L2RU: a Structured State Space Model with prescribed L2-bound

Structured state-space models (SSMs) have recently emerged as a powerful architecture at the intersection of machine learning and control, featuring layers composed of discrete-time linear time-invariant (LTI) systems followed by pointwise nonlinearities. These models combine the expressiveness of deep neural networks with the interpretability and inductive bias of dynamical systems, offering strong performance on long-sequence tasks with favorable computational complexity. However, their adoption in applications such as system identification and optimal control remains limited by the difficulty of enforcing stability and robustness in a principled and tractable manner. We introduce L2RU, a class of SSMs endowed with a prescribed $\mathcal{L}_2$-gain bound, guaranteeing input--output stability and robustness for all parameter values. The L2RU architecture is derived from free parametrizations of LTI systems satisfying an $\mathcal{L}_2$ constraint, enabling unconstrained optimization via standard gradient-based methods while preserving rigorous stability guarantees. Specifically, we develop two complementary parametrizations: a non-conservative formulation that provides a complete characterization of square LTI systems with a given $\mathcal{L}_2$-bound, and a conservative formulation that extends the approach to general (possibly non-square) systems while improving computational efficiency through a structured representation of the system matrices. Both parametrizations admit efficient initialization schemes that facilitate training long-memory models. We demonstrate the effectiveness of the proposed framework on a nonlinear system identification benchmark, where L2RU achieves improved performance and training stability compared to existing SSM architectures, highlighting its potential as a principled and robust building block for learning and control.