The missing link between the output and the H_2-norm of bilinear systems
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In this paper, we prove several new results that give new insights to bilinear systems. We show under which condition a bilinear system is asymptotically stable. Moreover, we provide a global characterization of reachability in bilinear systems based on a certain Gramian. Reachability energy estimates using the same Gramian have only been local so far. The main result of this paper, however, is the link between the output error and the H_2-error of two bilinear systems. This result has several consequences in the field of model order reduction. It explains why H_2-optimal model order reduction leads to good approximations in terms of the output error. Moreover, output errors based on the H_2-norm can now be proved for balancing related model order reduction schemes. These are given in this paper. All these new results are based on a connection between bilinear equations and a linear stochastic differential equations that is established here.
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