Error Analysis of the Linear Feedback Particle Filter American Control Conference, Milwaukee, June, 2018 Amirhossein Taghvaei Joint work with P. G. Mehta Coordinated Science Laboratory University of Illinois at Urbana-Champaign June 28, 2018
Outline Filtering problem in linear Gaussian setting Feedback Particle Filter (FPF) Stochastic and deterministic linear FPF Relation to the Ensemble Kalman filter Error analysis results Conclusion Error Analysis of the Linear FPF Amirhossein Taghvaei 1 / 13 Amirhossein
Filtering problem: Linear Gaussian setting Model: State process: d X t = AX t d t + σ B d B t , X 0 ∼ N ( m 0 , Σ 0 ) Observation process: d Z t = HX t d t + d W t , Filtering objective: Find prob. of X t given Z t := { Z s ; s ∈ [0 , t ] } Kalman filter: P X t |Z t is Gaussian N ( m t , Σ t ) Mean: d m t = Am t d t + K t ( d Z t − Hm t d t ) � �� � � �� � propagation correction dΣ t Variance: d t = Ric (Σ t ) (Ricatti equation) K t := Σ t H ⊤ Kalman gain: J. Xiong, An introduction to stochastic filtering theory, 2008. R. E Kalman and R. S Bucy. New results in linear filtering and prediction theory, 1961 Error Analysis of the Linear FPF Amirhossein Taghvaei 2 / 13 Amirhossein
Filtering problem: Linear Gaussian setting Model: State process: d X t = AX t d t + σ B d B t , X 0 ∼ N ( m 0 , Σ 0 ) Observation process: d Z t = HX t d t + d W t , Filtering objective: Find prob. of X t given Z t := { Z s ; s ∈ [0 , t ] } Kalman filter: P X t |Z t is Gaussian N ( m t , Σ t ) Mean: d m t = Am t d t + K t ( d Z t − Hm t d t ) � �� � � �� � propagation correction dΣ t Variance: d t = Ric (Σ t ) (Ricatti equation) K t := Σ t H ⊤ Kalman gain: J. Xiong, An introduction to stochastic filtering theory, 2008. R. E Kalman and R. S Bucy. New results in linear filtering and prediction theory, 1961 Error Analysis of the Linear FPF Amirhossein Taghvaei 2 / 13 Amirhossein
Filtering problem: Linear Gaussian setting Model: State process: d X t = AX t d t + σ B d B t , X 0 ∼ N ( m 0 , Σ 0 ) Observation process: d Z t = HX t d t + d W t , Filtering objective: Find prob. of X t given Z t := { Z s ; s ∈ [0 , t ] } Kalman filter: P X t |Z t is Gaussian N ( m t , Σ t ) Mean: d m t = Am t d t + K t ( d Z t − Hm t d t ) � �� � � �� � propagation correction dΣ t Variance: d t = Ric (Σ t ) (Ricatti equation) K t := Σ t H ⊤ Kalman gain: J. Xiong, An introduction to stochastic filtering theory, 2008. R. E Kalman and R. S Bucy. New results in linear filtering and prediction theory, 1961 Error Analysis of the Linear FPF Amirhossein Taghvaei 2 / 13 Amirhossein
Feedback particle filter (FPF) Overview A controlled interacting particle system to approximate the posterior dist. Numerical experiments: Stano, et. al. (2013) Tilton, et. al. (2013) (Yang, et. al. 2013) Berntorp, et. al. (2015) Surace, et. al. (2017) This work: Error analysis of the FPF algorithm for linear Gaussian setting T. Yang, P. G. Mehta, and S. P. Meyn. feedback particle filter, TAC , 2013 T. Yang, R. S. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariable feedback particle filter, Automatica , 2016 Error Analysis of the Linear FPF Amirhossein Taghvaei 3 / 13 Amirhossein
Feedback particle filter (FPF) Overview A controlled interacting particle system to approximate the posterior dist. (a) Error (b) Time FPF Numerical experiments: BPF Stano, et. al. (2013) Tilton, et. al. (2013) Berntorp, et. al. (2015) Surace, et. al. (2017) (Yang, et. al. 2013) This work: Error analysis of the FPF algorithm for linear Gaussian setting T. Yang, P. G. Mehta, and S. P. Meyn. feedback particle filter, TAC , 2013 T. Yang, R. S. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariable feedback particle filter, Automatica , 2016 Error Analysis of the Linear FPF Amirhossein Taghvaei 3 / 13 Amirhossein
Feedback particle filter (FPF) Overview A controlled interacting particle system to approximate the posterior dist. (a) Error (b) Time FPF Numerical experiments: BPF Stano, et. al. (2013) Tilton, et. al. (2013) Berntorp, et. al. (2015) Surace, et. al. (2017) (Yang, et. al. 2013) This work: Error analysis of the FPF algorithm for linear Gaussian setting T. Yang, P. G. Mehta, and S. P. Meyn. feedback particle filter, TAC , 2013 T. Yang, R. S. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariable feedback particle filter, Automatica , 2016 Error Analysis of the Linear FPF Amirhossein Taghvaei 3 / 13 Amirhossein
Feedback Particle Filter Overview Particles: { X 1 t , . . . , X N t } Mean-field process: ¯ X t N ≈ 1 � E [ f ( X t ) |Z t ] = E [ f ( ¯ f ( X i X t ) |Z t ] t ) N � �� � i =1 exactness T. Yang, P. G. Mehta, and S. P. Meyn. feedback particle filter, TAC , 2013 T. Yang, R. S. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariable feedback particle filter, Automatica , 2016 Error Analysis of the Linear FPF Amirhossein Taghvaei 4 / 13 Amirhossein
Feedback Particle Filter Overview Particles: { X 1 t , . . . , X N t } Mean-field process: ¯ X t N ≈ 1 � E [ f ( X t ) |Z t ] = E [ f ( ¯ f ( X i X t ) |Z t ] t ) N � �� � i =1 exactness T. Yang, P. G. Mehta, and S. P. Meyn. feedback particle filter, TAC , 2013 T. Yang, R. S. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariable feedback particle filter, Automatica , 2016 Error Analysis of the Linear FPF Amirhossein Taghvaei 4 / 13 Amirhossein
Feedback Particle Filter Overview Particles: { X 1 t , . . . , X N t } Mean-field process: ¯ X t N ≈ 1 � E [ f ( X t ) |Z t ] = E [ f ( ¯ f ( X i X t ) |Z t ] t ) N � �� � i =1 exactness T. Yang, P. G. Mehta, and S. P. Meyn. feedback particle filter, TAC , 2013 T. Yang, R. S. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariable feedback particle filter, Automatica , 2016 Error Analysis of the Linear FPF Amirhossein Taghvaei 4 / 13 Amirhossein
Stochastic and deterministic linear FPF Mean-field limit Stochastic linear FPF: [Bergemann, et. al. 2012] [Yang, et. al. 2013] K t ( d Z t − H ¯ X t + H ¯ m t d ¯ X t = A ¯ X t d t + σ B d ¯ + ¯ ¯ d t ) X 0 ∼ p 0 B t , 2 � �� � � �� � propagation correction (feedback control) Deterministic linear FPF: [Taghvaei, et. al. 2016] K t ( d Z t − H ¯ X t d t + 1 X t + H ¯ m t d ¯ X t = A ¯ 2 σ B σ ⊤ B ¯ Σ − 1 t ( ¯ m t ) d t + ¯ X t − ¯ d t ) 2 where the mean-field terms are m t := E [ ¯ X t |Z t ] (mean of ¯ ¯ X t ) Σ t := (covariance of ¯ ¯ X t ) K t := ¯ ¯ Σ t H ⊤ G. Evensen. Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics. 1994. K. Bergemann and S. Reich. An ensemble Kalman-Bucy filter for continuous data assimilation, 2012 A. Taghvaei, P. G. Mehta, An optimal transport formulation for the linear feedback particle filter, (ACC) 2016 Error Analysis of the Linear FPF Amirhossein Taghvaei 5 / 13 Amirhossein
Stochastic and deterministic linear FPF Mean-field limit Stochastic linear FPF: [Bergemann, et. al. 2012] [Yang, et. al. 2013] K t ( d Z t − H ¯ X t + H ¯ m t B t + ¯ d ¯ X t = A ¯ X t d t + σ B d ¯ ¯ d t ) , X 0 ∼ p 0 2 Deterministic linear FPF: [Taghvaei, et. al. 2016] K t ( d Z t − H ¯ X t d t + 1 X t + H ¯ m t d ¯ X t = A ¯ 2 σ B σ ⊤ B ¯ Σ − 1 t ( ¯ m t ) d t + ¯ X t − ¯ d t ) 2 where the mean-field terms are m t := E [ ¯ X t |Z t ] (mean of ¯ ¯ X t ) Σ t := (covariance of ¯ ¯ X t ) K t := ¯ ¯ Σ t H ⊤ G. Evensen. Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics. 1994. K. Bergemann and S. Reich. An ensemble Kalman-Bucy filter for continuous data assimilation, 2012 A. Taghvaei, P. G. Mehta, An optimal transport formulation for the linear feedback particle filter, (ACC) 2016 Error Analysis of the Linear FPF Amirhossein Taghvaei 5 / 13 Amirhossein
Stochastic and deterministic linear FPF Finite- N system Stochastic linear FPF: t + Hm ( N ) ( d Z t − HX i t + K ( N ) i.i.d d X i t = AX i t d t + σ B d B i t X i d t ) , ∼ p 0 0 t 2 for i = 1 , . . . , N Deterministic linear FPF: ( Z t − HX t + Hm ( N ) t d t + 1 − 1 ( X i B Σ ( N ) t − m ( N ) ) d t + K ( N ) d X i t = AX i 2 σ B σ ⊤ t d t ) t t t 2 where the mean-field terms are empirically approximated N := 1 � m ( N ) X i t (empirical mean) , t N i =1 N 1 � ) ⊤ (empirical covariance) Σ ( N ) ( X i t − m ( N ) )( X i t − m ( N ) := t t t N − 1 i =1 Objectives: Convergence m ( N ) Σ ( N ) → m t , → Σ t t t Convergence of the empirical distribution Error Analysis of the Linear FPF Amirhossein Taghvaei 6 / 13 Amirhossein
Stochastic and deterministic linear FPF Finite- N system Stochastic linear FPF: t + Hm ( N ) ( d Z t − HX i t + K ( N ) i.i.d d X i t = AX i t d t + σ B d B i t X i d t ) , ∼ p 0 0 t 2 for i = 1 , . . . , N Deterministic linear FPF: ( Z t − HX t + Hm ( N ) t d t + 1 − 1 ( X i B Σ ( N ) t − m ( N ) ) d t + K ( N ) d X i t = AX i 2 σ B σ ⊤ t d t ) t t t 2 where the mean-field terms are empirically approximated N := 1 � m ( N ) X i t (empirical mean) , t N i =1 N 1 � ) ⊤ (empirical covariance) Σ ( N ) ( X i t − m ( N ) )( X i t − m ( N ) := t t t N − 1 i =1 Objectives: Convergence m ( N ) Σ ( N ) → m t , → Σ t t t Convergence of the empirical distribution Error Analysis of the Linear FPF Amirhossein Taghvaei 6 / 13 Amirhossein
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