Team optimal decentralized state estimation Aditya Mahajan and - PowerPoint PPT Presentation

Team optimal decentralized state estimation Aditya Mahajan and Mohammad Afshari McGill University IEEE Conference on Decision and Control 19 December 2018

Let’s revisit separation of estimation and control in centralized systems

Decentralized estimation–(Afshari and Mahajan) 1 STANDARD LQG MODEL x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). min 𝔽 [ T ∑ t=1 [x(t) ⊺ Qx(t) + u(t) ⊺ Ru(t)]] Separation in estimation and control Choose u(t) = g t (y(1 : t), u(1 : t − 1)) to

Decentralized estimation–(Afshari and Mahajan) [x(t) ⊺ Qx(t) + u(t) ⊺ Ru(t)]] (L(t)x(t) + u(t)) ⊺ S(t)(L(t)x(t) + u(t)) + w(t) ⊺ P(t + 1)w(t)] t=1 ∑ T 𝔽 [ Total cost can be written as COMPLETION OF SQUARES t=1 1 ∑ T min 𝔽 [ y(t) = Cx(t) + v(t). x(t + 1) = Ax(t) + Bu(t) + w(t), STANDARD LQG MODEL Separation in estimation and control Choose u(t) = g t (y(1 : t), u(1 : t − 1)) to

Decentralized estimation–(Afshari and Mahajan) Total cost can be written as y(t) w(t) u(t) Linear System (L(t)x(t) + u(t)) ⊺ S(t)(L(t)x(t) + u(t)) + w(t) ⊺ P(t + 1)w(t)] t=1 ∑ T 𝔽 [ COMPLETION OF SQUARES 1 [x(t) ⊺ Qx(t) + u(t) ⊺ Ru(t)]] t=1 ∑ T min 𝔽 [ y(t) = Cx(t) + v(t). x(t + 1) = Ax(t) + Bu(t) + w(t), STANDARD LQG MODEL Separation in estimation and control Choose u(t) = g t (y(1 : t), u(1 : t − 1)) to

Decentralized estimation–(Afshari and Mahajan) T x(t) . ˜ ¯ y(t) w(t) u(t) Linear System (L(t)x(t) + u(t)) ⊺ S(t)(L(t)x(t) + u(t)) + w(t) ⊺ P(t + 1)w(t)] t=1 1 ∑ 𝔽 [ min 𝔽 [ STANDARD LQG MODEL x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Total cost can be written as Separation in estimation and control T ∑ t=1 [x(t) ⊺ Qx(t) + u(t) ⊺ Ru(t)]] COMPLETION OF SQUARES x(t) = part of state depending on u(1 : t) . x(t) = part of state depending on w(1 : t) . Choose u(t) = g t (y(1 : t), u(1 : t − 1)) to From linearity, x(t) = ¯ x(t) + ˜

Decentralized estimation–(Afshari and Mahajan) ∑ for total cost x(t) in expression z(t)−L¯ x(t) . ˜ ¯ y(t) w(t) u(t) Linear System 1 t=1 (L(t)x(t) + u(t)) ⊺ S(t)(L(t)x(t) + u(t)) + w(t) ⊺ P(t + 1)w(t)] T T STANDARD LQG MODEL x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). 𝔽 [ min 𝔽 [ ∑ t=1 [x(t) ⊺ Qx(t) + u(t) ⊺ Ru(t)]] COMPLETION OF SQUARES Total cost can be written as Separation in estimation and control x(t) = part of state depending on u(1 : t) . x(t) = part of state depending on w(1 : t) . Choose u(t) = g t (y(1 : t), u(1 : t − 1)) to From linearity, x(t) = ¯ x(t) + ˜ Substitute u(t) = ˆ

Decentralized estimation–(Afshari and Mahajan) w(t) 1 (L(t)˜ z(t)) ⊺ S(t)(L(t)˜ z(t)) + w(t) ⊺ P(t + 1)w(t)] Linear System u(t) y(t) T ¯ ˜ x(t) . z(t)−L¯ x(t) in expression for total cost ∑ t=1 = 𝔽 [ [x(t) ⊺ Qx(t) + u(t) ⊺ Ru(t)]] STANDARD LQG MODEL x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). min 𝔽 [ T (L(t)x(t) + u(t)) ⊺ S(t)(L(t)x(t) + u(t)) + w(t) ⊺ P(t + 1)w(t)] t=1 ∑ COMPLETION OF SQUARES Total cost can be written as 𝔽 [ T ∑ t=1 Separation in estimation and control x(t) = part of state depending on u(1 : t) . x(t) = part of state depending on w(1 : t) . Choose u(t) = g t (y(1 : t), u(1 : t − 1)) to From linearity, x(t) = ¯ x(t) + ˜ Substitute u(t) = ˆ x(t) + ˆ x(t) + ˆ

Decentralized estimation–(Afshari and Mahajan) ˜ 1 z(t)) + w(t) ⊺ P(t + 1)w(t)] Linear System u(t) w(t) y(t) ¯ x(t) . (L(t)˜ z(t)−L¯ x(t) in expression for total cost STATIC REDUCTION σ(y(1 : t), u(1 : t − 1)) = σ(˜ y(1 : t − 1)) . g t (˜ y(1 : t)) . z(t)) ⊺ S(t)(L(t)˜ Separation in estimation and control t=1 Total cost can be written as min 𝔽 [ T ∑ t=1 [x(t) ⊺ Qx(t) + u(t) ⊺ Ru(t)]] x(t + 1) = Ax(t) + Bu(t) + w(t), ∑ COMPLETION OF SQUARES 𝔽 [ T ∑ t=1 (L(t)x(t) + u(t)) ⊺ S(t)(L(t)x(t) + u(t)) + w(t) ⊺ P(t + 1)w(t)] = 𝔽 [ T STANDARD LQG MODEL y(t) = Cx(t) + v(t). Thus, wlog, consider ˆ z(t) = ˜ x(t) = part of state depending on u(1 : t) . x(t) = part of state depending on w(1 : t) . Choose u(t) = g t (y(1 : t), u(1 : t − 1)) to From linearity, x(t) = ¯ x(t) + ˜ Substitute u(t) = ˆ x(t) + ˆ x(t) + ˆ

Decentralized estimation–(Afshari and Mahajan) z(t)−L¯ z(t)) + w(t) ⊺ P(t + 1)w(t)] 1 u(t) w(t) y(t) ¯ ˜ x(t) . x(t) in expression (L(t)˜ for total cost STATIC REDUCTION σ(y(1 : t), u(1 : t − 1)) = σ(˜ y(1 : t − 1)) . g t (˜ y(1 : t)) . z(t) = −L 𝔽 [˜ y(1 : t)] z(t)) ⊺ S(t)(L(t)˜ Linear System t=1 Total cost can be written as STANDARD LQG MODEL x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). min 𝔽 [ T ∑ t=1 [x(t) ⊺ Qx(t) + u(t) ⊺ Ru(t)]] COMPLETION OF SQUARES Separation in estimation and control = 𝔽 [ ∑ 𝔽 [ T T ∑ t=1 (L(t)x(t) + u(t)) ⊺ S(t)(L(t)x(t) + u(t)) + w(t) ⊺ P(t + 1)w(t)] Thus, wlog, consider ˆ z(t) = ˜ x(t) = part of state depending on u(1 : t) . x(t) = part of state depending on w(1 : t) . Choose u(t) = g t (y(1 : t), u(1 : t − 1)) to x(t) | ˜ Thus, ˆ From linearity, x(t) = ¯ x(t) + ˜ Substitute u(t) = ˆ x(t) + ˆ x(t) + ˆ

Decentralized estimation–(Afshari and Mahajan) 2 Separation centralized stochastic control, the optimal control action depends on the solution of an estimation problem: 𝔽 [ T ∑ t=1 (L(t)˜ z(t)) ⊺ S(t)(L(t)˜ z(t))] Does the same happen in decentralized control? Motivation for current work x(t) + ˆ x(t) + ˆ

Decentralized estimation–(Afshari and Mahajan) 2 L 𝔽 [x(t) | I(t)] the best estimate? In decentralized estimation, is Does the same happen in decentralized control? z(t))] z(t)) ⊺ S(t)(L(t)˜ (L(t)˜ t=1 ∑ T 𝔽 [ estimation problem: optimal control action depends on the solution of an Separation centralized stochastic control, the Motivation for current work x(t) + ˆ x(t) + ˆ

Decentralized estimation–(Afshari and Mahajan) 2 and control. There is a long history of duality between estimation L 𝔽 [x(t) | I(t)] the best estimate? In decentralized estimation, is Does the same happen in decentralized control? z(t))] z(t)) ⊺ S(t)(L(t)˜ Motivation for current work (L(t)˜ t=1 ∑ T 𝔽 [ estimation problem: optimal control action depends on the solution of an Separation centralized stochastic control, the x(t) + ˆ x(t) + ˆ

Decentralized estimation–(Afshari and Mahajan) z(t))] certain applications. interesting in it’s own right in Decentralized estimation is estimation is interesting. Decentralized control is interesting. Ergo, decentralized and control. There is a long history of duality between estimation L 𝔽 [x(t) | I(t)] the best estimate? In decentralized estimation, is Does the same happen in decentralized control? Motivation for current work 2 z(t)) ⊺ S(t)(L(t)˜ (L(t)˜ t=1 ∑ T 𝔽 [ estimation problem: optimal control action depends on the solution of an Separation centralized stochastic control, the x(t) + ˆ x(t) + ˆ

DECENTRALIZED state estimation is fundamentally different from CENTRALIZED state estimation.

Decentralized estimation–(Afshari and Mahajan) 3 x ˆ z y x ∼ 𝒪(0, Σ x ) , y = Cx + v , v ∼ 𝒪(0, R) . z)] . Centralized estimation Choose ˆ z = g(y) to minimize 𝔽 [(Lx − ˆ z) ⊺ S(Lx − ˆ

Decentralized estimation–(Afshari and Mahajan) y = Cx + v , z = LKy , z)] . 3 v ∼ 𝒪(0, R) . Centralized estimation x ∼ 𝒪(0, Σ x ) , y z ˆ x OPTIMAL ESTIMATE : ˆ where K = Σ x C ⊺ (CΣ x C ⊺ + R) −1 . Choose ˆ z = g(y) to minimize 𝔽 [(Lx − ˆ z) ⊺ S(Lx − ˆ

Decentralized estimation–(Afshari and Mahajan) v ∼ 𝒪(0, R) . The optimal estimation strategy z = LKy , z)] . 3 Centralized estimation y = Cx + v , x ∼ 𝒪(0, Σ x ) , y z ˆ x OPTIMAL ESTIMATE : ˆ where K = Σ x C ⊺ (CΣ x C ⊺ + R) −1 . Choose ˆ z = g(y) to minimize 𝔽 [(Lx − ˆ z) ⊺ S(Lx − ˆ DOES NOT depend on the weight S .

Decentralized estimation–(Afshari and Mahajan) ⎦ z 1 ⊺ S [ z 1 ⎤ . 3 x ˆ z 1 ˆ z 2 y 1 y 2 ⎣[ Centralized estimation vs decentralized estimation y = Cx + v , x v ∼ 𝒪(0, R) . y z z)] . x ∼ 𝒪(0, Σ x ) , ˆ z = LKy , The optimal estimation strategy OPTIMAL ESTIMATE : ˆ where K = Σ x C ⊺ (CΣ x C ⊺ + R) −1 . Choose ˆ z = g(y) to minimize 𝔽 [(Lx − ˆ z) ⊺ S(Lx − ˆ DOES NOT depend on the weight S . y 1 = C 1 x + v 1 , y 2 = C 2 x + v 2 . Choose ˆ z 1 = g 1 (y 1 ) and ˆ z 2 = g 2 (y 2 ) to L 1 x − ˆ L 1 x − ˆ minimize 𝔽 ⎡ L 2 x − ˆ L 2 x − ˆ z 2 ] z 2 ]

Decentralized estimation–(Afshari and Mahajan) . 3 ⊺ S [ z 1 ⎤ ⎦ x ⎣[ ˆ z 1 ˆ z 2 y 1 y 2 ∑ i ∈ {1, 2}, z 1 Centralized estimation vs decentralized estimation v ∼ 𝒪(0, R) . The optimal estimation strategy x ∼ 𝒪(0, Σ x ) , z)] . y z y = Cx + v , z = LKy , ˆ x OPTIMAL ESTIMATE : ˆ where K = Σ x C ⊺ (CΣ x C ⊺ + R) −1 . Choose ˆ z = g(y) to minimize 𝔽 [(Lx − ˆ z) ⊺ S(Lx − ˆ DOES NOT depend on the weight S . OPTIMAL ESTIMATE : ˆ z i = F i y i , i ∈ {1, 2} , where y 1 = C 1 x + v 1 , j∈{1,2} [S ij F j Σ ji − S ij L j Θ i ] = 0, y 2 = C 2 x + v 2 . Choose ˆ z 1 = g 1 (y 1 ) and ˆ z 2 = g 2 (y 2 ) to and Σ ij = C i Σ x C ⊺ j + δ ij R i and Θ i = Σ x C ⊺ i . L 1 x − ˆ L 1 x − ˆ minimize 𝔽 ⎡ L 2 x − ˆ L 2 x − ˆ z 2 ] z 2 ]