Transcript Optimal Control Theory

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+BC
• 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
Results
Constants 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