Set-membership identifiability and estimation of parameters for - - PowerPoint PPT Presentation

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion

Set-membership identifiability and estimation of parameters for uncertain nonlinear systems

Nathalie Verdière1, Carine Jauberthie2, Louise Travé-Massuyès2

1LMAH, France 2LAAS-CNRS, France

15 juin 2011

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion

Outline

1

Set-membership identifiability Motivation Definitions

2

Methods to analyse set-membership identifiability and numerical applications Method 1 Method 2

3

Conclusion

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation

Identifiability of linear and nonlinear systems

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation

Identifiability of linear and nonlinear systems What about uncertain systems?

• The system has constant parameters but the knowledge

about the parameter values is uncertain:

corresponds to the plus/minus tolerance value provided by the builder of physical device parameters. The study of such systems can be brought back to the study

• f a family of constant parameter systems.
• Case for which parameter uncertainty comes from the fact

that parameters may vary across time.

This case is typical of devices that operate in different environmental conditions, which may affect parameter values.

In this work: only the first situation is considered.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation

Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. Solution: x1(t) = (−1 − t + et) cos(̟). Hypothesis: UP = [0, 2π], ̟ ∈ P∗ = π 2, 3π 2

• .

1.5708 3.1416 4.7124 6.2832

• 1

1 w cos(w)

w ∈ P*

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation

Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. Solution: x1(t) = (−1 − t + et) cos(̟). Hypothesis: UP = [0, 2π], ̟ ∈ P∗ = π 2, 3π 2

• .

1.5708 3.1416 4.7124 6.2832

• 1

1 w cos(w)

w ∈ P*

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation

Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. Solution: x1(t) = (−1 − t + et) cos(̟). Hypothesis: UP = [0, 2π], ̟ ∈ P∗ = π 2, 3π 2

• .

1.5708 3.1416 4.7124 6.2832

• 1

1 w cos(w)

w ∈ P*

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation

Why not use the set-membership methods?

Subject of a growing interest in various communities and applied for many tasks (for example: fault detection, diagnosis). A lot of works on set-membership (state, parameters) estimations. To our knowledge: no existing definition and method for the identifiability problem of error-bounded uncertain models.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation

Why not use the set-membership methods?

Subject of a growing interest in various communities and applied for many tasks (for example: fault detection, diagnosis). A lot of works on set-membership (state, parameters) estimations. To our knowledge: no existing definition and method for the identifiability problem of error-bounded uncertain models.

Two definitions of global set-membership identifiability are provided:

a conceptual definition, a definition relying on a measure µ (can be put in correspondence with operational set-membership estimation methods).

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

We consider the uncertain system: ΓP

1 =

           ˙ x(t, p) = f(x(t, p), u(t), p), y(t, p) = h(x(t, p), p), x(t0, p) = x0 ∈ X0, p ∈ P ⊂ UP, t0 ≤ t ≤ T, (1) where: x(t, p) ∈ Rn: state variables at time t, y(t, p) ∈ Rm: outputs at time t, u(t) ∈ Rr: input vector at time t, x0 ∈ X0, X0: a bounded set, f, h: real functions, analytic on M (an open set of Rn), p ∈ P ⊂ UP: vector of parameters, UP ⊂ Rp: an a priori known set of admissible parameters.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Conceptual definitions Notation: Y(P, u) (respectively Y(P)): the set of outputs, solution of ΓP

1 with the input u (resp. when u = 0)

Global set-membership identifiability Definition: Case of controlled systems The model ΓP

1 given by (1) is globally set-membership

identifiable for P∗ = ∅, P∗ ⊂ UP if there exists an input u such that Y(P∗, u) = ∅ and Y(P∗, u) ∩ Y(¯ P, u) = ∅, ¯ P ⊂ UP = ⇒ P∗ ∩ ¯ P = ∅.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Conceptual definitions Notation: Y(P, u) (respectively Y(P)): the set of outputs, solution of ΓP

1 with the input u (resp. when u = 0)

Global set-membership identifiability Definition: Case of controlled systems The model ΓP

1 given by (1) is globally set-membership

identifiable for P∗ = ∅, P∗ ⊂ UP if there exists an input u such that Y(P∗, u) = ∅ and Y(P∗, u) ∩ Y(¯ P, u) = ∅, ¯ P ⊂ UP = ⇒ P∗ ∩ ¯ P = ∅. Definition: Case of uncontrolled systems The model ΓP

1 given by (1) is globally set-membership

identifiable for P∗ = ∅, P∗ ⊂ UP if Y(P∗) = ∅ and Y(P∗) ∩ Y(¯ P) = ∅, ¯ P ⊂ UP = ⇒ P∗ ∩ ¯ P = ∅.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Conceptual definitions Notation: Y(P, u) (respectively Y(P)): the set of outputs, solution of ΓP

1 with the input u (resp. when u = 0)

Global set-membership identifiability Definition: Case of controlled systems The model ΓP

1 given by (1) is globally set-membership

identifiable for P∗ = ∅, P∗ ⊂ UP if there exists an input u such that Y(P∗, u) = ∅ and Y(P∗, u) ∩ Y(¯ P, u) = ∅, ¯ P ⊂ UP = ⇒ P∗ ∩ ¯ P = ∅. Definition: Case of uncontrolled systems The model ΓP

1 given by (1) is globally set-membership

identifiable for P∗ = ∅, P∗ ⊂ UP if Y(P∗) = ∅ and Y(P∗) ∩ Y(¯ P) = ∅, ¯ P ⊂ UP = ⇒ P∗ ∩ ¯ P = ∅. Extension to local set-membership identifiability for P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. Global set-membership identifiability: UP = [0, 2π], P∗ = π 2, 3π 2

• 1.5708

3.1416 4.7124 6.2832

• 1

1 w cos(w) Globally set-membership identifiable

P*

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. Global set-membership identifiability: UP = [0, 2π], P∗ = π 2, 3π 2

• 1.5708

3.1416 4.7124 6.2832

• 1

1 w cos(w) Globally set-membership identifiable

P* P

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. Non global set-membership identifiability: UP = [0, 2π], P∗ =

• π, 3π

2

• 1.5708

3.1416 4.7124 6.2832

• 1

1 w cos(w) non Globally set-membership identifiable

P*

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. Non global set-membership identifiability: UP = [0, 2π], P∗ =

• π, 3π

2

• 1.5708

3.1416 4.7124 6.2832

• 1

1 w cos(w) non Globally set-membership identifiable

P* P

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

µ-set-membership identifiability Let us now consider a bounded set Π of Rp. µ(Π) =diameter of Π= the least upper bound of {d(π1, π2), π1, π2 ∈ Π}, with d a classical metric on Rp. If Π is not bounded, µ(Π) = +∞. Afterwards, P∗ is supposed to be bounded.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

µ-set-membership identifiability Let us now consider a bounded set Π of Rp. µ(Π) =diameter of Π= the least upper bound of {d(π1, π2), π1, π2 ∈ Π}, with d a classical metric on Rp. If Π is not bounded, µ(Π) = +∞. Afterwards, P∗ is supposed to be bounded. µ-set-membership identifiability Definition The model ΓP

1 given by (1) is globally µ-set-membership

identifiable for P∗ = ∅, µ(P∗) as small as desired, if there exists an input u such that Y(P∗, u) = ∅ and Y(P∗, u) ∩ Y(¯ P, u) = ∅, ¯ P ⊂ UP = ⇒ P∗ ∩ ¯ P = ∅. If µ(P∗) ≥ ε, then we refer to ε-set-membership identifiability. ⇒ Practical importance of ε-set-membership identifiability.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Extension to the structural µ-set-membership identifiability (ΓP

1 is µ-set-membership identifiable for all P ∈ UP except

at a subset of points of zero measure in UP). Extension to local µ-set-membership identifiability for P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Proposition (Structural) global µ-set-membership identifiability for P∗ implies global set-membership identifiability for P∗ but the inverse is not true. Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. UP = [0, 2π], P∗ =

• π, 3π

2

• .

1.5708 3.1416 4.7124 6.2832

• 1

1 w cos(w) Globally set-membership identifiable

P* P

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions

Proposition (Structural) global µ-set-membership identifiability for P∗ implies global set-membership identifiability for P∗ but the inverse is not true. Example: ˙ x1 = x1 + t cos(̟), x1(0) = 0. UP = [0, 2π], P∗ =

• π, 3π

2

• .

1.5708 3.1416 4.7124 6.2832

• 1

1 w cos(w) Globally set-membership identifiable/non mu-set-membership identifiable

P P*

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Method 1: based on that proposed by Pohjanpalo y is supposed analytical ⇒ y is entirely caracterized by the value of its derivatives at 0 identifiability studied owing to the power series expansion

• f the solution y.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Method 1: based on that proposed by Pohjanpalo y is supposed analytical ⇒ y is entirely caracterized by the value of its derivatives at 0 identifiability studied owing to the power series expansion

• f the solution y.

We consider the system: ΓP

1 =

           ˙ x(t, p) = f(x(t, p), u(t), p), y(t, p) = h(x(t, p), p), x(0, p) = x0 ∈ X0, p ∈ P ⊂ UP, 0 ≤ t ≤ T. (2) Hypothesis: For all possible trajectories x(.), t → f(x(t, p), u(t), p) admits a Taylor series expansion on [0, T]. Notation: For each measurement vector y(.), we denote ak(.) = y(k)(.).

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Theorem If ΓP

1 is globally set-membership identifiable for P∗ = ∅ and an

input u and a0(.) ∈ Y(P∗, u), then the system: dk dtk [h(x(t, p), p)]|t=0 = ak(0), k = 0, 1, . . . , +∞, admits solutions in the connected set P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Theorem If ΓP

1 is globally set-membership identifiable for P∗ = ∅ and an

input u and a0(.) ∈ Y(P∗, u), then the system: dk dtk [h(x(t, p), p)]|t=0 = ak(0), k = 0, 1, . . . , +∞, admits solutions in the connected set P∗. , Theorem If ΓP

1 is globally µ-set-membership identifiable for P∗ = ∅ and an

input u and a0(.) ∈ Y(P∗, u), then the system: dk dtk [h(x(t, p), p)]|t=0 = ak(0), k = 0, 1, . . . , +∞, admits a unique solution in the connected set P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Theorem If ΓP

1 is globally set-membership identifiable for P∗ = ∅ and an

input u and a0(.) ∈ Y(P∗, u), then the system: dk dtk [h(x(t, p), p)]|t=0 = ak(0), k = 0, 1, . . . , +∞, admits solutions in the connected set P∗. , Theorem If ΓP

1 is globally µ-set-membership identifiable for P∗ = ∅ and an

input u and a0(.) ∈ Y(P∗, u), then the system: dk dtk [h(x(t, p), p)]|t=0 = ak(0), k = 0, 1, . . . , +∞, admits a unique solution in the connected set P∗. ⇒ can be used for proving the non (µ-)set-membership identifiability of some systems.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Theorem If there exists u such that the system dk dtk [h(x(t, p), p)]|t=0 = ak(0), k = 0, 1, . . . , +∞, admits for only solution the connected set P∗ = ∅ then ΓP

1 is

globally set-membership identifiable for P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Injectivity of a function (Lagrange and al., 2007) Consider a function f : A → B and any set A1 ⊆ A. The function f is said to be a partial injection of A1 over A, noted (A1, A)-injective, if ∀a1 ∈ A1, ∀a ∈ A, a1 = a ⇒ f(a1) = f(a). f is said to be A-injective if it is (A, A)-injective.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Injectivity of a function (Lagrange and al., 2007) Consider a function f : A → B and any set A1 ⊆ A. The function f is said to be a partial injection of A1 over A, noted (A1, A)-injective, if ∀a1 ∈ A1, ∀a ∈ A, a1 = a ⇒ f(a1) = f(a). f is said to be A-injective if it is (A, A)-injective. Theorem Suppose there exists u such that the system dk dtk [h(x(t, p), p)]|t=0 = ak(0), k = 0, 1, . . . , +∞, for a finite number d of equations admits for only solution the connected set P∗ = ∅. If the function φ : p ∈ P∗ → (h(x(0, p), p), . . . , dd−1h dtd−1 (x(t, p), p))|t=0 ∈ (I Rm)d is P∗-injective then ΓP

1 is µ-set-membership identifiable for P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Particular case of ΓP

1 :

ΓP

1 =

           ˙ x(t, p) = A(p)x(t, p) + B(p)u(t), y(t, p) = C(p)x(t, p) + D(p)u(t), x(0, p) = x0 ∈ X0, p ∈ P ⊂ UP, 0 ≤ t ≤ T, where A(p), B(p), C(p) and D(p): matrices depending on p. dk dtk [h(x(t, p), p)]|t=0 = ak(0), k = 0, 1, . . . , +∞, ⇔          C(p)x0 + D(p)u(0) = a0(0), C(p)(Ak(p)x0 +

k

• i=1

Ak−i(p)B(p)u(i−1)(0)) + D(p)u(k)(0) = ak(0), k = 1, . . . , +∞, (3)

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Application        ˙ x1 = −(k21 + k31)x1 + 1, x1(0) = 1, ˙ x2 = k21x1 − x2, x2(0) = x20, ˙ x3 = k31x1 − c13x3, x3(0) = 1, y = x2 + c13x3, (4) → k21, k31, c13 are parameters to be identified, → x20 ∈ [x20, x20],

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Application        ˙ x1 = −(k21 + k31)x1 + 1, x1(0) = 1, ˙ x2 = k21x1 − x2, x2(0) = x20, ˙ x3 = k31x1 − c13x3, x3(0) = 1, y = x2 + c13x3, (4) → k21, k31, c13 are parameters to be identified, → x20 ∈ [x20, x20], Theoretical results: Solutions of the following system (5) deduced from (3):        x20 + c13 = a0(0), k21 − x20 + c13(k31 − c13) = a1(0), k21(−k21 − k31) + x20 + c13(k31(−k21 − k31) − c13k31 +c132 + k31) = a2(0). (5)

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

According to the previous theorem, it is sufficient to find solutions of (5). In substituting k21 obtained with the second equation in the third equation, one gets the following system:                                  x20 + c13 = a0(0), (−a1(0)c13 + a1(0) − x20c13 + x20 − c3

13 + 2c2 13 − c13)

• =: α(c13)

k31 = a2(0) + a1(0)2 + 2a1(0)x20 + 2a1(0)c2

13

+x2

20 + 2x20c2 13 + c4 13 − x20 − c3 13

• =: β(c13)

k21 = a1(0) + x20 − c13(k31 − c13)

• γ(c13,k13)

.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

           x20 + c13 = a0(0), α(c13)k31 = β(c13) k21 = γ(c13, k13). (6) Identifiability conclusions: → First equation of (6): c13 ∈ [a0(0) − x20, a0(0) − x20]. → If 0 ∈ α(c13), one gets k31 ∈ [β(c13), β(c13)]/[α(c13), α(c13)] and k21 ∈ [γ(c13, k13), γ(c13, k13)]. → The system (5) admits for solution the only connected set P∗ = [γ(c13, k13), γ(c13, k13)] × [β(c13), β(c13)]/[α(c13), α(c13)] × [a0(0) − x20, a0(0) − x20]. → The system is globally set-membership identifiable for P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Furthermore, consider the function φ : (k21, k31, c13) → (a0(0), a1(0), a2(0)) = (x20 + c13, k21 − x20 + c13(k31 − c13), k21(−k21 − k31) + x20 + c13(k31(−k21 − k31) − c13k31 +c132 + k31)). (7) The function φ is P∗-injective ⇒ The system is µ-set membership identifiable for P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Numerical Application Numerical results

→ Simulation : Matlab/Intlab. → Real values of parameters: k21 = 0.1, k31 = 0.2, c13 = 0.1. → Simulated outputs: perturbed by the normal law with a mean equal to zero and a standard deviation equal to 0.01. → x20 = 0.1. x20 ∈ [0.09, 0.11].

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 1

Numerical Application Numerical results

→ Simulation : Matlab/Intlab. → Real values of parameters: k21 = 0.1, k31 = 0.2, c13 = 0.1. → Simulated outputs: perturbed by the normal law with a mean equal to zero and a standard deviation equal to 0.01. → x20 = 0.1. x20 ∈ [0.09, 0.11].

One gets:    k21 ∈ [−0.0625; 0.2843], k31 ∈ [−1.4072; 1.8365], c13 ∈ [0.0801; 0.1002]. P∗ := [−0.0625; 0.2843] × [−1.4072; 1.8365] × [0.0801; 0.1002].

→ The system is globally set-membership identifiable for P∗. → φ is P∗-injective ⇒ the system is µ-set membership identifiable for P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 2

Method 2: based on that proposed by D. Vidal et G. Joly-Blanchard: elimination order {p} < {y, u} < {x} (⇒ eliminate unobservable state variables), differential algebra approach (Kolchin and al., 1973)

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 2

Method 2: based on that proposed by D. Vidal et G. Joly-Blanchard: elimination order {p} < {y, u} < {x} (⇒ eliminate unobservable state variables), differential algebra approach (Kolchin and al., 1973) ⇒ relations between outputs and parameters: Ri(y, u, p) = θ0(y, u) +

ni

• k=1

θi

k(p)mk(y, u), i = 1, . . . , m

→ (θi

k)1≤k≤l are rational in p, θi u = θi v (u = v),

→ (mk)1≤k≤l are differential polynomials with respect to y and u and θ0 = 0. Size of the system = number of observations. Afterwards: i = 1 (⇒ one observation).

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 2

Consider R(y, u, p) = θ0(y, u) +

n

• k=1

θk(p)mk(y, u). Theorem If ∀(y, u), △(R)(y, u) = det(mk(y, u), k = 1, . . . , n) ≡ 0, and the function Φ : p ∈ P∗ → (θ1(p), . . . , θn(p)) ∈ (R)n is P∗-injective then ΓP

1 is µ-set-membership identifiable for P∗.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 2

Application Consider the following model (explore/quantify the capacity of the macrophage mannose receptor to endocytose soluble macromolecule):            ˙ x1 = α1(x2 − x1) − Vmx1

1+x1 ,

˙ x2 = α2(x1 − x2), x1(0) = 10/16, x2(0) = 0, y = x1. (8) x1 (resp. x2) is the enzyme concentration outside (resp. inside) the macrophage, p = (α1, Vm, α2): the unknown parameters which have to be identified.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 2

Theoretical results: The package diffalg of Maple gives the following output polynomial: R(y) = y2¨ y + 2¨ yy + ¨ y + γ1y ˙ y(y ˙ y + 2) + γ2 ˙ y + γ3y(y + 1), where γ = {α1 + α2, α1 + α2 + Vm, α2Vm}.

→ First hypothesis: Maple ⇒ ∆R(y) ≡ 0. → Second hypothesis: For all P∗ ⊂ [0, +∞[×[0, +∞[×[0, +∞[, Φ : (α1, Vm, α2) → (α1 + α2, α1 + α2 + Vm, α2Vm) is P∗-injective.

The system is µ-set-membership identifiable for all P∗ ⊂ [0, +∞[×[0, +∞[×[0, +∞[.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 2

Numerical application Numerical Method

→ Hypothesis: knowledge of y(tk)k=1,...,N and R(y) = y2¨ y + 2¨ yy + ¨ y + γ1y ˙ y(y ˙ y + 2) + γ2 ˙ y + γ3y(y + 1), → R(y(tk)) = 0 for k = 1, . . . , N ⇒ Aγ − b = 0 where Ak,1 = y(tk) ˙ y(tk)(y(tk) ˙ y(tk) + 2), Ak,2 = ˙ y(tk), Ak,3 = y(tk)(y(tk) + 1), bk = y2(tk)¨ y(tk) + 2¨ y(tk)y(tk) + ¨ y(tk). → least squares minimization of Aγ − b with respect to γ.

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Method 2

Numerical Results

→ Simulation: Matlab/Intlab. → Real values of parameters: α1 = 0.011, α2 = 0.02 and Vm = 0.1. → Simulated outputs: perturbed by heavy noise which is bounded by E = [−0.0004, 0.0004].

After 0.0834 seconds, the estimation algorithm leads to:    α1 + α2 = [0.0242, 0.0376], α1 + α2 + Vm = [0.1218, 0.1400], α2Vm = [0.0011, 0.0029] thus:    α1 = [−0.0099, 0.0281], α2 = [0.0095, 0.0342], Vm = [0.0843, 0.1157].

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion

Definitions of set-membership identifiability / µ-set-membership identifiability

provide a way to study identifiability for uncertain bounded-error systems have a role to play in many pratical problems (diagnosis/prognosis in uncertain environments) provide the guaranty that two situations corresponding to different parametrized setting are distinguishable

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion

Definitions of set-membership identifiability / µ-set-membership identifiability

provide a way to study identifiability for uncertain bounded-error systems have a role to play in many pratical problems (diagnosis/prognosis in uncertain environments) provide the guaranty that two situations corresponding to different parametrized setting are distinguishable

Methods to analyse (µ-)set-membership identifiability Applications on two examples

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion

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