This document must be cited according to its fjnal version which is published in a conference as: C. Afri, L. Bako, V. Andrieu, P. Dufour, "Adaptive identifjcation of continuous time MIMO state-space models", 54rd IEEE Conference on Decision and Control (CDC), Osaka, Japan, pp. 5677-5682, december 15-18, 2015 You downloaded this document from the CNRS open archives server, on the webpages of Pascal Dufour: http://hal.archives-ouvertes.fr/DUFOUR-PASCAL-C-3926-2008
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Adaptive identification of continuous-time MIMO state-space models C. Afri* L. Bako V. Andrieu P. Dufour Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Acknowledgments: * Ph.D. student, funded by the French Ministry of Research. 1/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Outline Identification problem 1 Estimation method 2 Exponential convergence 3 Simulation 4 Conclusion and future work 5 2/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Outline Identification problem 1 Estimation method 2 Exponential convergence 3 Simulation 4 Conclusion and future work 5 3/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work General schema Linear systems identification Identification Identification of the state models of the Input/Output models online offline Extended observers Adaptive observers Subspaces method based on SVD decomposition requires a parameterization of model matrices 4/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Studied approach Linear systems identification Identification Identification of the state models of the Input/Output models online offline Extended observers Adaptive observers Subspaces method based on SVD decomposition requires a parameterization of model matrices 5/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work MIMO (Multiple-Input Multiple-Output) systems Consider a linear system described by a state-space model of the form � x ( t ) = Ax ( t )+ Bu ( t ) ˙ S : y ( t ) = Cx ( t )+ Du ( t ) , 6/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work MIMO (Multiple-Input Multiple-Output) systems Consider a linear system described by a state-space model of the form � x ( t ) = Ax ( t )+ Bu ( t ) ˙ S : y ( t ) = Cx ( t )+ Du ( t ) , u ∈ R n u measured inputs signals y ∈ R n y measured outputs signals x ∈ R n , unknown state A ∈ R n × n , B ∈ R n × n u , C ∈ R n y × n and D ∈ R n y × n u are unknown parameters 6/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Systems assumptions & estimation problem System assumptions A1. The system S is stable → A is Hurwitz. A2. ( A , B , C ) is minimal → ( A , B ) is controllable and ( A , C ) is observable. A3. C is full row rank. 7/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Systems assumptions & estimation problem System assumptions A1. The system S is stable → A is Hurwitz. A2. ( A , B , C ) is minimal → ( A , B ) is controllable and ( A , C ) is observable. A3. C is full row rank. Estimation problem From the measurements of inputs u and outputs y → Estimate both the state x and the parameters ( A , B , C , D ) in an arbitrary state space basis. 7/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Outline Identification problem 1 Estimation method 2 Exponential convergence 3 Simulation 4 Conclusion and future work 5 8/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Main idea Search for a linear expression M ( A , B , C , D ) ϕ ( u , y ) = f ( u , y ) 9/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Main idea Search for a linear expression M ( A , B , C , D ) ϕ ( u , y ) = f ( u , y ) Use the Recursive Least Squares (RLS) method 9/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Main idea Search for a linear expression M ( A , B , C , D ) ϕ ( u , y ) = f ( u , y ) Use the Recursive Least Squares (RLS) method Search for a criterion V t ( M ) such that 9/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Main idea Search for a linear expression M ( A , B , C , D ) ϕ ( u , y ) = f ( u , y ) Use the Recursive Least Squares (RLS) method Search for a criterion V t ( M ) such that M ( t ) = argmin M V t ( M ) converges to M 0 the solution of the linear expression d dt ∇ V t ( M ) = 0 → a continuous-time recursive estimator. 9/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Main idea Search for a linear expression M ( A , B , C , D ) ϕ ( u , y ) = f ( u , y ) Use the Recursive Least Squares (RLS) method Search for a criterion V t ( M ) such that M ( t ) = argmin M V t ( M ) converges to M 0 the solution of the linear expression d dt ∇ V t ( M ) = 0 → a continuous-time recursive estimator. Extract A , B , C and D from M . 9/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
Identification problem Estimation method Exponential convergence Simulation Conclusion and future work Is there a linear relation M ( A , B , C , D ) ϕ ( y , u ) = f ( y , u )? 10/34 C. Afri: afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models
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