1. Optimal Control Theory Batch Beer Fermentation

2. General Case Min/max

3. General Case Min Φ = Endpoint cost L =Lagrangian u = Control X= State

4. General Case Min Φ = Endpoint cost- final product L =Lagrangian u = Control X= State

5. General Case Min Φ = Endpoint cost- final product L = Lagrangian – describes dynamics of system u = Control X= State

6. General Case Min Φ = Endpoint cost- final product L = Lagrangian – describes dynamics of system u = Control – what we can do to the system X= State

7. General Case Min Φ = Endpoint cost- final product L = Lagrangian – describes dynamics of system u = Control – what we can do to the system X= State – properties of the system

8. Case of Beer Min Φ = Endpoint cost- profit, quality X= State – properties of the system u = Control – what we can do to the system L = Lagrangian – describes dynamics of system

9. Case of Beer Min Φ = 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

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

11. Case of Beer Min Φ = 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.

12. Quadratic Case Chemical Reactions A+B  C Rate = k[A]^a[B]^b a and b are determined experimentally Used to determine mechanisms [] = concentration

13. Beer Basics

14. Fermentation Yeast 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.

15. Controls Commercial breweries can control Temperature – refrigeration (most important) Can be expensive pH, amino acids, sugar, yeast– initial conditions

16. Optimization Different methods have been used Sequential quadratic programming (SQP) Gradient method Dynamic programming Calculus of variations Neural Networks Multiple objectives to consider Professional results: Most conclusions end up at a very narrow region between 10-15*C SQP method found a rapid rise to 13*C then slow accent to 13.5*C Difference is 6.7% increase in ethanol production

17. Simple Model Assumptions Yeast is the only consumer of resources Sugar is the only growth limiting resource Wort is deoxygenated at t=0 Temperature and pressure are constant Production of side products are minimal/ignored

18. Simple model Relates yeast, alcohol and sugar levels. System of nonlinear ODEs dS=-m*Y*S dY=k*S*Y - d*Y^2 - p*A*Y dA=b*Y*S k, d, p, m, b = constants @ temp=T

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

20. Sources G.E. Carrillo-Ureta, P.D. Roberts, V.M. Becerra, Optimal Control of a Fermentation Process W. Fred Rameriz, Jan Maciejowski, Optimal Beer Fermentation Pascale B. Dengis, L.R. Ne´Lissen, 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. 2 http://en.wikipedia.org/wiki/Optimal_control Anatoly Zlotnik