Optimization And Its Application Involving Nonlinear Switched Delay Dynamic Systems Based On Some Experimental Data

Dynamic optimization in life science and industrial production can be described as a class of nonlinear switched delay systems based on some experiment data.Its optimization is a rising interdisciplinary science and covers biology,mathematics,engineering,chemistry,cybernetics and computer science.This dissertation is under the background of microbial batch fermenta-tion process,and mostly studies the theory and numerical algorithm for dynamic optimization of a class of nonlinear switched delay systems based on some experiment data.This problem subjected to a highly nonlinear and non-smooth dynamical system,is too complicated to admit analytical solutions.Therefore,this problem can be solved through the application of metabolic engineering,biological robustness,infinite dimensional optimization,non-differentiable opti-mization,control parametrization,time scaling transformation and variational principle.This dissertation can compensate the lack of efficient numerical algorithms of the practical dynamic optimization problem at present,and promote the studies of system biology,switching system,dynamic optimization and non-differentiable optimization.In addition,the obtained theoretical results have broad application prospects in biochemical process and industrial control field.The main contributions are summarized as follows.1.The bioconversion of glycerol to 1,3-propanediol(1,3-PD)is a complex bioprocess.In this chapter,based on cell growth of different characteristics at different periods,we consider a nonlinear switched dynamic system without equilibrium points involving unknown switching times and system parameters for formulating the multi-period cell growth in batch culture.Some important properties of the switched dynamic system are discussed.Our goal is to identify these switching times and system parameters with different initial state vectors.To this end,we present a parameter identification problem in which switching times and system parameters are decision variables and the cost function measures the discrepancy between experimental data and compu-tational results,subject to the switched dynamic system,continuous state inequality constraints,constraints of switching times and system parameters.The system sensitivity(the cost function's gradient,namely,the derivative of the cost function with respect to switching times and system parameters),which can be computed by solving an auxiliary initial value problem,can be regard-ed as the search direction of optimization algorithm.The identification problem is converted into a sequence of nonlinear programming subproblems through the application of the time-scaling transformation,the constraint transcription and local smoothing approximate techniques.Due to the highly complex nature of the identification problem,the computational cost is high.Thus,a parallel algorithm is proposed to solve these subproblems based on the novel combinations of system sensitivity and genetic algorithm.The switched dynamic system including optimal switching times and system parameters provides guidance to reasonably describe the cell growth of different characteristics at different periods.2.In this chapter,in consideration of the fact that the transport ways of 1,3-PD and glyc-erol with different weights across cell membrane are still unclear in batch culture,we consider a nonlinear enzyme-catalytic dynamic system without equilibrium points for formulating the batch process involving the concentration of intracellular substances.This system includes 121 possible metabolic pathways.Taking into account the difficulty in accurately measuring the concentration of intracellular substances and the absence of equilibrium points for the enzyme-catalytic system,the novel approach used here is to define quantitatively biological robustness of the intracellular substance concentrations for the overall process of batch culture.To determine the most possible metabolic pathway,we take the defined biological robustness as cost function and establish a bi-level dynamic optimization problem,in which 1452 system parameters(opti-mization variables of down level)and 484 pathway parameters(optimization variables of upper level)are involved.The optimization problem is subject to the enzyme-catalytic system,contin-uous state inequality constraints and box constraints.As such,solving the optimization problem by a serial program is a very complicated task.We propose a parallel migration particle swarm optimization algorithm(MPSO)capable of solving the identification model in conjunction with the constraint transcription and local smoothing approximation techniques.This provides guid-ance to understand the transport ways of 1,3-PD and glycerol with different weights across cell membrane.3.Time-delays are present in many real-word systems,including chemical processes,bio-logical systems,economic systems,and secure communication systems.In this chapter,we con-sider a nonlinear enzyme-catalytic time-delayed dynamical system without equilibrium points involving unknown system parameters and state-delays for describing the process of batch cul-ture with time-delayed phenomena.Taking account of the difficulty in accurately measuring the concentrations of intracellular substances and the absence of equilibrium points for the time-delay system,we define quantitatively biological robustness of the intracellular substance con-centrations for the entire process of batch culture to identify the unknown system parameters and state-delays.Taking the defined biological robustness as a cost function,we establish an opti-mization problem subject to the time-delayed system,continuous state inequality constraints and constraints of system parameters and parameter state-delays.By a penalty approach,this prob-lem can be converted into a sequence of nonlinear programming subproblems.We first show that the partial derivatives of the system state with respect to the time-delays can be computed by solving a set of auxiliary dynamic systems in conjunction with the governing time-delayed system.In consideration of both the difficulty in finding analytical solutions and the complexity of numerical solution to the nonlinear system,based on an improved simulated annealing and these detivatives,we develop a parallelized synchronous algorithm to solve these nonlinear pro-gramming subproblems.This verifies the appropriateness of the existence of state-delays.The nonlinear enzyme-catalytic time-delayed dynamical system including optimal system parame-ters and state-delays provides guidance to reasonably describe the process of batch culture with time-delayed phenomenon.4.Considering the fact that it is tedious to identify parameters(in Chapter 2)because of the existence of vast unknown parameters in a nonlinear switched dynamical system,this chap-ter considers a nonlinear enzyme-catalytic time-delayed switched dynamical system to describe batch culture of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumoniae.This system can not only predict the exponential growth phase but also the lag and the decreasing phases of batch culture since it contains two switching times for representing the ending momen-t of lag growth phase and the time when the cell specified growth rate reaches the maximum.Due to the difficulty in accurately measuring the concentrations of intracellular substances,a quantitative biological robustness for the concentrations of intracellular substances is defined by penalizing the expectation of the relative deviation between system outputs before and after the switching times are perturbed(in Chapters 3 and 4).The variance of the relative deviation is not taken into consideration in the definition.With this in mind,the biological robustness is expressed in terms of the expectation and variance of the relative deviation.Our aim is to iden-tify the switching times.To this end,a robust optimization problem is formulated,where the switching times are decision variables to be chosen such that the biological robustness measure is optimized.This problem governed by the nonlinear system is subject to a quality constraint and continuous state inequality constraints.By a hybrid time-scaling transformation,constraint transcription and local smoothing approximation techniques,a sequence of approximate robust optimization subproblems are obtained.The convergence analysis of this approximation is al-so investigated.We first show that the partial derivatives of the system state with respect to the switching times can be computed by solving a set of auxiliary dynamic systems in conjunc-tion with the governing time-delayed switched system.Owing to the highly complex nature of these subproblems,a parallel algorithm,based on simulated annealing and these derivatives,is proposed to solve these subproblems.The simpler nonlinear enzyme-catalytic time-delayed switched dynamical system including the obtained optimal switching times provides a better way to describe the process of batch culture of glycerol to 1,3-PD.