A Survey on Model Reduction of Coupled Systems
In this paper we give an overview of model order reduction techniques for coupled systems. We consider linear time-invariant control systems that are coupled through input-output relations and discuss model reduction of such systems using moment matching
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Institut f¨ur Mathematik, MA 4-5, Technische Universit¨at Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany [email protected]. Institut f¨ur Mathematik, MA 4-5, Technische Universit¨at Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany [email protected].
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Summary. In this paper we give an overview of model order reduction techniques for coupled systems. We consider linear time-invariant control systems that are coupled through input-output relations and discuss model reduction of such systems using moment matching approximation and balanced truncation. Structure-preserving approaches to model order reduction of coupled systems are also presented. Numerical examples are given.
1 Introduction Modelling and simulation of complex physical and technical processes yield coupled systems that consist of ordinary differential equations, differential-algebraic equations and partial differential equations. Such systems arise in many practical applications including very large system integrated (VLSI) chip design and microelectro-mechanical systems (MEMS), e.g. [10, 14, 21, 52, 58]. As the number and density of components on a single chip increase and feature sizes decrease, different physical effects such as thermal interaction, electromagnetic radiation, substrate noise and crosstalk cannot be ignored anymore. Furthermore, the design of microand nano-structures requires the development of new multi-physical models describing their complex internal behavior. Another application area of coupled systems is in subdomain decomposition. Partial differential equations on complicated spatial geometries may be represented as a system of partial differential equations on simpler domains coupled, for example, through boundary conditions. As the mathematical models get more detailed and different coupling effects have to be included, the development of efficient simulation and optimization tools for large-scale coupled systems is a challenging task. Such systems consist of several subsystems whose inputs and outputs are coupled via additional algebraic relations. The subsystems usually have a high number of internal variables that leads to large ∗
Supported by the DFG Research Center M ATHEON “Mathematics for key technologies” in Berlin.
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T. Reis and T. Stykel
memory requirements and computational complexity. To handle such large systems in simulation, control and optimization, their model order reduction (or reducedorder modelling) is indispensable. A general idea of model order reduction is to approximate a large-scale system by a reduced model of lower state space dimension that has the same behavior as the original system. In the last years, many different model reduction methods have been developed in computational fluid dynamics, control design and electrical and mechanical engineering, see [4,11,47] for books on this topic. In this paper we review recent progress in dimension reduction of coupled systems. In structural dynamics, model reduction methods based on subsystem structuring have been of inter
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