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.

翻译：结构化状态空间模型（SSM）近期作为融合机器学习与控制理论的强大架构崭露头角，其核心层由离散时间线性时不变（LTI）系统与逐点非线性激活函数级联构成。这类模型兼具深度神经网络的表达力与动力系统的可解释性及归纳偏置，在处理长序列任务时展现出优异性能与计算复杂度优势。然而，由于难以以系统化且可操作的方式实现稳定性与鲁棒性约束，该类模型在系统辨识与最优控制等领域的应用仍受局限。本文提出L2RU——一类具备预设ℒ₂增益界约束的SSM模型，可对所有参数值保证输入-输出稳定性与鲁棒性。L2RU架构源于满足ℒ₂约束的LTI系统的自由参数化设计，允许通过标准梯度方法进行无约束优化，同时保持严格的稳定性保证。具体而言，我们发展了两种互补参数化方案：非保守参数化实现对具有给定ℒ₂界的方形LTI系统的完整描述，保守参数化则将方法推广至一般（可能非方形）系统，并通过系统矩阵的结构化表示提升计算效率。两种参数化均支持高效初始化方案，可促进长记忆模型的训练。我们在非线性系统辨识基准上验证了所提框架的有效性，与现有SSM架构相比，L2RU在训练稳定性与性能提升方面表现更优，彰显其作为学习与控制领域系统化且鲁棒的基础模块的潜力。