On Empirical System Gramians

AbstractState‐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.