Stochastic control for underwater optimal trajectories CQFD & DCNS. Inria Bordeaux Sud Ouest & University of Bordeaux France

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1 Stochastic control for underwater optimal trajectories CQFD & DCNS Inria Bordeaux Sud Ouest & University of Bordeaux France CQFD Inria Bordeaux Sud Ouest Departamento de Engenharia Elétrica, USP São Carlos, 6 June 2013

2 Outline Presentation of CQFD Inria Themes Members Grants Trajectories optimization for submarines Problem Solution Examples How to join us Job opportunities at inria Departamento de Engenharia Elétrica USP São Carlos 6 June /24

3 Presentation of CQFD Inria What is Inria? I Public science and technology institution established in 1967 I Only public research body fully dedicated to computational sciences I Combining computer sciences with applied mathematics Inria s Research Centres Inria LILLE Nord Europe I I 8 research centers in France 3,400 researchers 1000 PhD students Inria PARIS - Rocquencourt Inria NANCY Grand Est Inria SACLAY Île-de-France Inria RENNES Bretagne Atlantique Inria GRENOBLE Rhône-Alpes Inria BORDEAUX Sud-Ouest Inria SOPHIA ANTIPOLIS Méditerranée Departamento de Engenharia Elétrica USP São Carlos 6 June /24

4 Presentation of CQFD Themes Inria team CQFD Inria research program on Stochastic methods and models Research themes Modeling of random complex systems Estimation of model parameters and performances Control performance optimization Tools Probability and statistics Departamento de Engenharia Elétrica USP São Carlos 6 June /24

5 Presentation of CQFD Members Members Permanent members François Dufour Marie Chavent Head of team Benoîte de Saporta Anne Gégout-Petit Jérôme Saracco Huilong Zhang PhD students Romain Azaïs Camille Baysse Isabelle Charlier Karim Claudio + 1 internship Raphaël Coudret Amaury Labenne Shuxian Li Laurent Vézard Christophe Nivot Departamento de Engenharia Elétrica USP São Carlos 6 June /24

6 Presentation of CQFD Grants Partnerships with industry Astrium (space transportation) stochastic models for crack propagation maintenance optimization for a structure subject to corrosion optimization of the assembly line of the new generation launcher DCNS (naval defense) optimization of trajectories for submarines EDF (electricity) modeling of the failures in the secondary circuit of a nuclear power plant Thales optronic (military equipment) health monitoring of an optronic equipment Departamento de Engenharia Elétrica USP São Carlos 6 June /24

7 Problem Outline Presentation of CQFD Trajectories optimization for submarines Problem Solution Examples How to join us Departamento de Engenharia Elétrica USP São Carlos 6 June /24

8 Trajectories optimization for submarines Problem Problem set by DCNS Submarine with sensors surrounded by targets Departamento de Engenharia Elétrica USP São Carlos 6 June /24

9 Trajectories optimization for submarines Problem Aim Optimization Propose an optimal trajectory for the submarine to hear the targets at best Departamento de Engenharia Elétrica USP São Carlos 6 June /24

10 Trajectories optimization for submarines Problem Difficulty: under water sound propagation Departamento de Engenharia Elétrica USP São Carlos 6 June /24

11 Solution Solution proposed by CQFD First step : Modeling position and speed of the submarine known position and speed of targets unknown random constant immersion and speed + noise Departamento de Engenharia Elétrica USP São Carlos 6 June /24

12 Solution Second step : discretization Aim Turn the problem in continuous time and space into a problem in discrete time and space Time discretization time step t = 1 minute Discretization of the submarine position Immersion grid with step z = 6 metres maximal possible variation in 1minute: ±4 z Departamento de Engenharia Elétrica USP São Carlos 6 June /24

13 Solution Discretization of targets position Approximate continuous (position and speed of targets) random variables by discrete ones in an intelligent way Departamento de Engenharia Elétrica USP São Carlos 6 June /24

14 Solution Discretization of targets position Approximate continuous (position and speed of targets) random variables by discrete ones in an intelligent way Departamento de Engenharia Elétrica USP São Carlos 6 June /24

15 Solution Discretization of targets position Approximate continuous (position and speed of targets) random variables by discrete ones in an intelligent way Departamento de Engenharia Elétrica USP São Carlos 6 June /24

16 Solution Algorithm Model simulator of trajectories grids Departamento de Engenharia Elétrica USP São Carlos 6 June /24

17 Solution Algorithm Model simulator of trajectories grids Departamento de Engenharia Elétrica USP São Carlos 6 June /24

18 Solution Algorithm Model simulator of trajectories grids Departamento de Engenharia Elétrica USP São Carlos 6 June /24

19 Solution Algorithm Model simulator of trajectories grids Departamento de Engenharia Elétrica USP São Carlos 6 June /24

20 Solution Algorithm Model simulator of trajectories grids New observed position nearest neighbor projection in the grid Departamento de Engenharia Elétrica USP São Carlos 6 June /24

21 Solution Third step : mathematical formulation of the problem Markov decision process (MDP) (X, A, {A(x), x X}, Q, c) X state space, possible positions for the submarine and targets A action space, possible maneuvers for the submarine A(x) possible actions at state x Q Markov kernel, gives the new relative positions of targets given the action chosen c performance function, acoustic loss Departamento de Engenharia Elétrica USP São Carlos 6 June /24

22 Solution Dynamic programming Optimal control problem for MDP Find the policy (a 0, a 2,..., a N 1 ) that minimizes the loss [ N 1 ] J (x 0 ) = min E c(x n, a n ) + c(x N ) (a 0,a 2,...,a N 1 ) Solution by dynamic programming n=0 J N (x) = c(x) J n (x) = min a A(x) E [ c(x, a) + J n+1 (y)q(x, a; dy) ] J 0 (x) = J (x) Departamento de Engenharia Elétrica USP São Carlos 6 June /24

23 Solution Fourth step: numerical solution Start with a simplified model and make it more realistic step by step First models studied one target, known constant immersion, submarine maneuvers only in immersion two targets, known constant immersion, submarine maneuvers only in immersion several targets, constant + noise mmersion, submarine maneuvers only in immersion Departamento de Engenharia Elétrica USP São Carlos 6 June /24

24 Examples Problem: one target immersion of target: 500m, of submarine: 300m initial distance between target and submarine: 35km initial relative speed of target wrt submarine: 10ms 1 computation horizon: 45min Departamento de Engenharia Elétrica USP São Carlos 6 June /24

25 Trajectories optimization for submarines Examples Results: one target zde = 12m Figure 2: Diagramme perte et contro le optimal pour z 6m 3: Diagramme de = perte et contro le optimal pour z Figure = 12.35m z= References sur le me me dessin, on trace (par un trait blanc) l historique de l immersion du porteur, le mouvement du bruiteur est repre sente ici par la distance relative enp. remarquer BERTSEKAS and J. N. TSITSIKLIS. Neuro-Dynamic tre porteur et bruiteur. Le re sultat semble tre s satisfaisant. [1] OnD.peut Optimization and neural computations series. Athena Scientific, B que sachusetts, dans un premier temps, le porteur e ectue un mouvement vers le [2] O. rapide HERNANDEZ-LERMA and J.B. LASSERRE. Discrete-time M processes,usp volume of Applications of mathematics. Springer-Verla de Engenharia Elétrica São 30 Carlos 6 June /24 haut afin d atteindre au plusdepartamento vite la zone rouge,

26 Examples Results: two targets Two targets immersion of targets: 500m and 100m, of submarine: 300m initial distance between targets and submarine: 35km 50km initial relative speed of target wrt submarine: 12ms 1 computation horizon: 45min Departamento de Engenharia Elétrica USP São Carlos 6 June /24

27 Examples Results: four targets Four targets immersion of targets: 600m, 300m, 100m et 400m, of submarine: 300m initial speed of targets 20 knots of submarine: 25 knots computation horizon: 45 minutes!"#$"!"#$"!"#$" %&"#$" Departamento de Engenharia Elétrica USP São Carlos 6 June /24

28 Examples Validation Simple models Advantages simpler problems to solve easy visual validation validation of practical feasability Drawbacks little realistic Departamento de Engenharia Elétrica USP São Carlos 6 June /24

29 Examples Further works 3D maneuvers for submarine long computation time Departamento de Engenharia Elétrica USP São Carlos 6 June /24

30 Examples Further works 3D maneuvers for submarine long computation time other types of mission: hear without being heard constraints Departamento de Engenharia Elétrica USP São Carlos 6 June /24

31 Examples Further works 3D maneuvers for submarine long computation time other types of mission: hear without being heard constraints ongoing work use data from tracking algorithms to update target positions sequence of short-term optimization problems Departamento de Engenharia Elétrica USP São Carlos 6 June /24

32 How to join us Job opportunities at inria How to join us Job opportunities at Inria internships PhD theses Post doc positions Permanent researcher positions Have a look at Inria website or contact the head of team François Dufour francois.dufour@math.u-bordeaux1.fr Departamento de Engenharia Elétrica USP São Carlos 6 June /24

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