PhD, Duisburg, 1990. 11. Staroswiecki, M. Generation of analytical redundancy relations in a linear interconnected system. In: Mathematical and Intelligent Models in System Simulation (J.C. Boltzer, Ed.), 1990, pp. 391-396. Scientific Publishing, IMACS.

12. Delmaire, G., J-P. Cassar and M. Staroswieski. Comparison of generalised least square identification and parity space techniques for FDI purpose in SISO systems. European Control Conf., 1995, pp. 2011-2016. 13. Nuninger, W., F. Kratz and J. Ragot. Observers and redundancy equations generation for systems with unknown inputs. CESA'96, Symp. on Control, Optimization and Supervision, 1, 1996, pp. 672-677. 14. Kudva, P., N. Viswanadham and A. Ramakrishanan. Observers for linear systems with unknown inputs. IEEE Trans. on Automatic Control, 25 (1), 1980, pp. 113-115. 15. Hou, M. and P.C. Müller. Design of observers for linear systems with unknown inputs. IEEE Trans. on Automatic Control, 37 (6), 1992, pp. 871-875. 16. Darouach, M., M. Zasadzinski and S.J. Xu. Fullorder observers for linear systems with unknown inputs. IEEE Trans. on Automatic Control, 39 (3), 1994, pp. 606-609. 17. Nuninger, W. Stratégie de diagnostic robuste à l’aide de la redondance analytique. Doctorat de l’Institut National Polytechnique de Lorraine, 29 octobre 1997, 240p. 18. Kratz, F., W. Nuninger and J. Ragot. Finite memory observer based method for failure detection in dynamic systems. SYSID'97, 3, 1997, pp. 1189-1194. 19. Gantmacher, F.R. The theory of matrices. Chelsea Publishing Company, 1977.

Appendix Lemma. The search the projection matrix that satisfies equation (10) is reduced to the search of the kernel of a binomial matrix. ♦ Proof. The proof is based on Gantmacher’s theory [19]. Searching the Kernel of a binomial matrix (A+λB) boils down to solve the following equation:

(A + λB)x = 0

(A.1)

where x is an unknown column matrix, A and B are matrices of dimension mxn. Without loss of generality we suppose that rank (A+λB) < n. Then a non-zero solution of equation (A.1) exists which determines a linear dependence among the column of (A+λB). In order to reduce the problem, only the polynomials (in λ) solutions x are considered. Moreover, among these solutions, the one of least possible degree are chosen. So let:

p

x(λ) =

∑x λ

i

(A.2)

i

i =0

As x is a solution of (A.1), the substitution of its expression (A.2) in equation (A.1) reduces the problem to equal to zero all the coefficients of the powers of λ. So only the following equation has to be solved:

A x0  B  x1  M p   = 0 with M p =  0 M     x  p 0

0 0 AO  O O 0  O B A  0 B

(A.3a,A.3b)

Gantmacher [19] proved that the number p is the least value of the index k for which the sign '