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spank[s](u, w)/spank[s](w). ) is torsion. An input-output system is a linear dynamics Λpert which is equipped with an output, i.e., a finite subset ypert = (y pert. 1.
1

Proc.

CESA Conf., July 2003

An algebraic approach to fault diagnosis for linear systems Michel FLIESS and C´edric JOIN Abstract— We give an intrinsic algebraic definition of fault detectability and isolability, parity equations, and residuals for time-invariant linear systems. The applicability and efficiency of our approach are illustrated by numerical simulations for three academic examples and a concrete case-study.

Keywords— Linear system, fault diagnosis, detectability, isolation, parity equations, residuals, modules, operational calculus.

I. Introduction This communication on model-based fault detection and isolation (FDI), which further develops some of the ideas sketched in [26], is devoted to a new standpoint. Most of today’s structural approaches to this important topic1 are either of analytic or geometric nature: • The analytic approach (see, e.g., [3], [17], [19], [30]) is further developing techniques from input-output inversion going back to Silverman [38]. • The geometric approach (see, e.g., [32], [33]) is based on Wonham’s theory of invariant subspaces [41]. Here are some salient features of our calculations: • It is not necessary to make any distinction between sensor and actuator fault variables. • The isolability of the fault variables in most of our examples of Sections III and IV could to the best of our knowledge not have been deduced from the existing theories, whereas it is here immediate. • There is no need to implement any asymptotic observer. • Robustness of residuals with respect to a large variety of additive disturbances is ensured. • Closed-loop diagnosis is achieved even with uncertain parameters. The utilisation of the algebraic theory of modules and of operational calculus, which is independent of any peculiar representation (transfer matrices, state-variable representations), is stemming from [8], [12], [14]. Note also the M. Fliess is with the Centre de Math´ ematiques et Leurs ´ Applications, Ecole Normale Sup´ erieure de Cachan, 61 avenue du Pr´ esident Wilson, 94235 Cachan, France, & Laboratoire ´ STIX/GAGE, Ecole polytechnique, 91128 Palaiseau, France. E-mail: [email protected]. C. Join (corresponding author) is with the Centre de Recherche en Automatique de Nancy (CRAN), CNRS UMR 7039, Universit´ e Henri Poincar´ e (Nancy I), BP 239, 54506 Vandœuvre-l` es-Nancy, France. E-mail: [email protected] 1 The recent books [2], [3], [19] are providing excellent overviews of this subject as well as quite complete references.

strong ties with recent identification [15] and state estimation [16] techniques. Our paper is organised as follows. After a short review of the algebraic theory of linear systems, Section II provides new definitions of fault detection and isolation, of parity equations, and of residuals. The first two academic examples of Section III permit not only to show the efficiency of our abstract definitions but also to construct strongly integral residuals which are easy to compute and which permits to bypass, as in [15], [16], unknown initial conditions and various types of noises. As demonstrated by the third example closed-loop fault diagnosis with uncertain parameters is made possible by our fast computations which are of purely algebraic nature (see, also, [15]). The classic benchmark of the three tank system (see [1] and, also, [40]) and the corresponding numerical simulations in Section IV is another indication of the practical interest of this mathematical machinery. Sections III and IV are written in such a way that any reader who is not familiar with the algebraic formalism might nevertheless grasp the fundamental ideas2 . Directions for possible future works are listed in a short conclusion which is also indicating some more methodological clues. II. An algebraic definition of linear fault diagnosis A. Basics on linear systems Let k be the field R or C of real or complex numbers. Introduce the commutative principal ideal P domain k[s] of linear differential operators of the form finite cν sν , cν ∈ k, where the indeterminate s stands for the usual operational symbol of derivation. A linear system3 Λpert is a finitely generated free k[s]-module4 . We distinguish in Λpert two finite subsets, the fault variables wpert = (w1pert , . . . , wqpert ) and the perturbation variables π = (π1 , . . . , πr ), which do not “interact”, i.e., spank[s] (wpert ) ∩ spank[s] (π) = {0}. The short exact sequence 0 → spank[s] (π) → Λpert → Λ → 0 defines the nominal system Λ = Λpert /spank[s] (π). The 2 We have preferred to discuss detailed computations and numerical simulations rather than in the limited space available for this conference communication to devote more pages for recalling the algebraic background. 3 See [8], [12], [14]. A review of the algebraic tools may be found in many standard textbooks (see, e.g., [24], [25]). 4 The freeness [14] is justified by the fact that any operational equation ax = 0, a ∈ k[s], yields x = 0. It means that the only torsion element is 0.

2

canonical image of any element λpert ∈ Λpert in Λ is written λ. A linear dynamics is a linear system Λpert , equipped with a control, i.e., a finite subset upert = (upert , . . . , upert m ) of Λ 1 which satisfies the following properties, where the first two conditions mean that the control variables do not interact with the perturbation and the fault variables: pert ) ∩ span • spank[s] (u k[s] (π) = {0} pert • spank[s] (u ) ∩ spank[s] (wpert ) = {0}, • the quotient module ³ ´ ³ ´ Λ/spank[s] (w) / spank[s] (u, w)/spank[s] (w) is torsion. An input-output system is a linear dynamics Λpert which is equipped with an output, i.e., a finite subset y pert = (y1pert , . . . , yppert ) of Λpert . From now on Λ will be an inputoutput system. System Λ is said to be controllable if, and only if, the quotient module Λ/spank[s] (w) is free. It is said to be observable if, and only if, Λ/spank[s] (w) = spank[s] (u, y, w)/spank[s] (w) B. Fault detectability and isolability B.1 Detectability pert The fault wι , ι = 1, . . . , q, is said to be detectable if, and only if, the quotient module

¯ ι) spank[s] (u, y, w)/spank[s] (u, w ¯ ι = w\{wι }, is not torsion. This condition is where w equivalent saying that wιpert is indeed “influencing” the output variables. B.2 Isolability and parity equations The fault wι is said to be isolable if, and only if, spank[s] (wι ) ∩ spank[s] (u, y) 6= {0} This is equivalent saying that the fault wι is related to the input and output variables by a parity equation X

aα sα wι =

finite

m X

X

i=1

finite

biβ sβ ui +

p X

X

cjγ sγ yj

j=1 finite

where aα , biβ , cjγ ∈ k are appropriate gains. The next property is clear: Proposition II.1: Any fault variable in wpert is isolable if, and only if, there exists a system of parity equations       w1 u1 y1       P  ...  = Q  ...  + R  ...  (1) wq

um

yp

where P ∈ k[s]q×q , Q ∈ k[s]q×m , R ∈ k[s]q×p , det P 6= 0.

C. Residuals Let k(s) be the quotient field of the ring k[s]. Denote by K the quotient skew field5 of the principal left ideal ring d k(s)[ ds ]. A residual associated to an isolable fault wιpert is a non-zero element rιpert ∈ K ⊗k[s] spank[s] (wιpert , π) d The presence of the algebraic derivative ds permits to get rid of the initial conditions and of the deterministic strucpert tured perturbations [15], [16]. The residual rι is said to d be weakly integral if, and only if, there exists $ ∈ k[s−1 , ds ], $ 6= 0, such that

$rιpert ∈ spank[s−1 , d ] (wιpert , π) ds

The next result is clear: Proposition II.2: Any residual is weakly integral. The residual rιpert is said to be strongly integral if, and only if, rιpert ∈ spank[s−1 , d ] (wιpert , π) ds

III. Three academic examples A. Actuator faults Consider the system   sxpert = −xpert + upert + w1pert  1 1 1   pert pert pert pert   sx2 = −x2 + u2 + w2 pert pert pert sx3 = 2x2 − x3   pert pert pert  = x1 + x2 + π1 y1    y pert = xpert + π 2 2 3 A.1 Fault detectability and isolability    −1 0 0 1 Set A =  0 −1 0 , B = 0 0 2 −1 0     µ ¶ 1 0 1 1 0 , F1 =  0 , F2 =  1 . The 0 0 1 0 0 matrix (see [18], [23], [31]) reads µ ¶ ¡ ¢ 1 1 D = CF1 , CF2 = 0 0

(2)

 0 1, C = 0 detectability

since the characteristic index of w1pert and w2pert is equal to 1. It follows from rk(D) = 1 < 2 that the detectability of the two fault variables cannot be deduced. Isolability of the two fault variables cannot be deduced from the (C, A)-invariant subspace algorithm (see [21], [33] and [28]). Straightforward computations yield however the parity system w1 w2

1 = (1 + s)y1 − (1 + 2s + s2 )y2 − u1 2 1 = (1 + 2s + s2 )y2 − u2 2

5 See, e.g., [34] for this utilisation of elementary noncommutative algebra.

3

The next result, which could not have been obtained from [33], [36], follows at once from the definitions of Section II-B: Proposition III.1: The two fault variables in system (2) are detectable and isolable. A.2 Determination of residuals and numerical simulations If we are starting at time t = 0, initial conditions have to be added to the above parity system (compare with [22], [35])

x1 (0) = 0.5, x2 (0) = 0.5, x3 (0) = 2, ( pert

w1

: (

pert

w2

:

pert

w1 = 0 if t < 40 pert w1 = 0.1 if t ≥ 40 w2pert = 0 if t < 75 w2pert = 0.25 if t ≥ 75

The computer-generated, zero-mean noises π1 and π2 are independent Gaussian white noises with a standard deviation of 0.05 approximately.

w1pert =(y1pert + π1 ) + s(y1pert + π1 ) − (y1pert (0) + π1 (0)) B. Actuator and sensor faults 1 − (y2pert + π2 ) − s(y2pert + π2 ) + (y2pert (0) + π2 (0)) Consider the system 2 1 2 pert  pert − (s (y2 + π2 ) − s(y2pert (0) + π2 (0)) sx1 = −xpert + upert + w1pert  2 1   sxpert = xpert − xpert − (y˙ 2pert (0) + π˙ 2 (0))) − upert 2 1 2 1 (3) pert pert pert  y = x + w + π1  1 1 2   pert y2 = xpert + π2 1 2 w2pert = (y2pert + π2 ) + s(y2pert + π2 ) − (y2pert (0) + π2 (0)) 2 1 ³ 2 pert The variables w1pert and w2pert are respectively actuator + s (y2 + π2 ) − s(y2pert (0) + π2 (0)) 2 fault and sensor faults. ´ − (y˙ 2pert (0) + π˙ 2 (0)) − u2

B.1 Fault detectability and isolability

The initial conditions, which may be unknown, and the derivatives, which are very sensitive to noises, are eliminated by introducing the following strongly integral resid2 pert uals rκW = s−2 ddsw2κ , κ = 1, 2 (compare with [22], [35]). They read in the time domain: pert

r1S

=

1 − t2 (y2pert (t) + π2 (t)) + 2

Z t³

2σ(y2pert (σ) + π2 (σ)) ´ pert pert − σ 2 (y2 (σ) + π2 (σ)) + σ 2 (y1 (σ) + π1 (σ)) dσ Z tZ σ³ + − (y2pert (λ) + π2 (λ)) + 2λ(y2pert (λ) + π2 (λ)) 0

0

0

1 − λ2 (y2pert (λ) + π2 (λ)) − λ2 upert (λ) 1 2 pert

− 2λ(y1

pert

(λ) + π1 (λ)) + λ2 (y1

pert r2S =

1 2 pert t (y2 (t) + π2 (t)) + 2

´ (λ) + π1 (λ)) dλdσ

Z t³

− 2σ(y2pert (σ) + π2 (σ)) Z tZ σ³ ´ + σ 2 (y2pert (σ) + π2 (σ)) dσ + (y2pert (λ) + π2 (λ)) 0

0

pert

− 2λ(y2

0

1 (λ) + π2 (λ)) + λ2 (y2pert (λ) + π2 (λ)) 2 ´

− λ2 upert (λ) dλdσ 2

For preventing wind-ups of iterated integrals, numerical pert pert simulations are given with rκSt = t12 rSκ , κ = 1, 2. There,

The next property is an immediate consequence of the parity system (1), which reads w1 w2

= =

(s2 + 2s + 1)y2 − u y1 − (s + 1) y2

Proposition III.2: The two fault variables in system (3) are detectable and isolable. As in Section III-A.1 the methods of [33], [36] do not yield similar results. B.2 Residuals and numerical simulations Strongly integral residuals where initial conditions do not d2 w

pert

dw

pert

2 appear are provided by s−2 ds12 and s−1 ds . Numerical simulations are given by dividing in the time domain those two expressions respectively by t2 and t in order to prevent wind-ups from iterated integrals:

pert r1S = µ Z t³ ´ 1 2 pert pert 2 pert t y (t) + −4σy (σ) + 2σ y (σ) dσ 2 2 2 t2 0 Z tZ σ³ pert pert + 2y2 (λ) − 4λy2 (λ) 0 0 ¶ ´ 2 pert 2 pert +λ y2 (λ) − λ u (λ) dλdσ pert r2S = µ Z t³ ´ ¶ 1 pert pert pert pert ty2 (t) + −σy1 (σ) − y2 (σ) + σy2 (σ) dσ t 0

4

4.5

1000

800

y1 y2 4

600

3.5 400

3 500

200

2.5 0

2 −200

1.5 0

−400

1 −600

0.5

−800

0

−0.5 0

10

20

30

40

50

60

70

80

90

100

−1000 0

10

20

30

40

50

60

70

80

90

100

−500 0

10

20

pert (b) Residual r1S

(a) Outputs 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

30

40

50

60

70

80

90

100

80

90

100

pert (c) Residual r2S

−0.8

−1 0

10

200

30

40

50

60

70

80

90

−1 0

100

10

20

pert (d) Residual r1St

30

40

50

60

70

80

90

100

pert (e) Residual r2St

Fig. 1. Simulation with output noise

5

70

y1 y2

60

60

50

4 50

40

3 40 30

2

30 20 20

1 10 10

0

0

0

−1 0

10

20

30

40

50

60

70

80

90

100

−10 0

10

20

30

40

50

60

70

80

90

100

−10 0

10

pert (b) Residual r1S

(a) Outputs

20

30

40

50

60

70

pert (c) Residual r2S

Fig. 2. Simulation with output noise and faults pert

Set x1

pert

(0) = 0.5, x2

(0) = 2 and

C. Fault diagnosis in closed-loop C.1 A PI controller for an uncertain system

½ pert

w1

(t) =

½ pert

w2

(t) =

0 if t < 70 0.2 if t ≥ 70

0 if t < 30 0.2 if t ≥ 30

Figures 3 and 4 are demonstrating good performances in spite of computer-generated random noises.

Consider the system  pert sx1 = −2xpert + aupert  1 1   sxpert = 4xpert − 3xpert + aupert + wpert 2 1 2 2 pert pert  y = x + π 1  1 1   pert y2 = xpert + π 2 2

(4)

where a ∈ (amin , amin ) ⊂ R is an uncertain parameter, and wpert an actuator fault. If the interval (amin , amin ) is small enough around the nominal value 1, a satisfactory PI controller for system (4) is: ! µ à ¶µ ¶ 2.1 0 −4.2 0 e upert 1 = e 4 1.5 0 −5.06 upert s 2

5

2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

y1 y2

1.5

1

0.5

0

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.5 0

10

20

30

40

50

60

70

80

90

100

−1 0

−0.8

10

20

30

40

50

60

70

80

90

−1 0

100

10

(b) Residual r1pert

(a) Outputs

20

30

40

50

60

70

80

90

100

(c) Residual r2pert

Fig. 3. Simulation in the fault free case

60

2

2

y1 y2

50

0

1.5

40 −2 1

30 −4 20

0.5

−6 10 0

−8

0

−0.5 0

10

20

30

40

50

60

70

80

90

(a) Outputs y(0) 6= 0

−10 0 100

10

20

30

40

50

60

70

80

90

100

−10 0

10

pert (b) Residual r1t

20

30

40

50

60

70

80

90

100

pert (c) Residual r2t

Fig. 4. Simulation in the fault case

where yιref , ι = 1, 2, in e = output.

! Ã pert y1ref − y1 pert , is a reference y2ref − y2

y ref (t) = 1

½

0 if t < 40 0.5 if t ≥ 40

C.2 A closed-loop residual with numerical simulations In the following strongly integral residual, which has been obtained as in the two previous examples, r = − y1pert (t) Z µ 1 t pert pert pert 4σy1 (σ) + y2 (σ) − 3σy2 (σ) + t 0 ¶ pert + ae σu2 (σ) dσ ae is the value of the uncertain parameter a given by the estimator (see [15]) ´ R t ³ pert pert pert ty1 − 0 y1 (σ) − 2σy1 (σ) dσ ae = R t pert σy1 (σ)dσ 0 The simulations are realised with the same output noise as before, xpert (0) = 0.5, xpert (0) = 2, and 1 2 ½ 0 if t < 70 wpert (t) = 0.2 if t ≥ 70 y1ref (t) =

½

1 if t < 20 2.5 if t ≥ 20

Figures 5-(c) and 6-(c) show the parameter estimation respectively in the fault free case and in the fault case. The residual evolution in figures 6-(d) ensures the detection in the fault case6 , with the closed-loop PI controller, which yields an excellent convergence7 to the reference output even with a fault apparition (see figure 6-(a)).

IV. A concrete case study: the three thank system

The pilot plant [1] depicted in figure 7 is composed of three identical cylindrical tanks. From the nonlinear model (see [42], and [27], [37]) we obtain the following linearised

6 A future publication will obtain similar results without having recourse to any estimator of the uncertain parameter a. 7 A flatness-based control strategy (see, e.g., [12]) would yield even better results.

6

6

3

u1 u2

y1 y2 2.5

4

2

2

1.5

0 1

−2 0.5

−4 0

−0.5 0

10

20

30

40

50

60

70

80

90

100

−6 0

10

20

30

(a) Outputs

40

50

60

70

80

90

100

70

80

90

100

(b) Inputs

2.5

1

2.4

0.8

2.3

0.6

2.2

0.4

2.1

0.2

2

0

1.9

−0.2

1.8

−0.4

1.7

−0.6

1.6

−0.8

1.5 0

10

20

30

40

50

60

70

80

90

−1 0

100

10

20

(c) Parameter estimation ae

30

40

50

60

(d) Residual

Fig. 5. Simulation in the fault free case

6

3

u1 u2

y1 y2 2.5

4

2

2

1.5

0 1

−2 0.5

−4 0

−0.5 0

10

20

30

40

50

60

70

80

90

100

−6 0

10

20

30

(a) Outputs

40

50

60

70

80

90

100

70

80

90

100

(b) Inputs

2.5

2

2.4 0

2.3 −2

2.2

2.1 −4

2 −6

1.9

1.8

−8

1.7 −10

1.6

1.5 0

10

20

30

40

50

60

70

80

90

100

−12 0

(c) Parameter estimation ae

10

20

30

40

50

60

(d) Residual

Fig. 6. Simulation in the fault case

system which is valid around an operating point:  0 pert 1 0 1 0 pert x1 x1 −α 0 α    pert C = @ 0 pert AB s B −(β + γ) β @ x2 A @ x2    pert pert α β −(α + β)  x3 x   0 1 3 011   0  S  pert @ 1 A pert @ A pert +

   

0 0

(u1

+ w1

)+

S

0

(u2

1 C A

pert + w2 )

The coefficients S, α, β, γ are related to the physical properties of the tanks. In contrast to the existing literature we are considering several actuator and sensor faults.

7

Pomp 2

Pomp 1

q1

q2

S S S x1

x3 q13

(S Tank 1

p

q32

, µ12 )

(S

p

x2

, µ 32 )

Tank 3

(S Tank 2

p

, µ 20 )

q20

Fig. 7. Diagram of the three-tank system

A. Fault detectability and isolability The parity system (1) reads here ·µ

r3pert = µ 1 t y3 (t) t α Z t³ ´ ¶ 1 α+β β + σy3 (σ) dσ − σy1 − σy2 (σ) − y3 (σ) + α α β 0

¶ µ 2 ¶ ¸ β s β w1 = S − s − β y2 + + (2 + )s + β y3 q − u1 α α α Numerical simulations were obtained around the operating point (0.4, 0.3, 0.2). Then S = 0.0154, α = β = 0.0227, w2 = S ((s + β + γ) y2 − βy3 ) − u2 γ = 0.0217. The random perturbations π1 , π2 , π3 are comw3 = (y1 + βy2 − (s + α + β) y3 ) puter generated zero-mean Gaussian white noises, with a standard deviation of 0.01 approximately. The next result, which follows at once, could here also not Figure 8 depicts the behaviour in the fault free case. Note have been derived by the techniques of [33], [36]. that residuals are approximatively 0 whereas the initial Proposition IV.1: The three fault variables in system (5) conditions (0.41, 0.22, 0.33) are different from the operating are detectable and isolable. point. With ½ 0 if t < 5 B. Residuals and numerical simulations w1pert (t) = 0.005 if t ≥ 5 The following residuals were obtained as in Sections III½ 0 if t < 15 A.2 and III-B.2: pert w2 (t) = 0.005 if t ≥ 15 ½ r1pert = 0 if t < 40 µ w3pert (t) = 1 S 2 0.1 if t ≥ 40 t y3 (t) t2 α figure 9 demonstrates the efficiency of our method. ¶ Z tµ βS 2 4S β 2 − σ y2 (σ) + − σy3 (σ) + S(2 + )σ y3 (σ) dσ + V. Conclusion α α α 0 Z tZ σ³ Interested readers have certainly noticed the complete 2βS 2S + λy2 (λ) − βSλ2 y2 (λ) + y3 (λ) absence of any optimisation and/or probabilistic technique α α 0 0 ¶ (see, also, [15], [16]). ´ βS Issues concerning robustness and feedback loops will be − (−4S − )λy3 (λ) + βSλ2 y3 (λ) − λ2 u1 dλdσ α further developed. The discrete-time analogue will be treated elsewhere8 . The extension to nonlinear systems of our intrinsic definitions of fault detectability and isolar2pert = bility, which were obtained without introducing any ad hoc µ 1 filters, will be employing differential algebra (see, e.g., [6], − Sty2 (t) t Z t³ ´ ¶ 8 Our algebraic setting has already been useful for studying discrete+ Sy2 (σ) − (β + γ)Sσy2 (σ) + βSσy3 (σ) + σu2 (σ) dσ time predictive control [13] and identification [39], and some types of 0

error-control codes [9].

8

0.45 y1 y2 y3 0.4

0.35

0.3

0.25

0.2

0.15 0

10

20

30

40

50

60

70

80

90

100

(a) Outputs 1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

−0.2

−0.2

−0.2

−0.4

−0.4

−0.4

−0.6

−0.6

−0.6

−0.8

−0.8

−1 0

10

20

30

40

50

60

70

80

90

100

−0.8

−1 0

10

(b) Residual r1pert

20

30

40

50

60

70

80

90

100

−1 0

10

(c) Residual r2pert

20

30

40

50

60

70

80

90

100

90

100

(d) Residual r3pert

Fig. 8. Simulation in the fault free case

4.5

0.5

0.1

4

0 0.05

3.5

−0.5 0

3

−1

2.5

−1.5 −0.05

2

−2 −0.1

1.5

1

−2.5

−3

−0.15

0.5

−3.5 −0.2

0

−0.5 0

−4

10

20

30

40

50

60

70

80

(a) Residual r1pert

90

100

−0.25 0

10

20

30

40

50

60

70

80

90

100

(b) Residual r2pert

−4.5 0

10

20

30

40

50

60

70

80

(c) Residual r3pert

Fig. 9. Simulation in the fault case

[7], [10]) and/or differential geometry of infinite jets (see, e.g., [11]). This latter case has been already studied by several authors (see, e.g., [3], as well as [4], [5], [20], [21], and [27], [28], [29]). References [1] [2] [3] [4] [5] [6]

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