Data-driven dissipativity analysis: application of the matrix S-lemma

Abstract

We study dissipativity of linear finite-dimensional input-state-output systems from a data-driven perspective. We assume that the system dynamics is unknown, in the sense that we do know the state space dimension and the input and output dimensions, but the system matrices are not known. Our contributions are the following. First, we prove that dissipativity of an unknown linear system can only be ascertained on the basis of the given data if a matrix constructed from measured states and inputs has full rank. In the noiseless data case, this implies that one can only verify dissipativity from data if the data-generating system is the only one that explains the data, in other words, if the true system is identifiable from the data. In this case, dissipativity of the unknown system can be ascertained by checking the feasibility of a given data-based linear matrix inequality. In the noisy data case, it turns out that one does not need identifiability. In order to check dissipativity in this case, we combine the matrix S-lemma with a basic dualization lemma to provide a data-driven test for dissipativity.

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