Optimization for Nonlinear Uncertain Switched Stochastic Systems with Initial State Difference in Batch Culture Process

Based on the deterministic description of batch culture expressed in form of switched ordinary differential equations, we introduce a switched stochastic counterpart system with initial state difference together with uncertain switching instants and system parameters to model the process of glycerol biodissimilation to 1,3-propanediol (1,3-PD) induced by Klebsiella pneumoniae (K. pneumoniae). Important properties of the stochastic system are discussed. Our aim is to obtain the unified switched instants and system parameters under the condition of different initial states. To do this, we will formulate a system identification problem in which these uncertain switched instants and system parameters are regarded as decision variables to be chosen such that the relative error between experimental data and computational results is minimized. Such problem governed by the stochastic system is subject to continuous state inequality constraints and box constraints. By performing a time-scaling transformation as well as introducing the constraint transcription and local smoothing approximation techniques, we convert such problem into a sequence of approximation subproblems. Considering both the difficulty of finding analytical solutions and the complex nature of these subproblems, we develop a parallelized differential evolution (DE) algorithm to solve these approximation subproblems. From an extensive simulation, we show that the obtained optimal switched instants and system parameters are satisfactory with initial state difference.