A Class Of Time-delayed Optimal Control Problems And Its Applications

If an optimal control model contains a time-delay term,then it is known as a timedelayed optimal control problem.In recent years,the time-delayed optimal control problem has been widely applied to industrial production,neural network,chemical reactions,etc.1,3-propanediol(1,3-PD)is an important chemical raw material.Microbial fermentation to produce 1,3-PD is a green and environmentally friendly production method using glycerola byproduct of biodiesel production.Therefore,it has attracted the attention of researcher all the world.Significant results have been achieved in the application of time-delayed optimal control in the production of 1,3-PD by microbial fermentation,and an optimization and control strategy has been provided for increasing the production of 1,3-PD in the actual fermentation process.For fed-batch process of fermentation in the production of 1,3-PD,in this paper,the optimal control theory and algorithms of a class of time-delay systems are studied.We mainly carried our research work in the following three aspects are studied,mainly carried out research work in the following three aspects:1.For 1,3-PD production in fed-batch process,the optimal control model involving nonlinear time-delay systems and its numerical algorithm are studied.According to the actual production process,we propose a nonlinear time-delay system to describe the production process.Then,a time-delay optimal control model with control and state constraints is proposed,in which the feeding rates of glycerol and alkali and the terminal moment of fermentation process are taken as control variables.Because the terminal time is not fixed but free variable,the optimal control problem is transformed into an equivalent time-delay optimal control model with fixed terminal time by time scale transformation.Furthermore,the equivalent problem is approximated by a series of nonlinear programming problems using the control parameterization method and the constraint transformation technique.At the same time,an improved particle swarm optimization algorithm is proposed to solve the corresponding nonlinear programming problems.Numerical results show that the productivity of 1,3-pd can be significantly increased by using the optimal control strategy.2.Considering the actual production process,influenced by the external environment factors,the time-delay in the reaction process is not fixed,but changes over time.An optimal control model with time-varying delay is proposed,which is controlled by time-varying delay function and kinetic parameters.The optimal time delay control problem is transformed into the corresponding parameter optimization problem by using B-spline function,polynomial function and cosine function to parameterize the time-varying delay control function.Furthermore,an improved particle swarm optimization algorithm is designed to solve the corresponding parameter optimization problem.Finally,the numerical results show that the B-spline function has the best approximate time delay function,and the effectiveness of the algorithm is proved.3.The estimation problem of constrained nonlinear systems with unknown delay and unknown system parameters is studied.The estimation problem minimizes the least squares error function between the system output and a group of noise measurements and satisfies the characteristic time constraint of the specified constraint.Firstly,the classical form of estimation is given,in which the expectation of the error function is taken as the objective function.Then,in order to obtain the robust estimation against measurement noise,we propose a robust estimation form,in which the objective function is the variances of the error function and an additional constraint shows the admissible sacrifices to the optimal expectation in the classical estimation problem.For both estimation problem,we show that the gradients of objective and constraint functions with respect to the time-delay and system parameters can be computed by solving a series of auxiliary time-delay systems.At the same time,an optimization algorithm based on gradient is designed to determine the optimal time delay and system parameters.Finally,two examples are given,including the parameter estimation in the process of microbial batch fermentation,to illustrate the effectiveness and applicability of the algorithm.