Optimal Control Theory

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Optimal Control Theory. Batch Beer Fermentation. General Case. Min/max. General Case. Min Φ = Endpoint cost L =Lagrangian u = Control X= State. General Case. Min Φ = Endpoint cost- final product L =Lagrangian u = Control X= State. General Case. Min - PowerPoint PPT Presentation

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Optimal Control Theory

Optimal Control TheoryBatch Beer FermentationGeneral CaseMin/max

General CaseMin = Endpoint cost L =Lagrangianu = Control X= State

General CaseMin = Endpoint cost- final product L =Lagrangianu = Control X= State

General CaseMin = Endpoint cost- final product L = Lagrangian describes dynamics of systemu = Control X= State

General CaseMin = Endpoint cost- final product L = Lagrangian describes dynamics of systemu = Control what we can do to the systemX= State

General CaseMin = Endpoint cost- final product L = Lagrangian describes dynamics of systemu = Control what we can do to the systemX= State properties of the system

Case of BeerMin = Endpoint cost- profit, quality X= State properties of the systemu = Control what we can do to the system L = Lagrangian describes dynamics of system

Case of BeerMin = Endpoint cost- profit, quality X= State concentrations of yeast and organic and inorganic chemical species.u = Control what we can do to the system L = Lagrangian describes dynamics of system

Case of BeerMin = Endpoint cost- profit, quality X= State concentrations of yeast and organic and inorganic chemical species.u = Control temperature L = Lagrangian describes dynamics of system

Case of BeerMin = Endpoint cost- profit, quality X= State concentrations of yeast and organic and inorganic chemical species.u = Control temperature L = Lagrangian equations relating state variables and controls.

Quadratic CaseChemical ReactionsA+BCRate = k[A]^a[B]^ba and b are determined experimentallyUsed to determine mechanisms[] = concentration

Beer Basics

FermentationYeast consume sugars and produce CO2 and ethanol.The yeast also produce other chemicals. Most side products are bad: ketones, aldehydes, sulfur compounds, other alcohols; however, esters are good.Main factors influencing side products are temperature, amino acids, and pH levels.ControlsCommercial breweries can controlTemperature refrigeration (most important)Can be expensivepH, amino acids, sugar, yeast initial conditions

OptimizationDifferent methods have been usedSequential quadratic programming (SQP)Gradient methodDynamic programming Calculus of variationsNeural NetworksMultiple objectives to considerProfessional results:Most conclusions end up at a very narrow region between 10-15*CSQP method found a rapid rise to 13*C then slow accent to 13.5*CDifference is 6.7% increase in ethanol productionSimple ModelAssumptionsYeast is the only consumer of resourcesSugar is the only growth limiting resourceWort is deoxygenated at t=0Temperature and pressure are constant Production of side products are minimal/ignored

Simple modelRelates yeast, alcohol and sugar levels. System of nonlinear ODEsdS=-m*Y*SdY=k*S*Y - d*Y^2 - p*A*YdA=b*Y*Sk, d, p, m, b = constants @ temp=TResults

Constants chosen for visible details not accuracy.Units on vertical axis are arbitrary and different for each plot.

SourcesG.E. Carrillo-Ureta, P.D. Roberts, V.M. Becerra, Optimal Control of a Fermentation ProcessW. Fred Rameriz, Jan Maciejowski, Optimal Beer Fermentation Pascale B. Dengis, L.R. NeLissen, Paul G. Rouxhet, mechanisms of yeast flocculation: comparison of top and bottom-fermenting strains, applied and environmental microbiology, Feb. 1995, p. 718-728, Vol. 61,No. 2http://en.wikipedia.org/wiki/Optimal_controlAnatoly Zlotnik