On Empirical System Gramians

Abstract

State-space realizations of input-output systems or control systems are a widely used class of models in engineering, physics, chemistry and biology. For the qualitative and quantitative classification of such systems, the system-theoretic properties of reachability and observability are essential, which are encoded in so-called system Gramian matrices. For linear systems these Gramians are computed as solutions to matrix equations, for nonlinear or parametric systems the data-driven empirical system Gramians approximate the actual system Gramians. These empirical Gramians have manifold applications, for example in model reduction or decentralized control of nonlinear systems, as well as sensitivity analysis, parameter identification and combined state and parameter reduction of parametric systems. Here, we demonstrate that empirical system Gramians are also useful for linear but hyperbolic input-output systems.While empirical system Gramians are designed for nonlinear and parametric systems, a data-driven computation can be advantageous even for linear systems (1). This is demonstrated using a principal application of (empirical) system Gramians: Model reduction. Gramian-based model reduction computes (Petrov-)Galerkin projection operators

Model order reduction for gas and energy networks

“…The (empirical) dominant subspaces method initially developed in [110], and originally named DSPMR (Dominant Subspace Projection Model Reduction), conjoins and compresses the dominant reachability and observability subspaces of an input-output system, such as the gas network model (8), obtained from (empirical) system Gramians. Heuristically, this method seems to be useful for hyperbolic input-output systems [49]. Here, we consider three variants: first, based on the empirical reachability and observability Gramians, second, based on the empirical cross Gramian and third, based on the empirical nonsymmetric cross Gramian.…”

“…For either test and training, five parameters are sampled. The input perturbations for the steady-state training scenario are selected to be a step function, which heuristically works well for hyperbolic systems [49].…”

Model Order Reduction for Gas and Energy Networks

“…The (empirical) dominant subspaces method initially developed in [100], and originally named DSPMR (Dominant Subspace Projection Model Reduction), conjoins and compresses the dominant reachability and observability subspaces of an input-output system, such as the gas network model ( 8), obtained from (empirical) system Gramians. Heuristically, this method seems to be useful for hyperbolic input-output systems [40]. Here, we consider three variants: first, based on the empirical reachability and observability Gramians, second, based on the empirical cross Gramian and third, on the empirical non-symmetric cross Gramian.…”

“…For either test and training, five parameters are sampled. The input perturbations for the steady-state training scenario are selected to be a step function, which heuristically works well for hyperbolic systems [40]. The empirical Gramian configuration uses default values except for constraining perturbations to only positive values.…”