Free Parametrization of L₂-Bounded Structured State-Space Controllers for Nonlinear Control with Stability Guarantees

Designing stabilizing control policies for nonlinear systems while optimizing complex objectives remains a formidable challenge. Neural networks (NNs), despite their expressive power, can be highly sensitive to small input perturbations and can easily destabilize the closed-loop system. Existing approaches often impose explicit constraints on the controller's parameters to ensure stability, but this typically leads to additional computational overhead. To address this issue, we leverage recently proposed structured state-space models (SSMs) to parametrize discrete-time control policies for nonlinear systems. Our key contribution is a new free parametrization of linear time-invariant (LTI) systems with a prescribed L2 gain. We use this result to construct the L2-Recurrent Unit (L2RU), an SSM layer that enforces the desired L2 bound by design. The resulting architecture can be used to guarantee closed-loop stability via the small-gain theorem or the so-called performance-boosting framework, independently of the controller's optimization parameters, thereby enabling fully unconstrained optimization of general nonlinear objectives. Furthermore, the structure induced by the proposed parametrization enables the efficient processing of long input sequences, as it is highly parallelizable through algorithms such as parallel scan. We demonstrate the effectiveness of this approach on a formation-control task for mobile robots, where the L2RU-based controller ensures collision and obstacle avoidance while maintaining stability and performance.