optimal control Beer

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