Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process

Developing suitable dynamic models of bioprocess is a difficult issue in bioscience. In this paper, considering the microbial metabolism mechanism, i.e., the production of new biomass is delayed by the amount of time it takes to metabolize the nutrients, in glycerol bioconversion to 1,3-propanediol, we propose a nonlinear time-delay system to formulate the fed-batch fermentation process. Some important properties are also discussed. Then, in view of the effect of time-delay and the high number of kinetic parameters in the system, the parametric sensitivity analysis is used to determine the key parameters. Finally, a parameter identification model is presented and a global optimization method is developed to seek the optimal key parameters. Numerical results show that the nonlinear time-delay system can describe the fed-batch fermentation process reasonably.     

Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and pH logic control

In this paper, we study the fed-batch fermentation of glycerol by Klebsiella pneumoniae with open loop glycerol input and pH logic control using a flow of alkali as manipulated variable. A nonlinear hybrid system is developed to describe this process. We prove the finiteness of switching numbers of the system in terms of bounded variation and explore the existence and uniqueness of the solutions. Additionally, a parameter identification problem is proposed and an asynchronous parallel particle swarm optimization (PSO) algorithm is constructed to solve it. Numerical results show the effectiveness of the algorithm and reveal that the proposed model could describe the fed-batch process properly.     

Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture

In this paper, we consider an optimal control problem of microbial fermentation process in which glycerol is converted to 1,3-propanediol by Klebsiella pneumoniae in fed-batch culture. During the period of reaction, the variation of pH value is monitored to determine glycerol replenishment quantity, guaranteeing that microorganism can always keep growing fast under enough nutrition. Every time pH value is lower than seven, the quantity of glycerol added is such that pH value returns seven again. Glycerol is poured into reactor at discrete time instant and the quantity is controllable. The problem is to determine for each discrete time instant the glycerol quantity to add and maximize the final concentration of 1,3-propanediol. We present a controlled explicit nonlinear impulsive dynamical system of fed-batch culture with state independent vector measures as controls and study the existence, uniqueness, boundedness, continuous dependence and Gâteaux differentiability of its solution with respect to controls. We then propose a multiple objective programming model and demonstrate the regularity of cost functionals and weak compactness of admissible control set. Finally we discuss the existence of optimal control and implement a hybrid particle swarm optimization algorithm to solve the model optimally. Computational results are presented on a numerical example.     

Modelling and properties of a nonlinear autonomous switching system in fed-batch culture of glycerol

A nonlinear autonomous switching system is proposed to describe the coupled fed-batch fermentation with the pH as the feedback parameter. We prove the non-Zeno behaviors of the switching system and some basic properties of its solution, including the existence, uniqueness, boundedness and regularity. Numerical simulation is also carried out, which reveals that the proposed system can describe the factual fermentation process properly.     

Existence of solutions for a nonlinear system with a parameter

In this paper, the existence of solutions for a system of nonlinear equations is considered. n² nonzero real solutions are obtained by using the critical point theory. Additionally, the Dirichlet boundary value problems of even order difference equations and partial difference equations are investigated.     

Existence and multiple solutions for a nonlinear system with a parameter

In this paper, the critical point theory and the Morse theory are employed to discuss the system of nonlinear equations of the form Au=λf(u)$Au=\lambda f\left(u\right)$, where the matrix A$A$ is not necessarily positive definite. Some existence and multiple solutions are obtained.     

Parameter identification and optimization algorithm in microbial continuous culture

In this study, a novelty mathematical model is established to formulate the continuous culture of glycerol to 1,3-Propanediol (1,3-PD) by Klebsiella pneumoniae, in which the inhibition of 3-hydroxypropionaldehyde (3-HPA) to cells growth and activity of some enzymes (such as glycerol dehydratase (GDHt) and 1,3-PD oxidoreductase (PDOR)), and the passive diffusion and active transport of glycerol and 1,3-PD across cell membrane are all taken into consideration. Taking the mean relative error between the experimental data and calculated values as the performance index, a parameter identification model involving multiple nonlinear dynamic systems is presented. The identifiability of the parameter identification model is also proved. Finally, an improved particle swarm optimization (PSO) algorithm is constructed to find the optimal parameters for the systems under substrate limitation and excess conditions, respectively. Numerical results not only show that the established model can be used to describe the continuous fermentation reasonably, but also the improved PSO algorithm is valid.     

Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol

A nonlinear dynamical system was established in our preceding work to describe the batch and continuous bioconversions of glycerol to 1,3-propanediol by Klebsiella pneumoniae. The purpose of this article is to analyze the sensitivity of kinetic parameters of the dynamical system and identify their values from experiment. A global sensitivity analysis approach is constructed by combining the local technique with the Monte Carlo method. With only those parameters of higher sensitivity as design variables, we propose a parameter identification model and solve it by a gradient-based simulated annealing algorithm. Numerical results show that our methods are feasible and efficient.     

Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation

The operational Tau method, a well-known method for solving functional equations, is employed to solve a system of nonlinear Volterra integro-differential equations with nonlinear differential part. In addition, an error estimation of the method is presented. Some cases of the mentioned equations are solved as examples to illustrate the ability and reliability of the method. The results reveal that the method is very effective and convenient.     

Nonlinear system identification and control using a real-coded genetic algorithm

A real-coded genetic algorithm (GA) applied to the system identification and control for a class of nonlinear systems is proposed in this paper. It is well known that GA is a globally optimal method motivated from natural evolutionary concepts. For solving a given optimization problem, there are two different kinds of GA operations: binary coding and real coding. In general, a real-coded GA is more suitable and convenient to deal with most practical engineering applications. In this paper, in the beginning we attempt to utilize a real-coded GA to identify the unknown system which its structure is assumed to be known previously. Next, according to the estimated system model an optimal off-line PID controller is optimally solved by also using the real-coded GA. Two simulated examples are finally given to demonstrate the effectiveness of the proposed method.     

Permanence of a discrete nonlinear N-species cooperation system with time delays and feedback controls

A discrete nonlinear N-species cooperation system with time delays and feedback controls is considered in this paper. Sufficient conditions which ensure the permanence of the system are obtained. It is shown that these conditions are weaker than those of Chen [F.D. Chen, Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. Math. Comput. 186(2007) 23-29], that is, our investigation shows that the additional condition in Chen’s paper is not necessary.     

Boundedness in a fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system $\left\{\begin{array}{cc}{u}_t=\nabla \cdot (D\left(u\right)\nabla u)-\nabla \cdot (G\left(u\right)\chi \left(v\right)\nabla v)+\nabla \cdot (H\left(u\right)\xi \left(w\right)\nabla w),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=d_1\Delta v+\alpha u-\beta v,\hfill & x\in \Omega ,t>0,\hfill \\ w_t=d_2\Delta w+\gamma u-\delta w,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset {\mathbb{R}}^n$ $(n\ge 1)$ is a bounded domain with smooth boundary, $d_1,d_2,\alpha ,\beta ,\gamma ,\delta >0$ are constants. Also, $D,G,H\in C^2\left([0,\infty )\right)$ fulfill that $a_0{(s+1)}^{m-1}\le D\left(s\right)\le a_1{(s+1)}^{m-1}$ with $a_0,a_1>0$ and $m\in \mathbb{R}$; $G\left(0\right)=0$, $0\le G\left(s\right)\le b_0{(s+1)}^{q-1}$ with $b_0>0$ and $q<min\{2,m+1\}$; $H\left(0\right)=0$, $0\le H\left(s\right)\le c_0{(s+1)}^{r-1}$ with $c_0>0$ and $r<min\{2,m+1\}$, and $\chi ,\xi$ satisfy that $0\le \chi \left(s\right)\le \frac{{\chi }_0}{s^{k_1}}$ with ${\chi }_0>0$ and $k_1>1$; $0\le \xi \left(s\right)\le \frac{{\xi }_0}{s^{k_2}}$ with ${\xi }_0>0$ and $k_2>1$. Global existence and boundedness in the case that $w=0$ were proved by Ding (2018). However, there is no work on the above fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system.     

An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification

The bioconversion of glycerol to 1,3-propanediol is a complex bioprocess. In this study, we aim to explore a novel model for describing the multistage cell growth in batch culture. This is achieved by a modification at the specific rate of cell growth in consideration of its time-dependent changes. The existence, uniqueness and boundedness of solutions to the system and the continuity of solutions with respect to the parameters are discussed. In addition, a parameter identification problem of the system is developed and a feasible optimization algorithm is constructed to solve it. Numerical result shows that the improved model could describe the batch culture well.     

On equivalence of the Gauss-Newton technique,the parameter influence coefficient technique,and the quasilinearization technique in dynamic system identification by least squares

The literature presents, among others, three general techniques for system identification by least squares. These techniques are the Gauss-Newton technique, the parameter influence technique and the quasilinearization technique. The purpose of this paper is to show the equivalence of all three techniques for the general case.     

Modelling and well-posedness of a nonlinear hybrid system in fed-batch production of 1,3-propanediol with open loop glycerol input and pH logic control

A fed-batch fermentation of glycerol by Klebsiella pneumoniae with open loop substrate input and pH logic control is considered in this paper. We propose a nonlinear hybrid system to describe this process, which consists of discrete variables to represent the flow rate of glycerol and alkali and continuous variables for substance concentrations. The hybrid system includes both time- and state-based switchings, which are respectively determined by a pre-assigned times sequence and an output equation of the pH. It is proved that the hybrid system is non-Zeno. Some basic properties of solutions to the hybrid system are also explored, including existence, uniqueness, boundedness and continuous dependence on initial state and parameters. Additionally, a numerical simulation is carried out to show that the proposed hybrid system can describe the fed-batch culture properly.     

Global solvability and asymptotic behavior for a nonlinear coupled system of viscoelastic waves with memory in a noncylindrical domain

In this paper we prove the exponential decay in the case n>2, as time goes to infinity, of regular solutions for a nonlinear coupled system of wave equations with memory and weak damping

     

Estimation of nonlinear systems without a priori structural information and its application in Hammerstein model identification with dynamic nonlinearities

Email: er-wei-bai{at}uiowa.edu Received on July 30, 2005; Accepted on July 23, 2006 In the paper, we discuss identification of a nonlinear systemwithout structural information and propose two methods, thekernel method and the orthonormal basis method. The convergenceresults are established for both methods without a priori structuralinformation. We then apply the results to identification ofHammerstein models with an unknown dynamic nonlinearity. Itis also shown that identification of the linear part in Hammersteinmodels is possible with no knowledge of the dynamic nonlinearity.     

On an inverse scattering problem for a class Dirac operator with discontinuous coefficient and nonlinear dependence on the spectral parameter in the boundary condition

On the positive semi‐infinite interval, we obtained a generalization of the Marchenko method for a Dirac equation system with a discontinuous coefficient and a quadratic polynomial on a spectral parameter in the boundary condition. In this connection, we use an new integral representation of the Jost solution of equation systems, which does not have a ‘triangular’ form. The scattering function of the problem is defined, and its properties are examined. The Marchenko‐type main equation is obtained, and it is shown that the potential is uniquely recovered in terms the scattering function. Copyright © 2012 John Wiley & Sons, Ltd.