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

Applications of computer algebra in the identifiability and diagnosability studies Nathalie Verdière 1 1 Normandie 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 2 Injection of macromolecules 1. Infection of a cell by an extracellular bacterium

My topics/works: Identifiability study and parameter estimation (testing an assumption in pharmacokinetic domain) - L. Denis-Vidal, G. Joly-Blanchard 2 Injection of macromolecules 1. Infection of a cell by an extracellular bacterium 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 2 Injection of macromolecules 1. Infection of a cell by an extracellular bacterium 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 2 Injection of macromolecules 1. Infection of a cell by an extracellular bacterium 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 � ˙ x ( t , p ) = f ( x ( t , p ) , u ( t ) , p ) , Γ p (1) y ( t , p ) = h ( x ( t , p ) , p ) . � x ( t , p ) ∈ R n : state variables at time t , � y ( t , p ) ∈ R m : output vector at time t , � u ( t ) ∈ R r : input vector at time t , � f , h : real functions, analytic on M (an open set of R n ), � p ∈ U P : vector of parameters, U P ⊂ R p : an a priori known set of admissible parameters.

Identifiability Definition � ˙ x ( t , p ) = f ( x ( t , p ) , u ( t ) , p ) , Γ p (1) y ( t , p ) = h ( x ( t , p ) , p ) . � x ( t , p ) ∈ R n : state variables at time t , � y ( t , p ) ∈ R m : output vector at time t , � u ( t ) ∈ R r : input vector at time t , � f , h : real functions, analytic on M (an open set of R n ), � p ∈ U P : vector of parameters, U P ⊂ R p : 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. Identification problem 2

Identifiability Definition � ˙ x ( t , p ) = f ( x ( t , p ) , u ( t ) , p ) , Γ p (1) y ( t , p ) = h ( x ( t , p ) , p ) . � x ( t , p ) ∈ R n : state variables at time t , � y ( t , p ) ∈ R m : output vector at time t , � u ( t ) ∈ R r : input vector at time t , � f , h : real functions, analytic on M (an open set of R n ), � p ∈ U P : vector of parameters, U P ⊂ R p : 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. Identification problem 2 ⇒ 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 ∈ U p , one gets  y ( t , p ) � = ∅ , ˜  ⇒ p = ¯ p . (2) ˜ y ( t , p ) ∩ ˜ y ( t , ¯ p ) � = ∅ , ∀ t ≥ 0 , ¯ p ∈ U p  The model is locally identifiable if it is globally identifiable in an open neighborhood v ( p ) ⊂ U p 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 ∈ U p , one gets  y ( t , p ) � = ∅ , ˜  ⇒ p = ¯ p . (2) y ( t , p ) ∩ ˜ ˜ y ( t , ¯ p ) � = ∅ , ∀ t ≥ 0 , ¯ p ∈ U p  The model is locally identifiable if it is globally identifiable in an open neighborhood v ( p ) ⊂ U p 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 ∈ U p , there exists t 1 > 0 such that if for all t ∈ [ 0 , t 1 ] , 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 ) ⊂ U p of p .

Identifiability Example Example :  k 12 ( x 2 − x 1 ) − k ν x 1 x 1 ˙ = ,   1 + x 1  ˙ x 2 = k 21 ( x 1 − x 2 ) ,   y = x 1 ,  1 Rosenfeld-Groebner algorithm with the elimination order [ y ] ≺ [ x 1 , x 2 ] : = { k 12 y 2 ˙ y + k 21 k ν y 2 + k 21 y 2 ˙ y + y 2 ¨ y + 2 k 12 y ˙ y + k 21 k ν y + 2 k 21 y ˙ C ( p ) y + k 12 ˙ y + k 21 ˙ y + k ν ˙ y + 2 y ¨ y + ¨ y , x 1 − y , k 12 x 2 y − k 12 y 2 + k 12 x 2 − k 12 y − k ν y − y ˙ y − ˙ y } .

Identifiability Example Example :  k 12 ( x 2 − x 1 ) − k ν x 1 x 1 ˙ = ,   1 + x 1  ˙ x 2 = k 21 ( x 1 − x 2 ) ,   y = x 1 ,  1 Rosenfeld-Groebner algorithm with the elimination order [ y ] ≺ [ x 1 , x 2 ] : = { k 12 y 2 ˙ y + k 21 k ν y 2 + k 21 y 2 ˙ y + y 2 ¨ y + 2 k 12 y ˙ y + k 21 k ν y + 2 k 21 y ˙ C ( p ) y + k 12 ˙ y + k 21 ˙ y + k ν ˙ y + 2 y ¨ y + ¨ y , x 1 − y , k 12 x 2 y − k 12 y 2 + k 12 x 2 − k 12 y − k ν y − y ˙ y − ˙ y } . 2 Keep the IO polynomial ( p = ( k 12 , k 21 , k ν ) , γ ( p ) = ( k 12 + k 21 , k 21 k ν , k 21 + k ν ) ): y + y 2 ¨ y + γ 1 ( p ) ( y 2 ˙ y ) + γ 2 ( p ) ( y 2 + y ) + γ 3 ( p ) ˙ P ( y , p ) = ¨ y + 2 y ¨ y + 2 y ˙ y

Identifiability Example Example :  k 12 ( x 2 − x 1 ) − k ν x 1 x 1 ˙ = ,   1 + x 1  ˙ x 2 = k 21 ( x 1 − x 2 ) ,   y = x 1 ,  1 Rosenfeld-Groebner algorithm with the elimination order [ y ] ≺ [ x 1 , x 2 ] : = { k 12 y 2 ˙ y + k 21 k ν y 2 + k 21 y 2 ˙ y + y 2 ¨ y + 2 k 12 y ˙ y + k 21 k ν y + 2 k 21 y ˙ C ( p ) y + k 12 ˙ y + k 21 ˙ y + k ν ˙ y + 2 y ¨ y + ¨ y , x 1 − y , k 12 x 2 y − k 12 y 2 + k 12 x 2 − k 12 y − k ν y − y ˙ y − ˙ y } . 2 Keep the IO polynomial ( p = ( k 12 , k 21 , k ν ) , γ ( p ) = ( k 12 + k 21 , k 21 k ν , k 21 + k ν ) ): y + y 2 ¨ y + γ 1 ( p ) ( y 2 ˙ y ) + γ 2 ( p ) ( y 2 + y ) + γ 3 ( p ) ˙ P ( y , p ) = ¨ y + 2 y ¨ y + 2 y ˙ y Suppose that y ( t , p ) = y ( t , ¯ p ) : 3 p )) ( y 2 ˙ P ( y , p ) − P ( y , ¯ ( γ 1 ( p ) − γ 1 (¯ y + 2 y ˙ p ) = y ) p )) ( y 2 + y ) +( γ 2 ( p ) − γ 2 (¯ +( γ 3 ( p ) − γ 3 (¯ p )) ˙ y = 0 . Remark: det ( y 2 ˙ y , ( y 2 + y ) , ˙ y + 2 y ˙ y ) = y 2 + y ... y 4 + ( 3 y 2 ¨ y ( y + 1 ) 2 ˙ y 2 ( y + 1 ) 2 ) �≡ 0. − 2 ˙ y ( − ˙ y + 5 y ¨ y + 2 ¨ y ) ˙ y − 3 y ¨

Identifiability Example Example :  k 12 ( x 2 − x 1 ) − k ν x 1 x 1 ˙ = ,   1 + x 1  ˙ x 2 = k 21 ( x 1 − x 2 ) ,   y = x 1 ,  1 Rosenfeld-Groebner algorithm with the elimination order [ y ] ≺ [ x 1 , x 2 ] : = { k 12 y 2 ˙ y + k 21 k ν y 2 + k 21 y 2 ˙ y + y 2 ¨ y + 2 k 12 y ˙ y + k 21 k ν y + 2 k 21 y ˙ C ( p ) y + k 12 ˙ y + k 21 ˙ y + k ν ˙ y + 2 y ¨ y + ¨ y , x 1 − y , k 12 x 2 y − k 12 y 2 + k 12 x 2 − k 12 y − k ν y − y ˙ y − ˙ y } . 2 Keep the IO polynomial ( p = ( k 12 , k 21 , k ν ) , γ ( p ) = ( k 12 + k 21 , k 21 k ν , k 21 + k ν ) ): y + y 2 ¨ y + γ 1 ( p ) ( y 2 ˙ y ) + γ 2 ( p ) ( y 2 + y ) + γ 3 ( p ) ˙ P ( y , p ) = ¨ y + 2 y ¨ y + 2 y ˙ y Suppose that y ( t , p ) = y ( t , ¯ p ) : 3 p )) ( y 2 ˙ P ( y , p ) − P ( y , ¯ ( γ 1 ( p ) − γ 1 (¯ y + 2 y ˙ p ) = y ) p )) ( y 2 + y ) +( γ 2 ( p ) − γ 2 (¯ +( γ 3 ( p ) − γ 3 (¯ p )) ˙ y = 0 . Remark: det ( y 2 ˙ y , ( y 2 + y ) , ˙ y + 2 y ˙ y ) = y 2 + y ... y 4 + ( 3 y 2 ¨ y ( y + 1 ) 2 ˙ y 2 ( y + 1 ) 2 ) �≡ 0. − 2 ˙ y ( − ˙ y + 5 y ¨ y + 2 ¨ y ) ˙ y − 3 y ¨ Study of γ ( p ) − γ (¯ 4 p ) = 0: γ ( p ) − γ (¯ p ) = 0 ⇒ p = ¯ p Conclusion: the model is identifiable.