Optimal Control Theory

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