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Gaussian Mixture Penalization for Trajectory Optimization Problems C. Rommel 1 , 2 , J. F. Bonnans 1 , B. Gregorutti 2 and P. Martinon 1 CMAP Ecole Polytechnique - INRIA 1 Safety Line 2 ISMP - July 2 nd 2018 ISMP - July 2 nd 2018 (CMAP, INRIA,

  1. Gaussian Mixture Penalization for Trajectory Optimization Problems C. Rommel 1 , 2 , J. F. Bonnans 1 , B. Gregorutti 2 and P. Martinon 1 CMAP Ecole Polytechnique - INRIA 1 Safety Line 2 ISMP - July 2 nd 2018 ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 1 / 30

  2. Motivation 20 000 airplanes — 80 000 flights per day, ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 2 / 30

  3. Motivation 20 000 airplanes — 80 000 flights per day, Should double until 2033, ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 2 / 30

  4. Motivation 20 000 airplanes — 80 000 flights per day, Should double until 2033, Responsible for 3% of CO 2 emissions, ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 2 / 30

  5. Motivation 20 000 airplanes — 80 000 flights per day, Should double until 2033, Responsible for 3% of CO 2 emissions, Accounts for 30% of operational cost for an airline, ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 2 / 30

  6. Motivation 20 000 airplanes — 80 000 flights per day, Should double until 2033, Responsible for 3% of CO 2 emissions, Accounts for 30% of operational cost for an airline, Rectilinear climb trajectories at full thrust. ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 2 / 30

  7. Motivation 20 000 airplanes — 80 000 flights per day, Should double until 2033, Responsible for 3% of CO 2 emissions, Accounts for 30% of operational cost for an airline, Rectilinear climb trajectories at full thrust. ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 2 / 30

  8. Optimal Control Problem � t f min C ( t , ✉ ( t ) , ① ( t )) dt , ( ① , ✉ ) ∈ X × U 0  ① = g ( t , ✉ , ① ) , ˙ for a.e. t ∈ [0 , t f ] , (OCP)   Φ( ① (0) , ① ( t f )) ∈ K Φ ,  s.t. ✉ ( t ) ∈ U ad , ① ( t ) ∈ X ad , for a.e. t ∈ [0 , t f ] ,   c ( ✉ ( t ) , ① ( t )) ≤ 0 , for all t ∈ [0 , t f ] .  ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 3 / 30

  9. Optimal Control Problem � t f min C ( t , ✉ ( t ) , ① ( t )) dt , ( ① , ✉ ) ∈ X × U 0  ① = g ( t , ✉ , ① ) , ˙ for a.e. t ∈ [0 , t f ] , (OCP)   Φ( ① (0) , ① ( t f )) ∈ K Φ ,  s.t. ✉ ( t ) ∈ U ad , ① ( t ) ∈ X ad , for a.e. t ∈ [0 , t f ] ,   c ( ✉ ( t ) , ① ( t )) ≤ 0 , for all t ∈ [0 , t f ] .  ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 3 / 30

  10. Optimal Control Problem � t f min C ( t , ✉ ( t ) , ① ( t )) dt , ( ① , ✉ ) ∈ X × U 0  ① = ˆ ˙ g ( t , ✉ , ① ) , for a.e. t ∈ [0 , t f ] , (OCP)   Φ( ① (0) , ① ( t f )) ∈ K Φ ,  s.t. ✉ ( t ) ∈ U ad , ① ( t ) ∈ X ad , for a.e. t ∈ [0 , t f ] ,   c ( ✉ ( t ) , ① ( t )) ≤ 0 , for all t ∈ [0 , t f ] .  ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 3 / 30

  11. Dynamics are learned from QAR data ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 4 / 30

  12. Dynamics are learned from QAR data ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 4 / 30

  13. Dynamics are learned from QAR data ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 4 / 30

  14. Dynamics are learned from QAR data See e.g. [Rommel et al., 2017a] and [Rommel et al., 2017b] ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 4 / 30

  15. ③ ① ✉ ③ Trajectory acceptability � t f min C ( t , ✉ ( t ) , ① ( t )) dt , ( ① , ✉ ) ∈ X × U 0  ① = ˆ ˙ g ( t , ✉ , ① ) , for a.e. t ∈ [0 , t f ] , (OCP)   Φ( ① (0) , ① ( t f )) ∈ K Φ ,  s.t. ✉ ( t ) ∈ U ad , ① ( t ) ∈ X ad , for a.e. t ∈ [0 , t f ] ,   c ( ✉ ( t ) , ① ( t )) ≤ 0 , for all t ∈ [0 , t f ] .  ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 5 / 30

  16. ③ Trajectory acceptability � t f min C ( t , ✉ ( t ) , ① ( t )) dt , ( ① , ✉ ) ∈ X × U 0  ① = ˆ ˙ g ( t , ✉ , ① ) , for a.e. t ∈ [0 , t f ] , (OCP)   Φ( ① (0) , ① ( t f )) ∈ K Φ ,  s.t. ✉ ( t ) ∈ U ad , ① ( t ) ∈ X ad , for a.e. t ∈ [0 , t f ] ,   c ( ✉ ( t ) , ① ( t )) ≤ 0 , for all t ∈ [0 , t f ] .  ⇒ ˆ ③ = (ˆ ① , ˆ ✉ ) solution of (OCP). ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 5 / 30

  17. Trajectory acceptability � t f min C ( t , ✉ ( t ) , ① ( t )) dt , ( ① , ✉ ) ∈ X × U 0  ① = ˆ ˙ g ( t , ✉ , ① ) , for a.e. t ∈ [0 , t f ] , (OCP)   Φ( ① (0) , ① ( t f )) ∈ K Φ ,  s.t. ✉ ( t ) ∈ U ad , ① ( t ) ∈ X ad , for a.e. t ∈ [0 , t f ] ,   c ( ✉ ( t ) , ① ( t )) ≤ 0 , for all t ∈ [0 , t f ] .  ⇒ ˆ ③ = (ˆ ① , ˆ ✉ ) solution of (OCP). Is ˆ ③ inside the validity region of the dynamics model ˆ g ? ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 5 / 30

  18. Trajectory acceptability � t f min C ( t , ✉ ( t ) , ① ( t )) dt , ( ① , ✉ ) ∈ X × U 0  ① = ˆ ˙ g ( t , ✉ , ① ) , for a.e. t ∈ [0 , t f ] , (OCP)   Φ( ① (0) , ① ( t f )) ∈ K Φ ,  s.t. ✉ ( t ) ∈ U ad , ① ( t ) ∈ X ad , for a.e. t ∈ [0 , t f ] ,   c ( ✉ ( t ) , ① ( t )) ≤ 0 , for all t ∈ [0 , t f ] .  ⇒ ˆ ③ = (ˆ ① , ˆ ✉ ) solution of (OCP). Is ˆ ③ inside the validity region of the dynamics model ˆ g ? Does it look like a real aicraft trajectory ? ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 5 / 30

  19. Trajectory acceptability 1 NATS UK air traffic control ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 6 / 30

  20. Trajectory acceptability Pilots acceptance 1 NATS UK air traffic control ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 6 / 30

  21. Trajectory acceptability Air Traffic Control 1 Pilots acceptance 1 NATS UK air traffic control ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 6 / 30

  22. Trajectory acceptability Air Traffic Control 1 Pilots acceptance How can we quantify the closeness from the optimized trajectory to the set of real flights? 1 NATS UK air traffic control ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 6 / 30

  23. Likelihood Let X be a random variable following an absolutely continuous probability distribution with density function f depending on a parameter θ . Then the function L ( θ | x ) = f θ ( x ) (1) considered as a function of θ , is the likelihood function of theta , given the outcome x of X . 0 Picture source: wikipedia, P-Value, author: Repapetilto CC. ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 7 / 30

  24. Likelihood Let X be a random variable following an absolutely continuous probability distribution with density function f depending on a parameter θ . Then the function L ( θ | x ) = f θ ( x ) (1) considered as a function of θ , is the likelihood function of theta , given the outcome x of X . 0 Picture source: wikipedia, P-Value, author: Repapetilto CC. ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 7 / 30

  25. Likelihood Let X be a random variable following an absolutely continuous probability distribution with density function f depending on a parameter θ . Then the function L ( θ | x ) = f θ ( x ) (1) considered as a function of θ , is the likelihood function of theta , given the outcome x of X . In our case: the optimized trajectory plays the role of θ , 0 Picture source: wikipedia, P-Value, author: Repapetilto CC. ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 7 / 30

  26. Likelihood Let X be a random variable following an absolutely continuous probability distribution with density function f depending on a parameter θ . Then the function L ( θ | x ) = f θ ( x ) (1) considered as a function of θ , is the likelihood function of theta , given the outcome x of X . In our case: the optimized trajectory plays the role of θ , the set of real flights plays the role of x , 0 Picture source: wikipedia, P-Value, author: Repapetilto CC. ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 7 / 30

  27. How to apply this to functional data? Assumption: We suppose that the real flights are observations of the same functional random variable Z = ( Z t ) valued in C ( T , E ), with E compact subset of R d and T = [0 , t f ]. ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 8 / 30

  28. How to apply this to functional data? Assumption: We suppose that the real flights are observations of the same functional random variable Z = ( Z t ) valued in C ( T , E ), with E compact subset of R d and T = [0 , t f ]. Problem: Computation of probability densities in infinite dimensional space is untractable... ISMP - July 2 nd 2018 (CMAP, INRIA, Safety Line) GMM Penalization for OCP 8 / 30

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