(236d) Optimum Experimental Design: An Online Approach

In order to understand and manipulate the behavior and performance of chemical processes, they are described by nonlinear models, usually systems of nonlinear differential equations. Only validated models produce realistic simulations of the real world behavior and reveal the potential to apply model-based process optimization methods. For validation, the unknown model parameters have to be calibrated to data from dynamic experiments by solving nonlinear constrained parameter estimation problems minimizing a weighted sum of squares of the deviations between data and model responses subject to the model equations. The errors of the experimental data cause errors of the parameter estimate. Its uncertainty is described by the variance-covariance matrix which can be computed from the sensitivities of the constrained least squares problem w.r.t. the parameters and which depends on the experimental design: the way how the experiments have been processed and which measurements have been chosen. For nonlinear models it further depends on the values of the parameter estimate. To obtain a parameter estimate with maximum reliability, an experimental design is computed minimizing a functional of the variance-covariance matrix subject to the model equations and constraints to experimental cost and operability. This results in intricate nonlinear non-standard constrained optimal control problems.

We have developed mathematical methods and numerical software, among others the packages PARFIT for parameter estimation and VPLAN for experimental design, to model and solve these intricate optimization problems. Due to their mutual dependence for nonlinear models, parameter estimates and experimental designs have to be computed in a sequential way maximizing the gain of information in each loop. In this contribution, we discuss a new approach: instead of collecting the data from entire runs of dynamic experiments and afterwards re-estimating the parameters and designing new entire dynamic experiments, we suggest, at runtime of the experiment, whenever a reasonable amount of new data has become available, to exploit this new information immediately by improving the parameter estimate and, based on that, computing a new design for the continuation of the experiment. The computations, the implementation of the experimental design and the collection of the data have to be performed online. In the talk, we present two case studies related to (bio-) chemical reaction systems where kinetic parameters have to be estimated. We compare an intuitive design to an off-line optimum design and an online design and can show that online methods can reduce the effort for model validation drastically.