1. IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse Inverse method for pyrolysable and ablative materials with optimal control formulation S.Alestra 1, J.Collinet 2, and F.Dubois 3 stephane.alestra@eads.netjean.collinet@astrium.eads.net 2 EADS ASTRIUM ST Les Mureaux, France 1 EADS Innovation Works Toulouse, FRANCE 3 Conservatoire National des Arts et Métiers Paris, France fdubois@cnam.fr

2. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p2 Atmospheric re-entry missions Reentry Aerothermodynamics Team at EADS ASTRIUM-ST  Multiphysics : aerodynamics, aerothermodynamics, plasma  design and sizing of the Thermal Protection System (TPS) of the aerospace vehicles  the identification of heat fluxes is of great industrial interest ARD Industrial problem Huygens probe (on Titan)

3. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p3 Internal energy balance Pyrolysis: Arrhenius equation Surface energy balance Surface Recession Decomposition and mass balance General equations of direct problem input data: heat fluxoutput data: temperature

4. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p4  Evaluate the heat fluxes from temperature measurements on thermal protection with ablation and pyrolysis Inverse Heat problem is hard !! : see theory, diffusion aspect,.. « Monopyro » 1D numerical tool, EADS ASTRIUM ST The inverse method  (t), t in [O,T] p(t)=   (t) ? T in [0,T] p(t)=   (t) ?

5. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p5 Direct Problem vector of temperature T and ablation s, functions of time t and position x. => system of coupled nonlinear time domain evolution differential equations:  The other variables described above are hidden in the formulation of F  System is rewritten in reduced variables s e T1T1 T2T2 T3T3  (t) X (t) p(t)=   (t) Heat Flux

6. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p6 Direct Discrete scheme  Heat Flux N~=2000  K number of one-dimensional grid points (~100), N number of time (N~=2000) iterations in the numerical scheme  The equation is written at time (n+1) :  Linearization at time n  forward time discrete linearized Euler scheme, with initial condition vanishing: stability

7. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p7 Cost Function  time domain unknown heat flux convection coefficient  Quadratic error or cost function j(p)  Measured temperature  Computed temperature  To minimize this quantity, by optimization algorithm  we need the derivatives of J(p), with respect to p. The inverse method

8. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p8 Adjoint System  Adjoint variable : dual multiplyer of  Lagrangian L + calculus of variations  Cancel the variations of  L with respect to   Direct system, forward in time, initial vanishing condition  Cancel the variations of  L with respect to  w  Adjoint system, backward in time, final vanishing condition The inverse method

9. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p9 Gradient computation  With this particular choice of , the gradient of the cost function is simply obtained by :  Variations  L function of  p  discrete gradients  apply an iterative inverse procedure minimizing J(p) to estimate the unknown parameter optimal function The inverse method

10. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p10 Gradient computation The inverse method Adjoint State Final Condition Gradients Direct State Initial condition time W = (T,s)  df / dp = complex, non linear df / dW = complex, non linear, tables Measurements

11. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p11 Optimization The inverse method Direct problem T(p) Optimization p+  p 1) Steepest Descent to explore 2) Quasi Newton to finish convergence (T(p)-  )**2 p0,  P optimal GradientApproximation of Hessian Direct + Adjoint system Can be computed by Automatic Differentiation tool

12. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p12 Automatic Differentiation TAPENADE, INRIA Sophia Antipolis, France The inverse method  Program (Fortran) : sequence of elementary arithmetic operations  Derivatives can be computed automatically  If the code is modified, it is more easy to compute new adjoints and new gradients Input function f cost function J(p) Output functions f’ gradient dJ/dp AD Tool

13. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p13 Automatic Differentiation The inverse method  Direct problem instruction  Cost Function  Adjoint system instructions : differentiation in reverse mode, with push, pop  Gradient computed by reverse mode time

14. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p14 Virgin material, low heat flux)  Pseudo measurements very well rebuilt (RMS<1K)  Automatic Differentiation (AD) sucessful Some applications Heat Flux without AD Heat Flux with AD Gradient Quasi Newton Cost Function

15. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p15 Carbon/Resin with ablation, pyrolysis & pseudo measurements  Results OK with pyrolysis and ablation (without and with AD)  Results OK with 2% noise on pseudo measurement  Tichonov regularization to stabilize the solution Some applications Convection (noise without regularization) Convection (noise with regularization) Cost Function Gradient Quasi Newton

16. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p16 ARD  1998 on Ariane Flight 503  First use of the inverse method for « in-flight » rebuilding during ARD post-flight analysis (1999)  Last improvements of the method OK Some applications

17. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p17 Plasma Torch Facility Material to be tested Nozzle Ablation compensation Fluxmeters Measurements: Laser (ablation) Pyrometer (surface temperature) TC1 TC2 TC3 TC4 TC5 TC6 TC7 TC8

18. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p18 Plasma torch: First results ONLY ONE SENSOR USED Influence of sensor used  Ablation restitution Influence of sensor used  Temperature at thermocouple

19. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p19 Missing sensors  Temperature at Thermocouple Missing sensors  convection coefficient restitution Influence of sensor used  Heat Flux restitution Influence of sensor used  Temperature at surface ONLY ONE SENSOR USED SEVERAL SENSORS USED AT THE SAME TIME

20. This document is the property of Astrium. It shall not be communicated to third parties without prior written agreement. Its content shall not be disclosed. Astrium Space Transportation IFAC Control of Distributed Parameter Systems, July 20-24, 2009, Toulouse 19/12/2006p20 Conclusion & perspectives  Conclusion:  Inverse method sucessfully implemented  First tests of Automatic Differentiation promising  Validation OK for pseudo-measurements (with or without noise)  Good results obtained on hard cases  Perspectives:  Theoretical aspects (observability, identificability) to be analyzed  Improve robustness of the method (initial guess,uncertainties on noise, regularization)  test on industrial re-entry problems  Improve automatic differentiation version for hard cases