Fractional Order Parameter Identification And Optimal Control For A Class Of Continuous Fermentation

A class of continuous fermentation problems of 1,3-propanediol(1,3-PD)production is considered in this paper.For this problem,we introduce the idea of fractional calculus,and establish the corresponding fractional microbial fermentation models.In order to verify the validity of the fractional systems,a fractional parameter identification is carried out.Furthermore,a fractional optimal control model is established to improve the concentration of 1,3-PD at the terminal time.The details of the research contents,methods and conclusions are summarized as follows.Considering the memory characteristics of fractional calculus,a fractional system is employed in our microbial continuous fermentation model of 1,3-PD production.Taking the orders and system parameters as the decision variables to be identified,and the relative errors between the calculated values and experimental values of state variables at the terminal time as the performance index,a corresponding fractional parameter identification model is established.Then,based on the co-state methods,the gradient formulas of the performance index and the state constraints with respect to the system parameters is discussed.And the trapezoid method and the predictor corrector method are used to solve the fractional order differential equations.In order to identify the orders and system parameters,a numerical optimization algorithm is formulated based on Particle Swarm Optimization and Sequential Quadratic Programming.By a large number optimization calculation,the model is solved numerically,and the orders and system parameters are obtained as well as the relative errors between the calculated values and the experimental values of the concentration of each substance at the terminal time.The numerical results show that the fractional order model is better than the existing integer order model in describing the continuous fermentation process.Based on the fractional order microbial fermentation model constructed in the above study,we introduce the sensitivity function to further optimize the concentration of 1,3-PD at the terminal time and reduce the influence of system parameters on the optimization process.On the basis of the sensitivity function,the Bola type of optimal control model of fractional order for continuous fermentation is established withglycerol dilution rate and glycerol feeding concentration as the input variables to be optimized and the maximum output of concentration of 1,3-PD at the end time coupling with minimum sensitivity of the state variables with respect to the system parameters as the performance index.We further discuss the computing methods of fractional order auxiliary equations with first and second order sensitivity and of the gradients of the performance index and the state constraints with respect to the input variables and system parameters.We combine trapezoid algorithm and predictor corrector method to solve the solutions of the state,co-state and auxiliary equations with fractional orders.Then,a parallel numerical optimization algorithm is constructed based on Particle Swarm Optimization and Sequential Quadratic Programming.By a large number optimization calculation,we obtain a new feeding strategy with dilution rate and concentration as well as the new system parameters that minimizes the sensitivity functions.The numerical results show that the maximum concentration of1,3-PD at the terminal time is better than the existing results under the new feeding strategy.