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

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

6. Sequential Method control profile (parameterization)

7. Sequential Method decision variables

8. Sequential Method decision variables

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

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

11. Wavelets Sequential Method NLP solver

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

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

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

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

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: where is the maximum level resolution. details control variable Vector of wavelets details Inner product ResolutionPosition

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. Control profile NLP solver Wavelets Analysis NLP solver

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

25. Wavelets Thresholding Analysis details

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

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

28. Thresholding strategies Thresholding: (i)decomposition of the data ; (ii)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: (i)decomposition of the data ; (ii)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): standard deviation of a white noise details coefficients

30. Thresholding strategies Thresholding: (i)decomposition of the data ; (ii)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): standard deviation of a white noise details coefficients

31. How Wavelets Work Control profile NLP solver Wavelets Sequential Algorithm 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.

32. Algorithm and Parameters 1.Integrator: Runge Kutta fourth order (ode45 Matlab); 2.Optimisation: Interior Point (Matlab) was used as NLP solver; 3.Wavelets: Routines of Matlab 7.6; 4.Stop Criteria

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

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

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

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

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

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

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

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

41. Case Studies

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

43. Control profile evolution

47. Results: Semi-batch Isothermal Reactor Uniform mesh with 128 stages Fixed Threshold 10 -4 Fixed Threshold 10 -5 Fixed Threshold 10 -7 Visushirink Iter. Total CPU ns Total CPU ns Total CPU ns Total CPU ns Total CPU ns 18888 216 322 2416 426303220 52836 30 632424034 7364642 8 5854 946 10 11 11280.76460.85580.91540.2234 Reference CPU time: Uniform mesh

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

49. Control profile evolution

55. Results: Bioreactor problem ) Uniform mesh with 128 stages Fixed Threshold 10 -1 Fixed Threshold 10 -4 Fixed Threshold 10 -7 Visushirink Iter. Total CPU ns Total CPU ns Total CPU ns Total CPU ns Total CPU ns 18888 216 326243220 430323826 538424634 644485846 748566664 8567472 97084 1 1280.8270 1.32741.58840.37 64 Reference CPU time: Uniform mesh

56. Mixture of Catalysts (BELL and SARGENT, 2000) Optimal Control Profile: 64 stages

57. Control profile evolution

60. Results: Bioreactor problem Uniform mesh with 64 stages Fixed Threshold 10 -1 Fixed Threshold 10 -4 Visushirink Iter. Total CPU ns Total CPU ns Total CPU ns Total CPU ns 1888 216 320 422 26 52830 1640.43280.56300.1426 Reference CPU time: Uniform mesh

61. Conclusions 1.Two threshold procedures were analyzed here: Fixed and Visuhrink (level dependent threshold). According to the results, Visushrink strategy is able to denoise the details in a more efficient way than Fixed strategy, that is more conservative. We have shown that the choice of a threshold procedure can improve the wavelet adaptation algorithm;

62. Conclusions 1.Two threshold procedures were analyzed here: Fixed and Visuhrink (level dependent threshold). According to the results, Visushrink strategy is able to denoise the details in a more efficient way than Fixed strategy, that is more conservative. We have shown that the choice of a threshold procedure can improve the wavelet adaptation algorithm; 2.Other examples from literature ( BINDER et al., 2000; SRINIVASAN et al., 2003; SCHLEGEL, 2004) was solved and have been presented similar results: a considerable improvement of CPU time when Visushrink is used.

63. Conclusions 1.Two threshold procedures were analyzed here: Fixed and Visuhrink (level dependent threshold). According to the results, Visushrink strategy is able to denoise the details in a more efficient way than Fixed strategy, that is more conservative. We have shown that the choice of a threshold procedure can improve the wavelet adaptation algorithm; 2.Other examples from literature ( BINDER et al., 2000; SRINIVASAN et al., 2003; SCHLEGEL, 2004) was solved and have been presented similar results: a considerable improvement of CPU time when Visushrink is used. 3.Other wavelets thresholding strategies (as Sureshrink and Minimaxi) has been investigated, however in some cased these strategies have undesirable results, with worse performance than Fixed strategy.

64. Future Works 1.This algorithm will be used to solve more complex problems with several control variables in order to improve the sequential adaptation of each control profile. Our expectation is to observe the intensification of threshold influence for these problems.

65. Future Works 1.This algorithm will be used to solve more complex problems with several control variables in order to improve the sequential adaptation of each control profile. Our expectation is to observe the intensification of threshold influence for these problems. 2.As observed here, wavelets are able to detect discontinuity points and therefore the location of different control arcs. A more sophisticated interpolation of control profile will be implemented in these regions with aims to reduce he number of stages and consequently decision variables.

66. Thank You