Robust optimal control for a batch nonlinear enzyme-catalytic switched time-delayed process with noisy output measurements

Abstract

In this paper, we consider a nonlinear switched time-delayed (NSTD) system with an unknown time-varying function describing the batch culture. The output measurements are noisy. According to the actual fermentation process, this time-varying function appears in the form of a piecewise-linear function with unknown kinetic parameters and switching times. The quantitative definition of biological robustness is given to overcome the difficulty of accurately measuring intracellular material concentrations. Our main goal is to estimate these unknown quantities by using noisy output measurements and biological robustness. This estimation problem is formulated as a robust optimal control problem (ROCP) governed by the NSTD system subject to continuous state inequality constraints. The ROCP is approximated as a sequence of nonlinear programming subproblems by using some techniques. Due to the highly complex nature of these subproblems, we propose a hybrid parallel algorithm, based on Nelder–Mead method, simulated annealing and the gradients of the constraint functions, for solving these subproblems. The paper concludes with simulation results.

Introduction

The evolution of a time delayed system is influenced by its state and/or control variables at some past time instants. Many practical real life systems can be modeled as time delayed systems [1], [2], [3], [4], [5]. Time-delays are sometimes deliberately introduced to help stabilize the system or force the system to track real data. However, in most cases, they appear naturally due to the nature of the problem concerned. The presence of time-delays in a system can cause undesirable behavior of the system or even causing it to become unstable. Thus, it has generated a strong interest among control community. For example, an optimization approach is proposed in [6] to the identification of state-delays. In [7], a class of optimal state-delay control problems is considered. Optimal control of switched systems with multiple time-delays and a cost on changing control is studied in [8]. A computational method is developed in [9] to solve time-delay optimal control problems with free terminal time. Delay independent stability criteria are derived for impulsive switched systems with time-invariant delays in [10]. A computation method is proposed in [11] to solve a class of dynamic optimization problems governed by switched time-delay systems with state-dependent switching conditions. There are many other studies on the control of time-delayed systems in the literature, such as those cited in the references of the papers mentioned above.

1,3-Propanediol (1,3-PD) is an important products used in chemical industry [12]. For the production of 1,3-PD, there are two methods: chemical synthesis; and microbial conversion of glycerol by Klebsiella pneumoniae (K. pneumoniae). This paper focuses on the second method because it is environmentally friendly, and high region specificity, and the recycling of feedstock is cheap. For the fermentation process, there are three main operation models: (i) batch culture [13] (all substrate is present at the beginning of the reaction and nothing is added or removed from the fermentor during the reaction); (ii) continuous culture (glycerol and alkali are continuously added during the reaction while removing old medium) [14]; and (iii) fed-batch culture (fresh medium is added discontinuously to the reactor at a constant rate to prevent nutrient depletion, but nothing is removed) [15]. The batch culture is a highly complicated real practically important problem. So the proposed solution approach is applicable to solve real practical problems. Other supporting reasons for the need of studying batch culture [16] are: (1) Batch culture can be expressed as an excessive metabolism process of glycerol. Thus, 1,3-PD yield, which is defined as the ratio between the formation of 1,3-PD and the consumption of glycerol, is high. This means that the high concentration of glycerol will lead to the high formation of 1,3-PD and low formation of by-product in batch culture of glycerol. It is worth noting that the high target product yield in batch culture can only be applied to the bioconversion of glycerol. However, not all the batch cultures are of the high target product yield; (2) Batch culture is a simple and easy operation mode when compared with fed-batch and continuous cultures; (3) Batch culture is basic for the understanding or controlling of fed-batch and continuous cultures. Therefore, batch culture has been extensively studied in the existing literature. There are many studies on the batch fermentation process being reported in the literature, such as bi-objective optimization [17], distributionally robust optimization [18], joint estimation [19], hybrid system [20], two-stage system [21] , strong stability [22], robust bi-objective optimal control [23], stochastic optimal control problem [24] and multiple characteristic time points [25]. However, the presence of time-delays is not considered in these studies.

In a dynamical system of microbial fermentation, the occurrences of time-delays are mainly due to two factors: (i) there exists the process of substrate absorption together with the inhibitions of substrate and multi-products across the cell membrane; (ii) cells are required to go through the growth process before they start to produce products [26]. In a recent paper [27], the identification of parameters of a nonlinear time-delay system in batch culture is under investigation, where the concentrations of intracellular substances are, however, ignored. In [28], a novel dynamic system, which takes into account the changes of the concentrations of both extracellular substances (biomass, glycerol, 1,3-PD, acetic acid and ethnol) and intracellular substances (glycerol, 1, 3-PD and 3-hydroxypropionaldehyde (3-HPA)), is proposed. According to the definition given in [29], the biological robustness is a property of a biological system that is insensitive to internal or external environmental influences. For a biological system, its biological robustness will ensure that its performance is maintained when it is subject to disturbances. For supporting arguments, see [30]. With the rapid development of systems biology, the research on biological robustness has been extensive [31], [32], [33], [34], [35]. In [20], [36], the definition of the quantitative robustness of batch fermentation organisms is given. However, it is only defined based on the expectation of the relative deviation of different system outputs caused by the perturbation of the decision variable in a given domain. The fluctuation of this relative deviation is not taken into account. In addition, the study being carried out in the above literature is based on the assumption that the experimental data is accurate. However, the experimental data cannot be completely accurate due to the influence of external temperature, humidity and pressure [37].

To construct an accurate model to describe the fermentation process, an unknown time varying function is required to be identified. However no information about this unknown time varying function can be extracted directly from the data set. Thus, optimization methods are required to be used to identify this time-varying function. In this paper, the batch culture of glycerol bioconversion to 1,3-PD induced by K. pneumonia is modeled as a nonlinear switched time-delayed (NSTD) system with an unknown time-varying function and noisy output measurements. According to the actual fermentation process, the time-varying function is in the form of a four-segment piecewise linear function with unknown kinetic parameters and switching times. To address the issue due to the lack of information on the unknown kinetic parameters and switching times, the quantitative biological robustness, which is inspired by the quantitative description of the biological robustness [29], [30], is proposed. It is expressed as a weighted sum of the expectation and variance of the relative deviation of the intracellular substance concentrations before and after the time-varying function is perturbed. To address the issue in regards of the noisy output measurements, the expectation of the least-squares error between the real system output (measured through experiments) and the predicted system output (generated from the mathematical model) of the extracellular substance concentration is defined. Our objective is to estimate the unknown kinetic parameters and switching times from the noisy experimental data and the quantitative biological robustness. In order to achieve this goal, a robust optimal control problem is proposed, in which the kinetic parameters and switching times are its decision variables, and the cost function is a weighted sum of the quantitative biological robustness of the intracellular substance concentrations and the least squares error of the expectation of the extracellular substance concentrations. This optimal control problem is governed by the NSTD system subject to the continuous state inequality constraints arising from the need that the concentrations of the extracellular and intracellular substances are to be limited to a specified region. By using the penalty function method, novel time scaling transformation and constraint transcription method, this optimal control problem is approximated by a sequence of nonlinear programming subproblems. Considering the complexity of these subproblems, a hybrid parallel algorithm, which is developed based on Nelder–Mead simplex search method (N–M simplex method), simulated annealing and the gradients of the constraint functions, is proposed to solve each of these subproblems. The numerical simulation shows that the optimal kinetic parameters and the switching times obtained are highly satisfactory.

The remainder of this paper is organized as follows. In Section 2, a NSTD system is formulated. In Section 3, a ROCP is proposed. In Section 4, a computational approach is developed. In Section 5, numerical simulations are carried out. In Section 6, some concluding remarks are made and the direction for future research is indicated.