Integral equation-based mathematical modeling and optimization of dynamic industrial process control systems
Keywords:
Volterra integral equations, dynamic process control, mathematical modeling, minimax optimization, calculus of variations, fixed-point theory, numerical quadrature.Abstract
Conventional ordinary differential models of dynamic industrial process control systems may only provide an approximate representation of the phenomena involved. The research presented in this paper introduces an approach to modeling and optimizing dynamic industrial processes based on the use of nonlinear integral Volterra-type equations within the framework of the theory of Banach spaces. The behavior of an industrial process is described by means of use of the generalized integral operator with the kernel that satisfies the Lipschitz continuity and boundedness conditions that allow the application of the Banach fixed-point theorem to prove the existence and uniqueness of the dynamic process behavior. A quadratic functional incorporating both control energy and tracking errors is determined, and the minimax control problem is defined to take into account the worst environmental impacts on the process. First-order conditions of optimality are derived through the use of the variational calculus resulting in a system of coupled integral adjoint equations. A numerical method for solving the problem based on product integration quadrature and iterative schemes of computation is given with an estimate for the convergence properties depending on the mesh size. An application of the methodology is illustrated with a nonlinear chemical reactor problem, where the results are compared with those obtained through the standard approaches of proportional-integral-derivative control and the modern model predictive control.MSC Subclassification: 45D05, 45G10, 49J20, 93C15, 90C47.
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