Cross-Gramian-based dominant subspaces
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Cross-Gramian-based dominant subspaces Peter Benner1,2
· Christian Himpe1
Received: 21 September 2018 / Accepted: 4 September 2019 / © The Author(s) 2019
Abstract A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an apriori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings. Keywords Controllability · Observability · Cross Gramian · Model reduction · Dominant subspaces · HAPOD · DSPMR Mathematics Subject Classification (2010) 93A15 · 93B11 · 93B20
1 Introduction Input-output systems map an input function to an output function via a dynamical system. The input excites or perturbs the state of the dynamical system and the output is some transformation of the state. Typically, these input and output functions are low dimensional while the intermediate dynamical system is high(er) dimensional. In
Communicated by: Anthony Nouy Christian Himpe
[email protected] Peter Benner [email protected] 1
Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
2
Faculty of Mathematics, Otto von Guericke University Magdeburg, Universit¨atsplatz 2, 39106 Magdeburg, Germany
P. Benner, C. Himpe
applications from natural sciences and engineering, the dimensionality of the dynamical system may render the numerical computation of outputs from inputs excessively expensive or at least demanding. Model reduction addresses this computational challenge by algorithms that provide surrogate systems, which approximate the input-output mapping of the original system with a low(er) dimensional intermediate dynamical system. Practically, the trajectory of the dynamical system’s state is constrained to a subspace of the original system’s state-space, for example, by using truncated projections. A standard approach for projection-based model reduction of input-output systems is balanced truncation [27], which transforms the state-space unitarily to a representation that is sorted (balanced) in terms of the input’s effect on the state (controllability) as well as the state’s effect on the output (observability) and discards (truncates) the least important states according to this measure. Instead of balancing, this work investigates a dominant subspaces approach [32] that conjoins the most controllable and most observable subspaces into a projection. This unbalanced model reduction method may yield larger or less accurate reduced order systems, yet allows a computationally advantageous formulation
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