Optimal model reduction by time-domain moment matching for Lur'e-type models

This paper considers the problem of model reduction for Lur'e-type models consisting of a feedback interconnection between linear dynamics and static nonlinearities. We propose an optimal variant of the time-domain moment-matching method in which the
$\mathcal {H}_{\infty }$
-norm of the error transfer-function matrix of the linear part of the model is minimised, while the static nonlinearities are inherited from the full-order model. We show that this approach also minimises an error bound on the
$\mathcal {L}_{2}$
-norm of the steady-state error between the responses of the full-order nonlinear model and the reduced-order nonlinear model. Furthermore, the proposed approach preserves both the Lur'e-type model structure as well as global stability properties. The problem is cast as an optimisation problem with bilinear matrix inequality constraints. This problem is then solved using a novel algorithm, although global convergence of the algorithm is not guaranteed. The effectiveness of the approach is illustrated in the reduction of a structural dynamics model of a linear beam with nonlinear supports.