Parametric Model Order Reduction for Linear and Nonlinear Systems

Model order reduction is known to be an efficient tool for replacing very large dynamical systems in numerical simulations by systems of much smaller dimension while keeping a desired accuracy in the approximation of the original system response. However, significant modifications to the underlying physical model such as geometric variations, changes in material properties, or alterations in boundary conditions are usually not reflected in the reduced-order system. This motivates the development of new model reduction methods which preserve the parametric dependence of the original system in the reduced-order model.

We derived a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems which do structurally depend (linear or nonlinear) on parameters. We are seeking for conditions under which the gradient and Hessian of the system response with respect to the system parameters are matched in the reduced-order model. Moreover, we will investigate the optimal choice of interpolation data for computing reduced-order models which are optimal with respect to a joint error measure (w.r.t. parameter and frequency domain).

Another aspect of this project is the investigation of a beneficial connection between linear parameter-varying (LPV) control systems and so-called bilinear control systems. Although the latter ones formally belong to the class of nonlinear control systems, many linear reduction techniques have been shown to possess bilinear analogues. Moreover, embedding LPV systems in the class of bilinear control systems allows for studying parametric model reduction techniques that automatically take care of a desired structure preservation of the underlying parametric process.

Another research focus was based  on Radial Basis Functions in a combination with classical Model Order Reduction techniques to create a truly parametric reduced order system. The goal is that the reduced order model for each individual parameter is close to the ℋ2 optimal one.

Simulation of transport networks is a challenging problem as the modeling results in nonlinear parametric Differential Algebraic Equations (DAE). Furthermore the system size can be rather large for real world problems such as a regions gas network, or a whole country's power line system. This leads to the interest in developing methods, which help to simulate such systems efficiently. Our research interest is to develop model order reduction techniques that can achieve that for transport network systems.

Related Publications

Related Poster

• Parametric Model Order Reduction;
  Evaluation Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg;
  Magdeburg, March 14, 2012.