Optimal control of nonlinear fed-batch process using direct transcription method

Highlights

• The control problem of the nonlinear fed-batch bioreactor is carried out using direct collocation method.

• The proposed performance index to be minimized has a more general form than the other classical ones. As a result, more design factors such as maximum production, minimum time, desired production and minimum production are provided.

• To avoid abrupt changes in feed rates, the higher-order derivatives of the control signals are constrained to keep limited.

• Trade-off between the objectives is explored and investigated.

• The proposed method is applied to the invertase and PHB production processes whose solutions are compared with the recent techniques suggested in the litrature.

Abstract

In this paper, we investigate the direct transcription method for dynamic optimization applied to fed-batch processes. The performance index proposed for optimization has a more general form than the classical ones. As a result, more design characteristics such as the minimum time, maximum production, desired production, and minimum energy are provided. As a matter of fact, abrupt changes in the feed profiles can lead to undesirable behaviors in bioprocesses, such as decreasing the productivity. To handle the mentioned drawback, a novel idea is presented that results in achieving smooth control policies by limiting the higher-order derivatives of the feed rates. Owing to the nonlinear model of the fed-batch processes, the presence of constraints and free final time analysis, the problem is reformulated as a nonlinear programming optimization by the direct trapezoidal method. Finally, the proposed method is applied to the invertase and PHB production processes and the effectiveness of the presented approach is demonstrated by comparing with several works in the literature.

Introduction

The use of biochemical reactors and related technologies is increasing due to their fruitful application in converting cells or biomass into chemical or pharmaceutical products, such as vaccines, beverages, antibiotics, and industrial solvents. Among various operation regions or classes of bioreactors, the fed-batch modes have received more attention owing to its considerable profits (Hwa Sung Shin, 2013). The major aim in such reactors is to reach a maximum or predetermined concentration of production at the end of the run which can be done by using the appropriate feed rates. Controlling the behavior of such units are one of the most important issues for industrial managers and especially for the process engineers.

The optimal control (OC) theory, as a modern approach among the control systems, deals with control problems in an optimization framework wherein the requirements of the system is translated into a performance index (PI) that should be optimized. Over the past few years, there has been a massive rise in applying the optimal approach to the various disciplines such as biology (Khalili et al., 2018), robotics (Carius et al., 2018), electrical machine (Wei et al., 2014), fractional order dynamical systems (Razminia, Asadizadehshiraz, Shaker, 2019a, Mohammadzadeh, Pariz, Hosseini Sani, Jajarmi, 2018), power systems (Das, Gurrala, Shenoy, 2018, Bian, Jiang, Jiang, 2015), and chemical engineering (Shi, Al-Durra, Boiko, 2018, Sun, He, Wang, Gui, Yang, Zhu, 2018, Cui, Tian, Qin, Wang, Zhao, 2018). In particular, an OC of the fed-batch operations has been a crucial subject in biotechnology that attracts many researchers (Liu, Gong, Teo, Sun, Caccetta, 2017, Liu, Gong, Shen, Feng, 2013).

The techniques of the OC are classified as dynamic programming, indirect and direct methods. The dynamic programming employs Hamilton-Jacobi-Bellman theory (Razminia et al., 2019b) to find an optimal solution which suffers from the curse of dimensionality. In the indirect method, the necessary conditions of optimality are derived and solved (Chachuat, 2007), however, satisfying such conditions or using adjoint variables, is known to be a daunting task in a vast majority of optimization problems (see (Fidanova, 2016) for other disadvantages of using indirect method). The direct method is another alternative of the OC that does not have the mentioned drawbacks of the dynamic programming and the indirect method. In this technique, a dynamic optimization problem is transferred to the nonlinear programming (NLP) one which can efficiently handle the nonlinearity and limiting constraints (Betts, 2010, Chachuat, 2007, Mehrpouya, Fallahi, 2015).

The main goal in the optimization of the fed-batch reactors is to maximize productivity by manipulating feed rates. As a matter of fact, finding an optimal policy for the chemical reactors is a daunting task owing to the nonlinearity and the presence of constraints. Consequently, a high-performance controller must be designed for the operation. Since the direct method can efficiently handle nonlinear dynamics and various types of constraints, it has been numerously employed by researchers for optimizing the fed-batch processes (Ochoa, 2016, López, Bucala, Villar, 2010). However, the resulting feed profiles in most of these works suffer from being non-smooth due to the use of piecewise constant or linear (Ochoa, 2016) for the parameterization of control signals in the direct method. The presence of abrupt variations in the cell’s environment can cause unpleasant effects on the cell’s metabolism, which in turn can considerably affect the production of the metabolites (Ochoa, 2016, Cheng, Wu, Chen, 2002, Godoy, Amorim, Lopes, Oliveira, 2008, Azevedo, Bragança, Simões, Cerqueira, Almeida, Keevil, Vieira, 2012). To handle the mentioned drawback, one can use higher-order piecewise polynomials for the parameterization of control signals which increase the number of decision variables and consequently increasing the computational time without converging to a feasible set. Recently, the work by Ochoa (2016), presents another alternate in which the feed rates are parameterized by sinusoidal functions that can provide smooth feed profiles. However, the used controller must be able to generate such control policies, the actuator must permit continuous changes and it can be only used in the direct approach.

In this manuscript, to achieve smooth control signals, a novel idea is presented that can be employed by all of the dynamical optimization techniques (e.g., direct, indirect and dynamic programming). In the proposed technique, the higher-order derivatives of control signals are constrained to keep limited by including them in the integrand of PI. In this case, to solve the optimization problem, the higher-order derivatives of the control signals except the highest one, are defined as new state variables and the highest order is now considered to be the control input to the augmented dynamical equation. If the higher-order derivatives of the control signals get minimized, smooth control policies will be obtained. Besides, most of the optimization methods that have been suggested in the literature for optimizing fed-batch processes, use the Mayer form of PI, which only concerns the maximization of the production at the end of process (Patkar, Seo, Lim, 1993, Peroni, Kaisare, Lee, 2005, Chaudhuri, Modak, 1998, Ochoa, 2016, López, Bucala, Villar, 2010). In this work, a PI is presented that covers most of the crucial objectives in the optimization of the fed-batch processes including minimizing energy and achieving a predefined amount of production.

In this manuscript, the dynamic optimization of the fed–batch reactors is proposed by employing the direct approach. Our main contributions in this work are given as follows.

• The control problem of the nonlinear fed–batch bioreactor is carried out using trapezoidal collocation method (TCM).

• The proposed PI to be minimized has a more general form than the classical ones. As a result, more design factors such as maximum production, minimum time, desired production, and minimum energy are provided.

• To avoid abrupt changes in feed rates, the higher-order derivatives of the control signals are constrained to keep limited.

• A trade–off between the objectives is explored.

• The proposed method is applied to the invertase and poly(β-hydroxybutyrate) (PHB) production processes whose solutions are compared with the recent techniques suggested in the literature.

The structure of this manuscript is as follows. The optimal control of the fed-batch process is presented in Section 2. To show the effectiveness of the presented theoretical formulation, the proposed method is applied to the invertase and PHB processes which are given in Section 3 and Section 4, respectively. Finally, some concluding remarks are provided in Section 5.