Open AccessOptimal feedback control for a class of fed-batch fermentation processes using switched dynamical system approach
Author(s) -
Xinru Wu,
Yuzhou Hou,
Kanjian Zhang
Publication year - 2022
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022510
Subject(s) - optimal control , mathematical optimization , mathematics , control theory (sociology) , optimization problem , interval (graph theory) , controller (irrigation) , convergence (economics) , integer (computer science) , quadratic equation , function (biology) , state (computer science) , dynamical systems theory , state variable , constraint (computer aided design) , computer science , control (management) , algorithm , physics , geometry , combinatorics , quantum mechanics , artificial intelligence , evolutionary biology , agronomy , economics , biology , programming language , economic growth , thermodynamics
This paper considers an optimal feedback control problem for a class of fed-batch fermentation processes. Our main contributions are as follows. Firstly, a dynamic optimization problem for fed-batch fermentation processes is modeled as an optimal control problem of switched dynamical systems, and a general state-feedback controller is designed for this dynamic optimization problem. Unlike the existing switched dynamical system optimal control problem, the state-dependent switching method is applied to design the switching rule, and the structure of this state-feedback controller is not restricted to a particular form. Then, this problem is transformed into a mixed-integer optimal control problem by introducing a discrete-valued function. Furthermore, each of these discrete variables is represented by using a set of 0-1 variables. By using a quadratic constraint, these 0-1 variables are relaxed such that they are continuous on the closed interval $ [0, 1] $. Accordingly, the original mixed-integer optimal control problem is transformed intoa nonlinear parameter optimization problem. Unlike the existing works, the constraint introduced for these 0-1 variables are at most quadratic. Thus, it does not increase the number of locally optimal solutions of the original problem. Next, an improved gradient-based algorithm is developed based on a novel search approach, and a large number of numerical experiments show that this novel search approach can effectively improve the convergence speed of this algorithm, when an iteration is trapped to a curved narrow valley bottom of the objective function. Finally, numerical results illustrate the effectiveness of this method developed by this paper.