Model-based design of experiments

The model-based design of experiments (including nonlinear experimental design or optimal experimental design is called (OVP)) a method for calculating optimal experiments for the quantitative description of processes using (nonlinear and dynamic) models. In contrast to the much more widespread statistical design of experiments , the unknown (and to be estimated) model parameters are not included linearly in the model equations. Often the models are systems of differential equations from nonlinear dynamics .

Model term

The key difference between statistical and model-based test planning is the concept of the model. The statistical design of experiments modeling the under examination system empirically by a direct relationship between influencing or control variables and observables accepts and parameterized as follows: ${\ displaystyle q}$ ${\ displaystyle h}$

{\ displaystyle h = \ sum _ {n} p_ {n} q ^ {n}}

The model-based test planning, however, introduces the model more generally as an indirect connection between and and introduces the (system) state . ${\ displaystyle q}$ ${\ displaystyle h}$ ${\ displaystyle x = (y, z)}$

This state is generally subject to (non-linear) dynamics and secondary conditions. One writes:

{\ displaystyle \ mathrm {d} y (t) / \ mathrm {d} t = f (t, y, z, p, q)}

{\ displaystyle 0 = g (t, y, z, p, q)}

{\ displaystyle h = h (t, y, z, p, q)}

Objective functions

The planning of experiments is a special problem of optimal control . Most often, a measure for the quality of the parameter estimation is optimized, which is based on a linearized evaluation of the result of the parameter estimation . The evaluation leads to the covariance matrix of the parameters and the following criteria are used as a measure. ${\ displaystyle \ mathrm {Cov} (p)}$

A criterion

The trace of the covariance matrix . ${\ displaystyle A = \ mathrm {tr} (\ mathrm {Cov} (p))}$

E criterion

The largest eigenvalue of the covariance matrix.

Geometric interpretation of the criteria

The covariance matrix of the parameters can be represented by a confidence ellipsoid in the parameter space. Then the A criterion corresponds to the mean dimension of the ellipsoid and the E criterion to the length of the largest semi-axis.

Application examples

Chemical reaction kinetics

Chemical reactions can be represented by rate equations. The simple reaction is described by the differential equation. In this case, the rate coefficient by Arrhenius exponentially dependent on the temperature: wherein the inverse thermal energy abbreviates: . It is now crucial that the parameter , the activation energy, enters the equations non-linearly. This means that the statistical test planning cannot be used here in this way (e.g. the optimal experiments are no longer in the corners of the test room). ${\ displaystyle A \ rightarrow B}$ ${\ displaystyle \ mathrm {d} c_ {A} (t) / \ mathrm {d} t = -kc_ {A} (t)}$ ${\ displaystyle k}$ ${\ displaystyle k = k_ {a} \ exp (- \ beta E_ {a})}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ beta = (RT) ^ {- 1}}$ ${\ displaystyle E_ {a}}$

software

At the moment there are only two packages that convert the described method into available software:

gPROMS is a commercial software from the manufacturer PSE
VPLAN is an academic software based on the developments of Stefan Körkel, Irene Bauer, Hans-Georg Bock and Johannes Schlöder (S. Körkel, I. Bauer, HG Bock, JP Schlöder. A sequential approach for nonlinear optimum experimental design in DAE systems. In F. Keil, W. Mackens, H. Voss, and J. Werther (eds.), Scientific Computing in Chemical Engineering II, Volume 2, Springer Verlag, Berlin, Heidelberg, 1999)

Web links

BASF Junior Research Group Optimum Experimental Design at Heidelberg University
Optimal Design of Experimens (ODOE) Community Website