State dimension reduction of recurrent equilibrium networks with contraction and robustness preservation

Recurrent equilibrium networks (RENs) are effective for learning the dynamics of complex dynamical systems with certified contraction and robustness properties through unconstrained learning. While this opens the door to learning large-scale RENs, deploying such large-scale RENs in real-time applications on resource-limited devices remains challenging. Since a REN consists of a feedback interconnection of linear time-invariant (LTI) dynamics and static activation functions, this article proposes a projection-based approach to reduce the state dimension of the LTI component of a trained REN. One of the two projection matrices is dedicated to preserving contraction and robustness by leveraging the already-learned REN contraction certificate. The other projection matrix is iteratively updated to improve the accuracy of the reduced-order REN based on necessary $h_2$-optimality conditions for LTI model reduction. Numerical examples validate the approach, demonstrating significant state dimension reduction with limited accuracy loss while preserving contraction and robustness.