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E. C. Biscaia Jr ., A. R. Secchi, L. S. Santos

Dynamic Optimisation Using Wavelets Bases. E. C. Biscaia Jr ., A. R. Secchi, L. S. Santos Programa de Engenharia Química (PEQ) – COPPE – UFRJ Rio de Janeiro - Brazil. Aims of the Contribution. Improve numerical methods for solving dynamic optimisation problems:. s.t. Sequential Method.

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E. C. Biscaia Jr ., A. R. Secchi, L. S. Santos

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  1. Dynamic Optimisation Using Wavelets Bases E. C. BiscaiaJr., A. R. Secchi,L. S. Santos Programa de Engenharia Química (PEQ) – COPPE – UFRJ Rio de Janeiro - Brazil

  2. Aims of the Contribution Improve numerical methods for solving dynamic optimisation problems: s.t

  3. Sequential Method Control variables are discretized and dynamic model is solved numerically at each iteration of the NLP

  4. Sequential Method Control variables are discretized and dynamic model is solved numerically at each iteration of the NLP

  5. Sequential Method Control variables are discretized and dynamic model is solved numerically at each iteration of the NLP discretization in time domain ns stages

  6. Sequential Method control profile (parameterization)

  7. Sequential Method decision variables

  8. Sequential Method decision variables

  9. Sequential Method NLP solver Calculates optimal control profile decision variables

  10. Sequential Method NLP solver NLP solver NLP solver Successive Refinement Calculates optimal control profile Initial profile Refinement decision variables

  11. Wavelets Sequential Method NLP solver

  12. Wavelets Sequential Method NLP solver Improving Adaptation of discrete points at each iteration Wavelets

  13. Wavelets Sequential Method NLP solver Improving Adaptation of discrete points at each iteration Wavelets

  14. Wavelets Sequential Method NLP solver Improving Adaptation of discrete points at each iteration Wavelets new mesh

  15. Wavelets Sequential Method NLP solver Improving Adaptation of discrete points at each iteration Wavelets new mesh

  16. Wavelets Analysis Considering a function , it can be transformed into wavelet domain as: details

  17. Wavelets Analysis Considering a function , it can be transformed into wavelet domain as: details control variable Inner product

  18. Wavelets Analysis Considering a function , it can be transformed into wavelet domain as: details control variable Inner product Vector of wavelets details Resolution Position where is the maximum level resolution.

  19. Wavelets Analysis Haar wavelet has been considered:

  20. Wavelets Analysis Haar wavelet has been considered:

  21. Wavelets Analysis Haar wavelet has been considered: Orthogonal basis

  22. Wavelets Analysis

  23. Wavelets Analysis NLP solver NLP solver Control profile

  24. How Wavelets Work NLP solver NLP solver Iteration 2 Wavelets Wavelets Iteration 1 Control profile

  25. Wavelets Thresholding Analysis details

  26. Wavelets Thresholding Analysis details Thresholding: some details are eliminated.

  27. Wavelets Thresholding Analysis details Thresholding: some details are eliminated. New thresholded control profile

  28. Thresholding strategies • Thresholding: • decomposition of the data ; • comparing detail coefficients with a given threshold value and shrinking coefficients close to zero, eliminating data noise effects (DONOHO and JOHNTONE, 1995):

  29. Thresholding strategies • Thresholding: • decomposition of the data ; • comparing detail coefficients with a given threshold value and shrinking coefficients close to zero, eliminating data noise effects (DONOHO and JOHNTONE, 1995): • Visushrink (DONOHO, 1992): details coefficients standard deviation of a white noise

  30. Thresholding strategies • Thresholding: • decomposition of the data ; • comparing detail coefficients with a given threshold value and shrinking coefficients close to zero, eliminating data noise effects (DONOHO and JOHNTONE, 1995): • Visushrink (DONOHO, 1992): • Fixed user specified (SCHLEGEL and MARQUARDT,2004 and BINDER, 2000): details coefficients standard deviation of a white noise

  31. How Wavelets Work NLPsolver NLP solver Incorporate the Visushrink threshold procedure and compare with other fixed threshold parameters; Observe if the CPU is affected by changes of threshold rule. Improve, at each iteration, the estimate of control profile. Sequential Algorithm Wavelets Wavelets Control profile

  32. Algorithm and Parameters • Integrator: Runge Kutta fourth order (ode45 Matlab); • Optimisation: Interior Point (Matlab) was used as NLP solver; • Wavelets: Routines of Matlab 7.6; • Stop Criteria

  33. Flowsheet of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor (Srinivasanet al. ,2003) Constant by parts interpolation

  34. Flowsheet of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor (Srinivasanet al. ,2003)

  35. Flowsheet of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor (Srinivasanet al. ,2003)

  36. Flowsheet of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor (Srinivasanet al. ,2003) Visushrink Threshold Fixed Threshold

  37. Flowsheet of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor (Srinivasanet al. ,2003) Visushrink Threshold Fixed Threshold

  38. Flowsheet of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor (Srinivasanet al. ,2003) Visushrink Threshold Fixed Threshold

  39. Flowsheet of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor (Srinivasanet al. ,2003) Locations of discontinuity points ~ large details coefficients Visushrink Threshold Fixed Threshold

  40. Flowsheet of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor (Srinivasanet al. ,2003) Visushrink Threshold Fixed Threshold

  41. Case Studies

  42. Semi-batch Isothermal Reactor (Srinivasanet al., 2003)) Optimal Control Profile: 128 stages

  43. Control profile evolution

  44. Control profile evolution

  45. Control profile evolution

  46. Control profile evolution

  47. Results: Semi-batch Isothermal Reactor Reference CPU time: Uniform mesh

  48. Bioreactor problem (CaNTOet al. , 2001)) M: monomer S: substrate Optimal Control Profile: 128 stages

  49. Control profile evolution

  50. Control profile evolution

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