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 Verdière 1 , Carine Jauberthie 2 , Louise Travé-Massuyès 2 1 LMAH, France 2 LAAS-CNRS, France 15 juin 2011

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Outline Set-membership identifiability 1 Motivation Definitions Methods to analyse set-membership identifiability and 2 numerical applications Method 1 Method 2 Conclusion 3

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 of 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 � ˙ x 1 = x 1 + t cos ( ̟ ) , Example: x 1 ( 0 ) = 0 . Solution: x 1 ( t ) = ( − 1 − t + e t ) cos ( ̟ ) . � π � 2 , 3 π Hypothesis: U P = [ 0 , 2 π ] , ̟ ∈ P ∗ = . 2 1 w ∈ P * cos(w) 0 -1 0 1.5708 3.1416 4.7124 6.2832 w

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation � ˙ x 1 = x 1 + t cos ( ̟ ) , Example: x 1 ( 0 ) = 0 . Solution: x 1 ( t ) = ( − 1 − t + e t ) cos ( ̟ ) . � π � 2 , 3 π Hypothesis: U P = [ 0 , 2 π ] , ̟ ∈ P ∗ = . 2 1 w ∈ P * cos(w) 0 -1 0 1.5708 3.1416 4.7124 6.2832 w

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Motivation � ˙ x 1 = x 1 + t cos ( ̟ ) , Example: x 1 ( 0 ) = 0 . Solution: x 1 ( t ) = ( − 1 − t + e t ) cos ( ̟ ) . � π � 2 , 3 π Hypothesis: U P = [ 0 , 2 π ] , ̟ ∈ P ∗ = . 2 1 w ∈ P * cos(w) 0 -1 0 1.5708 3.1416 4.7124 6.2832 w

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:  ˙ x ( t , p ) = f ( x ( t , p ) , u ( t ) , p ) ,    y ( t , p ) = h ( x ( t , p ) , p ) ,   Γ P x ( t 0 , p ) = x 0 ∈ X 0 , 1 = (1)  p ∈ P ⊂ U P ,     t 0 ≤ t ≤ T , where: x ( t , p ) ∈ R n : state variables at time t , y ( t , p ) ∈ R m : outputs at time t , u ( t ) ∈ R r : input vector at time t , x 0 ∈ X 0 , X 0 : a bounded set, f , h : real functions, analytic on M (an open set of R n ), p ∈ P ⊂ U P : vector of parameters, U P ⊂ R p : 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 Γ P 1 with the input u (resp. when u = 0) Global set-membership identifiability Definition: Case of controlled systems The model Γ P 1 given by (1) is globally set-membership identifiable for P ∗ � = ∅ , P ∗ ⊂ U P if there exists an input u such that Y ( P ∗ , u ) � = ∅ and ⇒ P ∗ ∩ ¯ Y ( P ∗ , u ) ∩ Y (¯ P , u ) � = ∅ , ¯ P ⊂ U 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 Γ P 1 with the input u (resp. when u = 0) Global set-membership identifiability Definition: Case of controlled systems The model Γ P 1 given by (1) is globally set-membership identifiable for P ∗ � = ∅ , P ∗ ⊂ U P if there exists an input u such that Y ( P ∗ , u ) � = ∅ and ⇒ P ∗ ∩ ¯ Y ( P ∗ , u ) ∩ Y (¯ P , u ) � = ∅ , ¯ P ⊂ U P = P � = ∅ . Definition: Case of uncontrolled systems The model Γ P 1 given by (1) is globally set-membership identifiable for P ∗ � = ∅ , P ∗ ⊂ U P if Y ( P ∗ ) � = ∅ and ⇒ P ∗ ∩ ¯ Y ( P ∗ ) ∩ Y (¯ P ) � = ∅ , ¯ P ⊂ U 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 Γ P 1 with the input u (resp. when u = 0) Global set-membership identifiability Definition: Case of controlled systems The model Γ P 1 given by (1) is globally set-membership identifiable for P ∗ � = ∅ , P ∗ ⊂ U P if there exists an input u such that Y ( P ∗ , u ) � = ∅ and ⇒ P ∗ ∩ ¯ Y ( P ∗ , u ) ∩ Y (¯ P , u ) � = ∅ , ¯ P ⊂ U P = P � = ∅ . Definition: Case of uncontrolled systems The model Γ P 1 given by (1) is globally set-membership identifiable for P ∗ � = ∅ , P ∗ ⊂ U P if Y ( P ∗ ) � = ∅ and ⇒ P ∗ ∩ ¯ Y ( P ∗ ) ∩ Y (¯ P ) � = ∅ , ¯ P ⊂ U 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: ˙ x 1 = x 1 + t cos ( ̟ ) , x 1 ( 0 ) = 0 . Global set-membership identifiability: � π � 2 , 3 π U P = [ 0 , 2 π ] , P ∗ = 2 Globally set-membership identifiable 1 P* cos(w) 0 -1 0 1.5708 3.1416 4.7124 6.2832 w

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions Example: ˙ x 1 = x 1 + t cos ( ̟ ) , x 1 ( 0 ) = 0 . Global set-membership identifiability: � π � 2 , 3 π U P = [ 0 , 2 π ] , P ∗ = 2 Globally set-membership identifiable 1 P P* cos(w) 0 -1 0 1.5708 3.1416 4.7124 6.2832 w

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions Example: ˙ x 1 = x 1 + t cos ( ̟ ) , x 1 ( 0 ) = 0 . Non global set-membership identifiability: � � π, 3 π U P = [ 0 , 2 π ] , P ∗ = 2 non Globally set-membership identifiable 1 P* cos(w) 0 -1 0 1.5708 3.1416 4.7124 6.2832 w

Set-membership identifiability Methods to analyse set-membership identifiability and numerical applications Conclusion Definitions Example: ˙ x 1 = x 1 + t cos ( ̟ ) , x 1 ( 0 ) = 0 . Non global set-membership identifiability: � � π, 3 π U P = [ 0 , 2 π ] , P ∗ = 2 non Globally set-membership identifiable 1 P P* cos(w) 0 -1 0 1.5708 3.1416 4.7124 6.2832 w

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 R p . µ (Π) = diameter of Π = the least upper bound of { d ( π 1 , π 2 ) , π 1 , π 2 ∈ Π } , with d a classical metric on R p . If Π is not bounded, µ (Π) = + ∞ . Afterwards, P ∗ is supposed to be bounded.