Modelling and parameter identification of a nonlinear enzyme-catalytic time-delayed switched system and its parallel optimization

In this paper, we propose a nonlinear enzyme-catalytic time-delayed switched system with unknown state-delays, switching times and system parameters for describing the process of batch culture of glycerol bioconversion to 1,3-propanediol (1,3-PD) induced by Klebsiella pneumoniae (K. pneumoniae). Some important properties of the time-delayed switched system are discussed. In consideration of the difficulty in accurately measuring the concentration of intracellular substances and the absence of equilibrium points for the time-delayed switched system, we quantitatively define biological robustness of the intracellular substance concentrations for the entire process of batch culture. Our goal is to identify the unknown quantities. To this end, we formulate an optimization problem in which the cost function minimizes the defined biological robustness and the unknown state-delays, switching times and system parameters are regarded as decision variables. This optimization problem is subject to the time-delayed switched system, continuous state inequality constraints and parameter constraints. The hybrid time-scaling transformation, constraint transcription and local smoothing approximation techniques are used to convert this optimization problem to a sequence of approximate subproblems. In consideration of both the difficulty of finding analytical solutions and the highly complex nature of this optimization problem, we develop a parallel particle swarm optimization (PPSO) algorithm to solve these approximate subproblems. Finally, we explore the appropriateness of the optimal estimates for the state-delays, switching times and system parameters as well as the effectiveness of the parallel algorithm via numerical simulations.