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

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

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

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

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

Sequential Method control profile (parameterization)

Sequential Method decision variables

Sequential Method decision variables

Sequential Method decision variables NLP solver Calculates optimal control profile

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

Wavelets Sequential Method NLP solver

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

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

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

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

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

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

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

Wavelets Analysis Haar wavelet has been considered:

Wavelets Analysis Haar wavelet has been considered:

Wavelets Analysis Haar wavelet has been considered: Orthogonal basis

Wavelets Analysis

Control profile NLP solver Wavelets Analysis NLP solver

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

Wavelets Thresholding Analysis details

Wavelets Thresholding Analysis Thresholding: some details are eliminated. details

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

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):

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

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

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.

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

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

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

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

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

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

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

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

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

Case Studies

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

Control profile evolution

Results: Semi-batch Isothermal Reactor Uniform mesh with 128 stages Fixed Threshold Fixed Threshold Fixed Threshold Visushirink Iter. Total CPU ns Total CPU ns Total CPU ns Total CPU ns Total CPU ns Reference CPU time: Uniform mesh

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

Control profile evolution

Results: Bioreactor problem ) Uniform mesh with 128 stages Fixed Threshold Fixed Threshold Fixed Threshold Visushirink Iter. Total CPU ns Total CPU ns Total CPU ns Total CPU ns Total CPU ns Reference CPU time: Uniform mesh

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

Control profile evolution

Results: Bioreactor problem Uniform mesh with 64 stages Fixed Threshold Fixed Threshold Visushirink Iter. Total CPU ns Total CPU ns Total CPU ns Total CPU ns Reference CPU time: Uniform mesh

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;

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.

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.

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.

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.

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