Optimal control of continuous fermentation processes

Three layer control structure is proposed for optimal control of continuous fermentation processes. The start-up optimization problems are solved as a first step for optimization layer building. A steady state optimization problem is solved by a decomposition method using prediction principle. A discrete minimum time optimal control problem with state delay is formulated and a decomposition method, based on an augmented Lagrange's function is proposed to solve it. The problem is decomposed in time domain by a new coordinating vector. The obtained algorithms are used for minimum time optimal control calculation of Baker's Yeast fermentation process.List of Symbols x(t) g/l biomass concentration - s(t) g/l limiting substrate concentration - x ⁰ g/l inlet biomass concentration - s ⁰(t) g/l inlet substrate concentration - D(t) h^–1 dilution rate - (t) h^–1 specific growth rate - Y g/g yield coefficient - (t) h^–1 specific limiting substrate consumption rate - k _D h^–1 disappearing constant - w ₁, w ₂ known constant or piece-wise disturbances - _m h^–1 maximum specific growth rate - k _s g/l Michaelis-Menten's parameter - h time delay - x ₀, s ₀ g/l initial concentrations - ¯x, ¯s, ¯D optimal steady state value - V _min , V _max , v=x,s,d,t bounds of variables - t h sampling period - K number of steps in the optimization horison - Js, J _d performance indexes - L _s Lagrange's function - L _d Lagrange's functional - ₀ weighting coefficient for the amount of the limiting substrate throwing out of the fermentor - ₁, ₂ dual variables of Lagrange's function - steps in steady state coordination procedure - errors values for steady state coordination process - _v , v=x, s conjugate variables of Lagrange's functional - _v , v=x,s penalty coefficients of augmented Lagrange's functional - _v , v=x, s interconnections of the time - e _v , v=x,s, D, _x , _s gradients of Lagrange's functional - j, l indexes of calculation procedures - values of errors in calculations The researches was supported by National Scientific Research Foundation under grants No NITN428/94 and No NITN440/94