A Max–Min Control Problem Arising in Gradient Elution Chromatography

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.