Applications of computer algebra in the identifiability and - - PowerPoint PPT Presentation

Applications of computer algebra in the identifiability and diagnosability studies

Nathalie Verdière1

1Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France

CUNY, 2019

My topics/works: Identifiability study and parameter estimation (testing an assumption in pharmacokinetic domain) - L. Denis-Vidal, G. Joly-Blanchard

• 1. Infection of a cell by an extracellular bacterium

2 Injection of macromolecules

My topics/works: Identifiability study and parameter estimation (testing an assumption in pharmacokinetic domain) - L. Denis-Vidal, G. Joly-Blanchard

• 1. Infection of a cell by an extracellular bacterium

2 Injection of macromolecules Fault diagnosability, fault diagnosis (malfunction in a navigation system) - L. Travé-Massuyès, C. Jauberthie, S Orange

My topics/works: Identifiability study and parameter estimation (testing an assumption in pharmacokinetic domain) - L. Denis-Vidal, G. Joly-Blanchard

• 1. Infection of a cell by an extracellular bacterium

2 Injection of macromolecules Fault diagnosability, fault diagnosis (malfunction in a navigation system) - L. Travé-Massuyès, C. Jauberthie, S Orange Identifiabiliy, identification, diagnosability in the context of bounded-error uncertain models - L. Travé-Massuyès, C. Jauberthie

My topics/works: Identifiability study and parameter estimation (testing an assumption in pharmacokinetic domain) - L. Denis-Vidal, G. Joly-Blanchard

• 1. Infection of a cell by an extracellular bacterium

2 Injection of macromolecules Fault diagnosability, fault diagnosis (malfunction in a navigation system) - L. Travé-Massuyès, C. Jauberthie, S Orange Identifiabiliy, identification, diagnosability in the context of bounded-error uncertain models - L. Travé-Massuyès, C. Jauberthie Identifiability in PDE’s models (D. Manceau, L. Denis-Vidal, S. Zhu)

Outline

1 Identifiability Definition Example General result 2 Fault diagnosability and fault diagnosis Definitions Example Link between identifiability and diagnosability Multiple faults Example of a two coupled water tank 3 Conclusion and perspectives 4 Bibliography

Identifiability Definition

Γp ˙ x(t, p) = f(x(t, p), u(t), p), y(t, p) = h(x(t, p), p). (1) x(t, p) ∈ Rn: state variables at time t, y(t, p) ∈ Rm: output vector at time t, u(t) ∈ Rr: input vector at time t, f, h: real functions, analytic on M (an open set of Rn), p ∈ UP: vector of parameters, UP ⊂ Rp: an a priori known set of admissible parameters.

Identifiability Definition

Γp ˙ x(t, p) = f(x(t, p), u(t), p), y(t, p) = h(x(t, p), p). (1) x(t, p) ∈ Rn: state variables at time t, y(t, p) ∈ Rm: output vector at time t, u(t) ∈ Rr: input vector at time t, f, h: real functions, analytic on M (an open set of Rn), p ∈ UP: vector of parameters, UP ⊂ Rp: an a priori known set of admissible parameters. Two problems can be considered: The forward problem: given p, u, find x and y. The inverse problem: given y and u, estimate p.

1

Identifiability problem : From the output(s) of the model, is it possible to estimate uniquely the parameter vector p? If the answer is YES, then the model is said identifiable.

2

Identification problem

Identifiability Definition

Γp ˙ x(t, p) = f(x(t, p), u(t), p), y(t, p) = h(x(t, p), p). (1) x(t, p) ∈ Rn: state variables at time t, y(t, p) ∈ Rm: output vector at time t, u(t) ∈ Rr: input vector at time t, f, h: real functions, analytic on M (an open set of Rn), p ∈ UP: vector of parameters, UP ⊂ Rp: an a priori known set of admissible parameters. Two problems can be considered: The forward problem: given p, u, find x and y. The inverse problem: given y and u, estimate p.

1

Identifiability problem : From the output(s) of the model, is it possible to estimate uniquely the parameter vector p? If the answer is YES, then the model is said identifiable.

2

Identification problem ⇒ Input-output method based on the Rosenfeld-Groebner algorithm (implemented in Maple by F . Boulier, CRIStAL) and based on differential algebra approach (Kolchin and al., 1973)

Identifiability Definition

Formalization

Controlled models (u = 0) WITHOUT initial condition; (˜ x, ˜ y) = unique set of solutions The model is globally identifiable if there exists an input u such that, for all p ∈ Up,

• ne gets

˜ y(t, p) = ∅, ˜ y(t, p) ∩ ˜ y(t, ¯ p) = ∅, ∀ t ≥ 0, ¯ p ∈ Up    ⇒ p = ¯ p. (2) The model is locally identifiable if it is globally identifiable in an open neighborhood v(p) ⊂ Up of p.

Identifiability Definition

Formalization

Controlled models (u = 0) WITHOUT initial condition; (˜ x, ˜ y) = unique set of solutions The model is globally identifiable if there exists an input u such that, for all p ∈ Up,

• ne gets

˜ y(t, p) = ∅, ˜ y(t, p) ∩ ˜ y(t, ¯ p) = ∅, ∀ t ≥ 0, ¯ p ∈ Up    ⇒ p = ¯ p. (2) The model is locally identifiable if it is globally identifiable in an open neighborhood v(p) ⊂ Up of p. Controlled model (u = 0) WITH initial conditions; (x, y) unique solution The model is globally identifiable if there exists an input u such that, for all p, ¯ p ∈ Up, there exists t1 > 0 such that if for all t ∈ [0, t1], the equalities y(t, p) = y(t, ¯ p) implies that p = ¯ p. The model is locally identifiable if it is globally in an open neighborhood v(p) ⊂ Up

• f p.

Identifiability Example

Example:        ˙ x1 = k12(x2 − x1) − kνx1 1 + x1 , ˙ x2 = k21(x1 − x2), y = x1,

1

Rosenfeld-Groebner algorithm with the elimination order [y] ≺ [x1, x2]: C(p) = {k12 y2 ˙ y + k21 kν y2 + k21 y2 ˙ y + 2 k12 y ˙ y + k21 kν y + 2 k21 y ˙ y + y2 ¨ y +k12 ˙ y + k21 ˙ y + kν ˙ y + 2 y ¨ y + ¨ y, x1 − y, k12 x2 y − k12 y2 + k12 x2 − k12 y − kν y − y ˙ y − ˙ y}.

Identifiability Example

Example:        ˙ x1 = k12(x2 − x1) − kνx1 1 + x1 , ˙ x2 = k21(x1 − x2), y = x1,

1

Rosenfeld-Groebner algorithm with the elimination order [y] ≺ [x1, x2]: C(p) = {k12 y2 ˙ y + k21 kν y2 + k21 y2 ˙ y + 2 k12 y ˙ y + k21 kν y + 2 k21 y ˙ y + y2 ¨ y +k12 ˙ y + k21 ˙ y + kν ˙ y + 2 y ¨ y + ¨ y, x1 − y, k12 x2 y − k12 y2 + k12 x2 − k12 y − kν y − y ˙ y − ˙ y}.

2

Keep the IO polynomial (p = (k12, k21, kν), γ(p) = (k12 + k21, k21 kν, k21 + kν)): P(y, p) = ¨ y + 2 y ¨ y + y2 ¨ y + γ1(p) (y2 ˙ y + 2y ˙ y) + γ2(p) (y2 + y) + γ3(p) ˙ y

Identifiability Example

Example:        ˙ x1 = k12(x2 − x1) − kνx1 1 + x1 , ˙ x2 = k21(x1 − x2), y = x1,

1

Rosenfeld-Groebner algorithm with the elimination order [y] ≺ [x1, x2]: C(p) = {k12 y2 ˙ y + k21 kν y2 + k21 y2 ˙ y + 2 k12 y ˙ y + k21 kν y + 2 k21 y ˙ y + y2 ¨ y +k12 ˙ y + k21 ˙ y + kν ˙ y + 2 y ¨ y + ¨ y, x1 − y, k12 x2 y − k12 y2 + k12 x2 − k12 y − kν y − y ˙ y − ˙ y}.

2

Keep the IO polynomial (p = (k12, k21, kν), γ(p) = (k12 + k21, k21 kν, k21 + kν)): P(y, p) = ¨ y + 2 y ¨ y + y2 ¨ y + γ1(p) (y2 ˙ y + 2y ˙ y) + γ2(p) (y2 + y) + γ3(p) ˙ y

3

Suppose that y(t, p) = y(t, ¯ p): P(y, p) − P(y, ¯ p) = (γ1(p) − γ1(¯ p)) (y2 ˙ y + 2y ˙ y) +(γ2(p) − γ2(¯ p)) (y2 + y) +(γ3(p) − γ3(¯ p)) ˙ y = 0. Remark: det(y2 ˙ y + 2 y ˙ y, (y2 + y), ˙ y) = −2 ˙ y (− ˙ y4 + (3 y2 ¨ y + 5 y ¨ y + 2 ¨ y) ˙ y2 + y ... y (y + 1)2 ˙ y − 3 y ¨ y2 (y + 1)2) ≡ 0.

Identifiability Example

Example:        ˙ x1 = k12(x2 − x1) − kνx1 1 + x1 , ˙ x2 = k21(x1 − x2), y = x1,

1

Rosenfeld-Groebner algorithm with the elimination order [y] ≺ [x1, x2]: C(p) = {k12 y2 ˙ y + k21 kν y2 + k21 y2 ˙ y + 2 k12 y ˙ y + k21 kν y + 2 k21 y ˙ y + y2 ¨ y +k12 ˙ y + k21 ˙ y + kν ˙ y + 2 y ¨ y + ¨ y, x1 − y, k12 x2 y − k12 y2 + k12 x2 − k12 y − kν y − y ˙ y − ˙ y}.

2

Keep the IO polynomial (p = (k12, k21, kν), γ(p) = (k12 + k21, k21 kν, k21 + kν)): P(y, p) = ¨ y + 2 y ¨ y + y2 ¨ y + γ1(p) (y2 ˙ y + 2y ˙ y) + γ2(p) (y2 + y) + γ3(p) ˙ y

3

Suppose that y(t, p) = y(t, ¯ p): P(y, p) − P(y, ¯ p) = (γ1(p) − γ1(¯ p)) (y2 ˙ y + 2y ˙ y) +(γ2(p) − γ2(¯ p)) (y2 + y) +(γ3(p) − γ3(¯ p)) ˙ y = 0. Remark: det(y2 ˙ y + 2 y ˙ y, (y2 + y), ˙ y) = −2 ˙ y (− ˙ y4 + (3 y2 ¨ y + 5 y ¨ y + 2 ¨ y) ˙ y2 + y ... y (y + 1)2 ˙ y − 3 y ¨ y2 (y + 1)2) ≡ 0.

4

Study of γ(p) − γ(¯ p) = 0: γ(p) − γ(¯ p) = 0 ⇒ p = ¯ p Conclusion: the model is identifiable.

Identifiability General result

Afterwards, 1 observation (i = 1). P(y, u, p) = m0(y, u) +

q

• k=1

γk(p)mk(y, u) = 0, (γk(p))1≤k≤q is called the exhaustive summary. Proposition (L. Denis-Vidal) If (mk(y, u))1≤k≤q are linearly independent then the model is globally identifiable at p if for all ¯ p ∈ Up ∀k = 1, . . . , q, γk(¯ p) = γk(p) ⇒ p = ¯ p. (3)

Identifiability General result

Afterwards, 1 observation (i = 1). P(y, u, p) = m0(y, u) +

q

• k=1

γk(p)mk(y, u) = 0, (γk(p))1≤k≤q is called the exhaustive summary. Proposition (L. Denis-Vidal) If (mk(y, u))1≤k≤q are linearly independent then the model is globally identifiable at p if for all ¯ p ∈ Up ∀k = 1, . . . , q, γk(¯ p) = γk(p) ⇒ p = ¯ p. (3) Remarks If φ(p) = (γk(p))k=1,...,q, (3) consists in verifying that φ is injective. This injectivity study can provide:

the strategy to reparametrize unidentifiable ODE models into identifiable

• nes (Evans 2000, Meshkat 2011)

the strategy to determine the key parameters to estimate in order to obtain the identifiability of some important biological parameters (Verdière 2018) the biological protocols to put in place to estimate parameters in the case of a model containing an important number of parameters (Csercsik 2012).

Identifiability General result

Afterwards, 1 observation (i = 1). P(y, u, p) = m0(y, u) +

q

• k=1

γk(p)mk(y, u) = 0, (γk(p))1≤k≤q is called the exhaustive summary. Proposition (L. Denis-Vidal) If (mk(y, u))1≤k≤q are linearly independent then the model is globally identifiable at p if for all ¯ p ∈ Up ∀k = 1, . . . , q, γk(¯ p) = γk(p) ⇒ p = ¯ p. (3) Remarks If φ(p) = (γk(p))k=1,...,q, (3) consists in verifying that φ is injective. Wronskian of the sequence of functions (φ1, . . . , φs) is defined by: Wronskian = Det(φ1, . . . , φs) =

• φ1

. . . φs ˙ φ1 . . . ˙ φs . . . . . . . . . φ(s−1)

1

. . . φ(s−1)

s

• .

(4) The functions (mk(y, u))k=1,...,q are linearly independent if the functional determinant is not identically equal to zero. It is sufficient to find a time point.

Identifiability General result

Afterwards, 1 observation (i = 1). P(y, u, p) = m0(y, u) +

q

• k=1

γk(p)mk(y, u) = 0, (γk(p))1≤k≤q is called the exhaustive summary. Proposition (L. Denis-Vidal) If (mk(y, u))1≤k≤q are linearly independent then the model is globally identifiable at p if for all ¯ p ∈ Up ∀k = 1, . . . , q, γk(¯ p) = γk(p) ⇒ p = ¯ p. (3) Remarks If φ(p) = (γk(p))k=1,...,q, (3) consists in verifying that φ is injective. Wronskian of the sequence of functions (φ1, . . . , φs) IO polynomial can be used to have a first estimation of (γk(p))k=1,...,q and so the parameters.

Fault diagnosability and fault diagnosis Definitions

• N. Verdière, C. Jauberthie, Louise Travé-Massuyès, Functional diagnosability and

detectability of nonlinear models based on analytical redundancy relations, Journal of Process Control, Vol. 35, 1-10; 2014.

• N. Verdière, S. Orange, Diagnosability in the case of multi-faults in nonlinear

models, Journal of Process Control, Vol 69, pp. 1-7, 2018. Definitions A fault is an unpermitted deviation of at least one parameter of the system from the acceptable standard condition. Fault diagnosability establishes which faults can be discriminated using the available sensors in a system. Fault diagnosis consists in fault detection of the malfunction of a system and the fault isolation of the faulty component.

Fault diagnosability and fault diagnosis Definitions

• N. Verdière, C. Jauberthie, Louise Travé-Massuyès, Functional diagnosability and

detectability of nonlinear models based on analytical redundancy relations, Journal of Process Control, Vol. 35, 1-10; 2014.

• N. Verdière, S. Orange, Diagnosability in the case of multi-faults in nonlinear

models, Journal of Process Control, Vol 69, pp. 1-7, 2018. Definitions A fault is an unpermitted deviation of at least one parameter of the system from the acceptable standard condition. Fault diagnosability establishes which faults can be discriminated using the available sensors in a system. Fault diagnosis consists in fault detection of the malfunction of a system and the fault isolation of the faulty component. The models          ˙ x(t, p, f) = g(x(t, p), u(t), f, p), y(t, p, f) = h(x(t, p), u(t), f, p), x(t0, p, f) = x0, t0 ≤ t ≤ T. (5) f = 0 means no fault. In the case of uncontrolled models u = 0.

Fault diagnosability and fault diagnosis Example

Bernoulli equation: ˙ y(t) = β1y(t) + β2y(t)2, for t ∈ [0, 5], y(0) = −1. (6) Assumption: two positive single faults f1 and f2 impact additively the two parameters β1 and β2 respectively; p = (β1, β2)T , f = (f1, f2)T . ˙ y(t) = (β1 + f1)y(t) + (β2 + f2)y(t)2, for t ∈ [0, 5], y(0) = −1.

Fault diagnosability and fault diagnosis Example

Bernoulli equation: ˙ y(t) = β1y(t) + β2y(t)2, for t ∈ [0, 5], y(0) = −1. (6) Assumption: two positive single faults f1 and f2 impact additively the two parameters β1 and β2 respectively; p = (β1, β2)T , f = (f1, f2)T . ˙ y(t) = (β1 + f1)y(t) + (β2 + f2)y(t)2, for t ∈ [0, 5], y(0) = −1. Method

1

IO relation (or ARRs for Analytical redundancy relations) IO polynomial: P(¯ y, p, f) = w0(¯ y, p) − w1(¯ y, f, p) = 0 with w0(¯ y, p) = ˙ y(t) − β1y(t) − β2y(t)2 w1(¯ y, f, p) = f1y(t, p, f) + f2y(t, p, f)2. (residual computation form) (residual internal form).

Fault diagnosability and fault diagnosis Example

Bernoulli equation: ˙ y(t) = β1y(t) + β2y(t)2, for t ∈ [0, 5], y(0) = −1. (6) Assumption: two positive single faults f1 and f2 impact additively the two parameters β1 and β2 respectively; p = (β1, β2)T , f = (f1, f2)T . ˙ y(t) = (β1 + f1)y(t) + (β2 + f2)y(t)2, for t ∈ [0, 5], y(0) = −1. Method

1

IO relation (or ARRs for Analytical redundancy relations) IO polynomial: P(¯ y, p, f) = w0(¯ y, p) − w1(¯ y, f, p) = 0 with w0(¯ y, p) = ˙ y(t) − β1y(t) − β2y(t)2 w1(¯ y, f, p) = f1y(t, p, f) + f2y(t, p, f)2. (residual computation form) (residual internal form).

2

Define fault signatures

• Classical signature: Sig(f1) = Sig(f2) = 1
• Functional signature (f[1] = (f1, 0)T , f[2] = (0, f2)T ):

FSig(f1) = w1(¯ y, f[1]) = f1y(., p, f[1]) FSig(f2) = w1(¯ y, f[2]) = f2y(., p, f[2])2. (y(., p, f[i]) denotes the output impacted by fi only).

Fault diagnosability and fault diagnosis Example

Bernoulli equation: ˙ y(t) = β1y(t) + β2y(t)2, for t ∈ [0, 5], y(0) = −1. (6) Assumption: two positive single faults f1 and f2 impact additively the two parameters β1 and β2 respectively; p = (β1, β2)T , f = (f1, f2)T . ˙ y(t) = (β1 + f1)y(t) + (β2 + f2)y(t)2, for t ∈ [0, 5], y(0) = −1. Method

1

IO relation (or ARRs for Analytical redundancy relations) IO polynomial: P(¯ y, p, f) = w0(¯ y, p) − w1(¯ y, f, p) = 0 with w0(¯ y, p) = ˙ y(t) − β1y(t) − β2y(t)2 w1(¯ y, f, p) = f1y(t, p, f) + f2y(t, p, f)2. (residual computation form) (residual internal form).

2

Define fault signatures

• Classical signature: Sig(f1) = Sig(f2) = 1
• Functional signature (f[1] = (f1, 0)T , f[2] = (0, f2)T ):

FSig(f1) = w1(¯ y, f[1]) = f1y(., p, f[1]) FSig(f2) = w1(¯ y, f[2]) = f2y(., p, f[2])2. (y(., p, f[i]) denotes the output impacted by fi only).

3

Study fault signatures to know if the faults are discriminable If FSig(f1) = FSig(f2), then the two faults are said discriminable and the model diagnosable.

Fault diagnosability and fault diagnosis Example

Bernoulli equation: ˙ y(t) = β1y(t) + β2y(t)2, for t ∈ [0, 5], y(0) = −1. (7) β1 = 1, β2 = 2. Output is disturbed by a Gaussian noise so that the relative error has a maximal value of 0.1. A permanent fault appears at time t = 1.5s and its value varies between 0.1 and 1. Residual: ρ := w0(¯ y, p) = ˙ y(t) − β1y(t) − β2y(t)2.

Figure: Bernoulli residual when a

fault acts on β1 (= FSig(f1))

Figure: Bernoulli residual when a

fault acts on β2 (= FSig(f2)) Remark: The two faults are detectable at t = 1.5s (ρ = 0).

Fault diagnosability and fault diagnosis Example

Rosenfeld-Groebner algorithm: Pi(¯ y, ¯ u, p) = w0,i(¯ y, ¯ u, p) − w1,i(¯ y, ¯ u, f, p) = 0 (8) with w1,i(¯ y, ¯ u, f, p) =

ni

• k=1

γi

k(f, p)mk,i(¯

y, ¯ u). Definitions - Functional fault signature, functionally discriminable/diagnosable Functional fault signature: FSig(fj) = (w1,i(¯ y, ¯ u, f[j], p))i=1,...,m. Two faults fj and fl are functionally discriminable if for all input u, there exists at least

• ne index i∗ and a finite time t1 ∈]t0, T] such that for all t ∈ [t0, t1],

FSig(i∗)(fj) = FSig(i∗)(fl). When all the faults are functionally discriminable, the model is said functionally diagnosable. Definition - Functionally detectable The fault fj is functionally detectable if the functional signature FSig(fj) is not equal to the null vector.

Fault diagnosability and fault diagnosis Link between identifiability and diagnosability

Proposition: Link between the identifiability and the diagnosability If a model is identifiable with respect to the faults then it is diagnosable. The reciprocal is false.

Fault diagnosability and fault diagnosis Link between identifiability and diagnosability

Proposition: Link between the identifiability and the diagnosability If a model is identifiable with respect to the faults then it is diagnosable. The reciprocal is false. Example: Bernoulli equation ˙ y(t) = β1y(t) + β2y(t)2. (9) Assumption: two positive single faults f1 and f2 impact additively the two parameters β1 and β2 respectively; p = (β1, β2)T , f = (f1, f2)T .

1

IO polynomial P(y, p, f) = ˙ y(t) − β1y(t) − β2y(t)2 − f1y(t, p, f) − f2y(t, p, f)2

2

det(y, y2) ≡ 0

3

Let φ(f) = (−f1, −f2), φ is injective. The model is identifiable with respect to f, then it is diagnosable.

Fault diagnosability and fault diagnosis Link between identifiability and diagnosability

Proposition: Link between the identifiability and the diagnosability If a model is identifiable with respect to the faults then it is diagnosable. The reciprocal is false. Example: ˙ y(t, p) = (p1 − p2)y(t, p), y(0) = 1. Solution: y(t, p) = e(p1−p2)t, t ∈ R, p = (p1, p2)T . Assumption: two positive and additive single faults f1 and f2 impact the two parameters p1 and p2 respectively. ˙ y(t, p) = (p1 + f1 − p2 − f2)y(t, p), y(0) = 1. Diagnosability: w1(¯ y, f, p) = (f1 − f2)y(t, p, f) ⇒ FSig(f1) = f1y(t, p, f[1]) = f1e(p1+f1−p2)t > 0, FSig(f2) = −f2y(t, p, f[2]) = −f2e(p1−p2−f2)t < 0. (10) FSig(f1) = FSig(f2) ⇒ The model is diagnosable. Identifiability: φ : (f1, f2) → f1 − f2 is not injective ⇒ the model is not identifiable with respect to the faults.

Fault diagnosability and fault diagnosis Multiple faults

Example: Mass (m = 1) attached to an elastic spring (force k) u external force (≡ 0), d ≥ 1 ¨ x + k x − du = 0 ⇔ ˙ x1 = x2, ˙ x2 = −kx1 + du. (11) f1 ∈ [0, 2), f2 ∈ [0, 2) two faults such that:    ˙ x1 = x2, ˙ x2 = −k(f1 − 1)2x1 + (d + f2)u, y = x1 (12) IO polynomial: P(y, p, f) = ¨ y + k (f1 − 1)2 y − (d + f2)u = 0 Exhaustive summary: φ(f) =

• k(f1 − 1)2, −d − f2
• Signatures:
• Functional fault signature: FSig(f1) = −k(f12 − 2f1)y, FSig(f2) = f2u.
• Algebraic signature: ASig(f) =
• k(f1 − 1)2, −d − f2
• .

Fault diagnosability and fault diagnosis Multiple faults

N, N ′: subsets of {1, . . . , e} (e = 2, N = {1}, N ′ = {2}) fN the multiple fault vector whose components fi are not equal to 0 if i ∈ N and equal to 0 otherwise (f{1} = (f1, 0), f{2} = (0, f2)) fN belongs to FN = {f ∈ F|fi = 0 if i ∈ N and fi = 0 if i / ∈ N }, F subset of Re Definitions ASig is a function which associates to f a vector of algebraic expressions admitting f1, · · · , fe as indeterminates The multiple faults of FN and of FN ′ are said algebraically discriminable if there exists an algebraic signature such that, for all input u, ASig(FN ) ∩ ASig(FN ′) = ∅. This equality is in particular satisfied when there exists an index i such that ASigi(FN ) ∩ ASigi(FN ′) = ∅. If all the multiple faults are algebraically discriminable, the model is said algebraically diagnosable.

Fault diagnosability and fault diagnosis Multiple faults

N, N ′: subsets of {1, . . . , e} (e = 2, N = {1}, N ′ = {2}) fN the multiple fault vector whose components fi are not equal to 0 if i ∈ N and equal to 0 otherwise (f{1} = (f1, 0), f{2} = (0, f2)) fN belongs to FN = {f ∈ F|fi = 0 if i ∈ N and fi = 0 if i / ∈ N }, F subset of Re Definitions ASig is a function which associates to f a vector of algebraic expressions admitting f1, · · · , fe as indeterminates The multiple faults of FN and of FN ′ are said input-strongly algebraically discriminable (IAD) if, there exists an algebraic signature such that, for all input u, ASig(FN ) ∩ ASig(FN ′) = ∅. This equality is in particular satisfied when there exists an index i such that ASigi(FN ) ∩ ASigi(FN ′) = ∅. If all the multiple faults are algebraically discriminable, the model is said algebraically diagnosable. Example: ¨ x + k(f1 − 1)2 x − (d + f2)u = 0, f1 ∈ [0, 2), f2 ∈ [0, 2) Algebraic signature: ASig(f) =

• k(f1 − 1)2, −d − f2
• .

ASig(f∅) = (k, −d), ASig(f{1}) = (k(f1 − 1)2, −d), ASig(f{2}) = (k, −d − f2) and ASig(f{1,2}) = (k(f1 − 1)2, −d − f2). ASig(f{1}) ∩ ASig(f{1, 2}) = ∅ since for any f2 ∈ (0, 2), −d = −d − f2.

Fault diagnosability and fault diagnosis Multiple faults

N, N ′: subsets of {1, . . . , e} (e = 2, N = {1}, N ′ = {2}) fN the multiple fault vector whose components fi are not equal to 0 if i ∈ N and equal to 0 otherwise (f{1} = (f1, 0), f{2} = (0, f2)) fN belongs to FN = {f ∈ F|fi = 0 if i ∈ N and fi = 0 if i / ∈ N }, F subset of Re Definitions ASig is a function which associates to f a vector of algebraic expressions admitting f1, · · · , fe as indeterminates The multiple faults of FN and of FN ′ are said input-strongly algebraically (IAD) discriminable if, there exists an algebraic signature such that, for all input u, ASig(FN ) ∩ ASig(FN ′) = ∅. This equality is in particular satisfied when there exists an index i such that ASigi(FN ) ∩ ASigi(FN ′) = ∅. If all the multiple faults are IAD discriminable, the model is said IAD diagnosable. Remark The algebraic signature defined by the exhaustive summary is not sufficiently discriminant!

Fault diagnosability and fault diagnosis Multiple faults

P(y, u, p, f) = m0(y, u) +

q

• k=1

γk(p, f)mk(y, u) = 0, The considered system:      γ1(p, f) = φ1, . . . γq(p, f) = φq, (13) Algorithm Algebraic-Signature Groebner basis computation Algebraic signature: each of its component depends only on φk and the parameters of the system By construction, each component of the algebraic signature vanishes when at least one specific (multiple) fault occurs.

Fault diagnosability and fault diagnosis Multiple faults

P(y, u, p, f) = m0(y, u) +

q

• k=1

γk(p, f)mk(y, u) = 0, The considered system:      γ1(p, f) = φ1, . . . γq(p, f) = φq, (13) Algorithm Algebraic-Signature Groebner basis computation Algebraic signature: each of its component depends only on φk and the parameters of the system By construction, each component of the algebraic signature vanishes when at least one specific (multiple) fault occurs. Example: ¨ x + k(f1 − 1)2 x − (d + f2)u = 0, f1 ∈ [0, 2), f2 ∈ [0, 2) Exhaustive summary: φ(f) = (k(f1 − 1)2, −d − f2) = (φ1, φ2). Algorithm Algebraic_signature: ASig(f) = (φ1 − k, φ2 + d).

Fault diagnosability and fault diagnosis Multiple faults

ASig : Re − → (R[φ1, . . . , φN])l f → (ASig1(φ), . . . , ASigl(φ)) . Remark: Cp,f = set of all algebraic equations and inequalities verified by p and f. SN = {γ1

1(p, f) = φ1, . . . , γns s (p, f) = φN} ∪ Cp,f ∪ {vifi = 1|i ∈ N} ∪ {fi = 0|i /

∈ N} Criterion Two criterion to discriminate multiple fault signatures lying on the emptiness of semialgebraic sets: Does the kth component of ASig(fN ) vanishes (resp. never vanishes) for at least

• ne real value of the multiple fault fN ? (SN ∪ {ASigk(fN ) = 0} = ∅?)

When ASigj(f) = 0 is equivalent to fi = 0?

Fault diagnosability and fault diagnosis Multiple faults

ASig : Re − → (R[φ1, . . . , φN])l f → (ASig1(φ), . . . , ASigl(φ)) . Remark: Cp,f = set of all algebraic equations and inequalities verified by p and f. SN = {γ1

1(p, f) = φ1, . . . , γns s (p, f) = φN} ∪ Cp,f ∪ {vifi = 1|i ∈ N} ∪ {fi = 0|i /

∈ N} Criterion Two criterion to discriminate multiple fault signatures lying on the emptiness of semialgebraic sets: Does the kth component of ASig(fN ) vanishes (resp. never vanishes) for at least

• ne real value of the multiple fault fN ? (SN ∪ {ASigk(fN ) = 0} = ∅?)

When ASigj(f) = 0 is equivalent to fi = 0? Example: ¨ x + k(f1 − 1)2 x − (d + f2)u = 0, f1 ∈ [0, 2), f2 ∈ [0, 2) ASig(f) = (φ1 − k, φ2 + d), Cp,f = {0 < k < 4, 1 ≤ d, 0 ≤ f1 < 2, 0 ≤ f2 < 2} . f ASig1(f) ASig2(f) f{} f{1} f{2} f{1,2}

Fault diagnosability and fault diagnosis Multiple faults

ASig : Re − → (R[φ1, . . . , φN])l f → (ASig1(φ), . . . , ASigl(φ)) . Remark: Cp,f = set of all algebraic equations and inequalities verified by p and f. SN = {γ1

1(p, f) = φ1, . . . , γns s (p, f) = φN} ∪ Cp,f ∪ {vifi = 1|i ∈ N} ∪ {fi = 0|i /

∈ N} Criterion Two criterion to discriminate multiple fault signatures lying on the emptiness of semialgebraic sets: Does the kth component of ASig(fN ) vanishes (resp. never vanishes) for at least

• ne real value of the multiple fault fN ? (SN ∪ {ASigk(fN ) = 0} = ∅?)

When ASigj(f) = 0 is equivalent to fi = 0? Example: ¨ x + k(f1 − 1)2 x − (d + f2)u = 0, f1 ∈ [0, 2), f2 ∈ [0, 2) ASig(f) = (φ1 − k, φ2 + d), Cp,f = ∅. f ASig1(f) ASig2(f) f{} f{1} f{2} f{1,2}

Fault diagnosability and fault diagnosis Example of a two coupled water tank

x1 x2 u f1 f2 f3 f4 y1 y2 y1

Sensor faults clogging Actuator fault

Model of a two coupled water-tank                    ˙ x1(t, p) = p1 (u(t) + f1) −p2 (1 − f3)

• x1(t, p),

˙ x2(t, p) = p3 (1 − f3)

• x1(t, p)

−p4

• x2(t, p),

y(t, p) = p5

• x1(t, p) + f2,

x = (x1, x2)T : level in each tank (vary between 0 and 10) f1: unknown additive fault on the actuator signal f2: additive fault on the sensor at the output of the first water tank f3 ∈ [0; 1]: clogging fault. If f3 = 1: fully clogged pipe situation; 0 < f3 < 1: a partial clogging.

Fault diagnosability and fault diagnosis Example of a two coupled water tank

x1 x2 u f1 f2 f3 f4 y1 y2 y1

Sensor faults clogging Actuator fault

Model of a two coupled water-tank                            ˙ x1(t, p) = p1 (u(t) + f1) −p2 (1 − f3) z1(t, p), ˙ x2(t, p) = p3 (1 − f3)

• x1(t, p)

−p4 z2(t, p), z1(t, p)2 = x1(t, p), z2(t, p)2 = x2(t, p), y(t, p) = p5

• x1(t, p) + f2,

(z1(t, p) =

• x1(t, p), z2(t, p) =
• x2(t, p))

x = (x1, x2)T : level in each tank (vary between 0 and 10) f1: unknown additive fault on the actuator signal f2: additive fault on the sensor at the output of the first water tank f3 ∈ [0; 1]: clogging fault. If f3 = 1: fully clogged pipe situation; 0 < f3 < 1: a partial clogging.

Fault diagnosability and fault diagnosis Example of a two coupled water tank

           ˙ x1(t, p) = p1 (u(t) + f1) − p2 (1 − f3)

• x1(t, p),

˙ x2(t, p) = p3 (1 − f3)

• x1(t, p) − p4
• x2(t, p),

y(t, p) = p5

• x1(t, p) + f2,

(16) IO relation: 2 y ˙ y − p5 (f3 − 1)2 (p1 p5 f1 + p2 f2) − p1 p2

5 (f3 − 1)2 u + p2 p5 (f3 − 1)2 y − 2 f2 ˙

y = 0 Exhaustive summary: φ(f1, f2, f3) = (−p5 (f3 − 1)2 (p1 p5 f1 + p2 f2), −p1 p2

5 (f3 − 1)2, p2 p5 (f3 − 1)2, −2 f2).

Algebraic signature using the Algorithm Algebraic_signature: ASig(f) = (φ1, φ4, p1 p2

5 + φ2, −p2 p5 + φ3, −φ3 φ4 + 2 φ1, −p2 p5 φ4 + 2 φ1).

Fault diagnosability and fault diagnosis Example of a two coupled water tank

Discrimination: Cp,f = {0 < p1, . . . , 0 < p5, 0 ≤ f3 < 1} .

ASig1(f) ASig2(f) ASig3(f) ASig4(f) ASig5(f) ASig6(f) f{} f{1} f{2} f{3} f{1,2} f{1,3} f{2,3} f{1,2,3}

Numerical Expected Values of the Algebraic Signatures

Conclusion and perspectives

Conclusion

Identifiability, fault diagnosability, fault diagnosis Identifiabiliy, identification, fault diagnosis in the context of bounded-error uncertain models Identifiability in PDE’s models

Perspectives

Identifiability on integro-differential models (F . Boulier, CRIStAL) Identifiability and diagnosability in networks (S. Orange, LMAH).

Bibliography

Identifiability

• N. Verdière, S. Orange, A strategic algorithmic tool for doing an a priori

identifiability study of dynamical nonlinear models, Mathematical Biosciences 308, 2018, pp. 105-113.

• S. Zhu, N. Verdière, L. Denis-Vidal, D. Kateb, Identifiability analysis and parameter

estimation of an epidemiologic model in a spatially continuous domain, Ecological Complexity, Available online 8 January 2018. Boulier F . and al., Identifiability, Integro-Differential Equations and Neurobiology, Journées Annuelles du GT BIOSS, March 2017, Montpellier, France. Diagnosability

• N. Verdière, C. Jauberthie, Louise Travé-Massuyès, Functional diagnosability and

detectability of nonlinear models based on analytical redundancy relations, Journal of Process Control, Septembre 2014, Vol. 35, 1-10.

• N. Verdière, S. Orange, Diagnosability in the case of multi-faults in nonlinear

models, Journal of Process Control, Vol 69, pp. 1-7, 2018. THANK YOU FOR YOUR ATTENTION!