Optimal control of batch processes involving simultaneous enzymatic and microbial reactions

Optimal enzyme feed rate profiles have been calculated, based on a model for a fed-batch simultaneous enzymatic and microbial reaction (SEMR) process. The model parameters corresponded to a relatively slow citric acid fermentation. The profiles were calculated using an iterative algorithm based on the minimum principle. Penalty functions were used to enforce inequality constraints on the enzyme feed rate. Significant improvements in the objective function relative to that for the best constant enzyme feed rate were found. The effect on the optimal profiles of changes in the parameters of the model and the objective function were investigated, as was the effect of introducing the stationary state assumption to eliminate glucose concentration as a state variable. Major differences between bang-bang control variable profiles and singular arcs were found, with the singular arc solution slightly better than the optimal bang-bang control.List of Symbols a N-vector of initial conditions - b₁–b₁₀ parameters defined in Table 2 - c vector of cost parameters - c₁–c₆ penalty function parameters - E enzyme concentration (U/l) - f N-vector of functions - F enzyme feed rate (U/l-h) - g N-vector of functions - G glucose concentration (g/l) - H Hamiltonian - J objective function - J^* modified objective function - L number of integration steps per time interval - L number of control variables - M number of time intervals - n iteration index - N number of state variables - P product concentration (g/l) - r₁ glucose formation rate (g/l-h) - r₂ product formation rate (g/l-h) - t time (h) - T final time (h) - u L-vector of control variables - x N-vector of state variables - z N-vector of adjoint variables - Z total enzyme fed (U/l)Greek convergence parameterThe support of one of the authors by the National Science Foundation (Grant CBT-84-20552) is gratefully acknowledged.