The main research fields are Modelling and Model Reduction of nonlinear processes in chemical engineering and of cellular systems. Detailed models of high complexity are reduced by means of mathematical methods and tools to simpler ones. These can be treated analytically or can be solved numerically with an computational effort that is acceptable. The reduced models still show the precision required for the specific task under consideration. The concepts for process synthesis and process control that are obtained from reduced models are proven to be so efficient that they can be applied to the underlying real process with success. In the Systems and Control Theory Group (SCT) we develop such reduction algorithms. Emphasis is put on error bounds that are needed for the validation of the reduced models.
System biology tries to pave the road towards a holistic comprehension of complex biological systems and thus towards a predictive biology. The complex functionality of metabolic and regulatory networks and of cell cycles needs to be mapped out. In general it is impossible to comprehend the complexity in an intuitive way. Based on system theory and the interdisciplinary cooperation with biologists and computer scienctists, mathematical models of amenable size and structure are established. The decomposition (modularisation) of a network and the analysis of the resulting subnets (modules) and their interconnections present an approach that promises to be successful in tackling the manifold aspects of biological networks. The SCT group develops concepts, methods and tools that enable researchers to treat the problems they are is confronted with in system biology in an adequate manner.
Always having in mind the specific problem formulation, e.g.
one is thus interested in simplified models of the underlying process. These simplified or reduced models should still be capable to reflect the essentials of the dynamical behaviour and should allow the construction of effective and computable control strategies.
We briefly address the reduction by the geometric theory of invariant manifolds:
The detection of suitable scales of time, variables and parameters justifying a generalized quasistationarity assumption is of particular interest.
Hereby, the decompsition into slow and fast variables, into masters and slaves may vary within phase space.
The derivation of quantitative statements is a conditio sine qua non: it is essential to establish bounds on how slow and how fast
certain dynamics need to be for proving the usefulness of abstract theorems and to have bounds on how good a chosen approximation needs
to be for achieving reliable numerical solutuions.
Such error bounds are necessary for the validation of the reduced models and their numerically computed solutions.
For example, given a slow/fast decomposition in a reaction-diffusion or a transport-reaction system of parabolic partial differential equations, a Galerkin-type reduction or a more general reduction, based on empirical eigenfunctions (Karhunen-Loève method), often allow the derivation of a finite dimensional ordinary differential equation on an approximate inertial manifold. The synthesis of state- or output-feedback for the partial differential equation is then performed for the reduced ordinary differential equation. For given input constraints, the locations of the actuators and sensors are to be chosen carefully with regard to these constraints. Moreover, any input bound may restrict the set of admissible initial conditions for the partial differential equation. Of course, any concept for synthesis and control that is obtained from the reduced finite dimensional model needs to be proven so efficient that it can be applied to the underlying real process safely and with success.
For an Introduction to the Geometric Theory of ODEs with Applications to Chemical Processes see Chapter 1 in the monograph Large-Scale Networks in Engineering and Life Sciences, edited by Birkhaeuser in 2014.