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

• Φ = Endpoint cost- final product

• L = Lagrangian – describes dynamics of system

• u = Control

• X= State

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

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

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

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

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

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.

Quadratic Case

• Chemical Reactions• A+BC• Rate = k[A]^a[B]^b• a and b are determined experimentally• Used to determine mechanisms• [] = concentration

Beer Basics

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.

Controls

• Commercial breweries can control• Temperature – refrigeration (most important)• Can be expensive

• pH, amino acids, sugar, yeast– initial conditions

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

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

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

ResultsConstants chosen for visible details not accuracy.

Units on vertical axis are arbitrary and different for each plot.

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