Geodesic Density Tracking with Applications to Data Driven Modeling

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1 Geodesic Density Tracking with Applications to Data Driven Modeling Abhishek Halder and Raktim Bhattacharya Department of Aerospace Engineering Texas A&M University College Station, TX American Control Conference June 4, 2014

2 Motivating Applications Process industry applications Source: Chu et.al. (2011), Model Predictive Control and Optimization for Papermaking Process, doi: / NMR spectroscopy and MRI applications

3 Prior Work on Density Based Modeling and Control Covariance control: R.E. Skelton et. al. ( mid 1990s) Asymptotic density control: H. Wang et. al. (1999, 2001, 2005); Forbes, Forbes, Guay (2003) Ensemble control: Li and Khaneja (2007, 2009) Our contribution: Optimal mass transport framework for finite time feedback density control, data-driven modeling and refinement

4 Template Problem Statement Given: a sequence of observed joint densities η j for output vectors y j R d at times t j, j = 0, 1,..., M. Find: dynamical systems S j+1 : y j y j+1, over each horizon [t j, t j+1 ).

5 Template Problem Statement Given: a sequence of observed joint densities η j for output vectors y j R d at times t j, j = 0, 1,..., M. Find: dynamical systems S j+1 : y j y j+1, over each horizon [t j, t j+1 ). Minimum effort constraint: S j+1 minimizes total transportation cost y j+1 y j 2 R 2d l 2(R d ) ρ (y j, y j+1 ) dy j dy j+1 over all transportation policy ρ (y j, y j+1 ), such that y j η j and y j+1 η j+1.

6 Three Variants of the Template Problem Finite horizon feedback control of densities: Find state or output feedback u ( ) with pre-specified control structure on S j+1 Data driven d th order modeling: Find S j+1 with no a priori knowledge Density based model refinement: Think of source density η j as nominal model prediction, and target density η j+1 as true observation, at the same physical time. Find refined model from the nominal/baseline model.

7 Template Problem Optimal Mass Transport Gaspard Monge (1781), Leonid Kantorovich (1942): move a pile of soil from an excavation to another site through minimum work Defines Wasserstein distance W, a metric on the space of densities W 2 = optimal transport cost = inf y ŷ 2 ϱ P 2 (ρ, ρ) R 2d l 2(R d ) ϱ (y, ŷ) dydŷ ] = inf [ E y ŷ 2 ϱ P 2 (ρ, ρ) l 2(R d ) }{{} J 1 (ϱ) Equivalently, W 2 = inf β (ŷ) ŷ 2 β( ) l 2(R Ŷ d ) ρ (ŷ) dŷ, subject to }{{} J 2 (β) c (β) = det ( β) ρ β (ŷ) ρ (ŷ) = 0.

8 Optimal Mass Transport Background Brenier (1991): optimal β ( ) exists and is unique. Further, β ( ) = ψ. Here ψ : R d R, and is convex. Benamou & Brenier (2001): Consider the space-time variational T formulation T inf ϕ (ŷ, s) v (ŷ, s) 2 (ϕ,v) R d l 2(R 0 d ) dŷds subject to }{{} J 3 (ϕ,v) ϕ s + (ϕv) = 0, ϕ (, 0) = η, ϕ(, T ) = η. Then J 3 = W 2 and v is gradient flow. W 2 = inf J 1 (ϱ) ϱ P 2 (ρ, ρ) }{{} infinite dimensional LP T inf (ϕ,v) J 3 (ϕ, v) }{{} Nonsmooth convex optimization = inf J 2 (β) = β:c(β)=0 }{{} Nonlinear nonconvex optimization

9 Finite Horizon Feedback Control of Densities Theorem: Consider tracking Gaussians η j = N (µ j, Σ j ), under LTI structure x j+1 = Ax j + Bu j. Let ( ) θ j = µ j+1 µ j, Θ j = Σ 1 2 j+1 Σ 1 2 j+1 Σ j Σ j+1 Σ 2 j+1. The state feedback u j u (x j ) guaranteeing optimal transport 1. exists iff (Θ j A), θ j ker ( I BB ) 2. if exists, then must be affine form u j = K jx j + κ j, where K j = B (Θ j A) ( I BB ) R, and κ j = B θ j ( I BB ) r 3. is unique, if B is full rank.

10 Data Driven d th Order Modeling Duffing vector field (unknown to modeler) to generate data: ẋ 1 = x 2, ẋ 2 = αx 3 1 βx 1 δx 2, y = {x 1, x 2 }, α = 1, β = 1, δ = 0.5 Density propagation with 500 samples from initial density ξ 0 = U ([ 2, 2] 2) 10 snapshot data {t j, η j } 10 j=1 Subdivided each of the 10 intervals [t j, t j+1 ), j = 0,..., 9 into 60 sub-intervals. Want to compute optimal transport vector field v j j+1 for each of those intervals

11 Data Driven d th Order Modeling

12 Data Driven d th Order Modeling of v 1 2

13 Data Driven dth Order Modeling of v8 9 Abhishek Halder (TAMU) Geodesic Density Tracking ACC 2014, Portland, OR

14 Data Driven d th Order Modeling: Duffing transport vs. Optimal Transport for [t 1, t 2 )

15 Density Based Model Refinement

16 Density Based Model Refinement: Formulation Startegy: Only refine the output model (why?) For example, consider proposed model x = f ( x), ŷ = ĥ ( x) Call ŷ j ŷ (t j ). We know η j and η j. ( ) We seek β j : R no R no, so that ŷ j + = β j ŷj satisying ŷ j + η j and ŷ j η j Then the refined model is: x = f ( x), ŷ + j = β j ĥ ( x) Seek optimal push-forward: inf β j (ŷj subject to η j = β j η j. β( ) Ŷ ) ŷ j 2 l 2 (R no ) η jdŷ j } {{ } J 2 (β),

17 Linear Gaussian Model Refinement Theorem: Consider discrete-time deterministic LTI pairs: (A, C), (Â, Ĉ), sarting with ξ 0 = N (µ 0, Σ 0 ). Then refined model is: x j+1 = Â x j, ŷ j + = Θ j Ĉ x j + θ j. ( ) Θ j = Σ 1/2 j Σ 1/2 j Σ j Σ 1/2 1/2 1/2 j Σ j, and θ j = µ j µ j. The s th synthetic time PDF at j th physical time is: N ( µŷ y (s), Σŷ y (s) ), where [ µŷ y (s) = (1 s) ĈÂj + s CA j] µ 0, Σŷ y (s) = [(1 s) I + s Θ (j)] ( (ĈÂ j) Σ 0 (ĈÂ j) ) [(1 s) I + s Θ (j)].

18 Linear Gaussian Model Refinement Theorem: Consider discrete-time deterministic LTI pairs: (A, C), (Â, Ĉ), sarting with ξ 0 = N (µ 0, Σ 0 ). Then refined model is: x j+1 = Â x j, ŷ j + = Θ j Ĉ x j + θ j. ( ) Θ j = Σ 1/2 j Σ 1/2 j Σ j Σ 1/2 1/2 1/2 j Σ j, and θ j = µ j µ j. The s th synthetic time PDF at j th physical time is: N ( µŷ y (s), Σŷ y (s) ), where [ µŷ y (s) = (1 s) ĈÂj + s CA j] µ 0, Σŷ y (s) = [(1 s) I + s Θ (j)] ( (ĈÂ j) Σ 0 (ĈÂ j) ) [(1 s) I + s Θ (j)]. [ ] [ ] Example: A =, Â =, [ ] [ ] [ ] C =, Ĉ =. µ = {1, 3}, Σ 0 = 6 7

19 Linear Gaussian Model Refinement: Example

20 Summary Optimal transport framework for systems with density level observation Closed form feedback control for minimum effort linear Gaussian tracking Convex optimization framework for data driven modeling Closed form solution for linear Gaussian model refinement

21 Summary Optimal transport framework for systems with density level observation Closed form feedback control for minimum effort linear Gaussian tracking Convex optimization framework for data driven modeling Closed form solution for linear Gaussian model refinement Funding support: NSF CSR Award # Thank you.