A Max–Min Control Problem Arising in Gradient Elution Chromatography

Abstract

Gradient elution chromatography is an industrial process used to separate and purify multi-component chemical mixtures. In this article, we consider an optimal control problem in which manipulative variables in the chromatographic process need to be determined to maximize separation efficiency. This problem has two nonstandard characteristics: (i) the objective function is nonsmooth, and (ii) each state variable is defined over a different time horizon. The final time for each state variable, the so-called retention time, is not fixed and actually depends on the control variables. To solve this optimal control problem, we first introduce a set of auxiliary decision variables to govern the ordering of the retention times. Then, we approximate the control by a piecewise-constant function and apply a novel time-scaling transformation to map the retention times and control switching times to fixed points in a new time horizon. The retention times and control switching times become decision variables in the new time horizon. On this basis, the optimal control problem is reduced to an approximate nonlinear optimization problem that can be solved using a recently developed exact penalty method. Numerical results show that our approach is both accurate and efficient.

Optimal Control of Nonlinear Switched Systems: Computational Methods and Applications

“…The general switched system framework also encapsulates dynamic systems in which some (or all) of the control variables assume values in a discrete set. Such discrete-valued control variables arise in many applications, including submarine control [11], hybrid power systems [54], micro-robots [4], sensor scheduling [63], switching power amplifiers [57], subway trains [55], and gradient-elution chromatography [12]. An optimal control problem involving discrete-valued control variables is called an optimal discrete-valued control problem [27,68].…”

A computational algorithm for a class of non-smooth optimal control problems arising in aquaculture operations

“…Extensive numerical testing has shown that the optimal solution obtained is sometimes sensitive to the values chosen for λ and γ. It would be of great benefit to develop a general guide as to the choice of values of these parameters, especially considering the vast array of applications of the exact penalty approach [1,19].…”

“…MISER automatically calculates the objective function gradients by integrating a costate system backwards in time; for more details see [1,2,6,11].…”

“…Following their lead, we also ignore the feeding rate f (t) and consider only the harvesting fractions and the corresponding harvesting times as decision variables. However, note that the computational approach in this paper can be easily extended to also optimize the feeding rate f (t) using the control parameterization technique described in [1,4,6].…”

“…The time-scaling transformation and exact penalty approach result in a problem that can be readily solved using MISER 3.3 [2], which is an optimal control software based on the control parameterization technique [1,4,6]. The approach described in this paper can also be readily extended to more general optimal control problems involving discontinuous objective functions.…”