Implicit Neural Networks as Static Controllers: Certificates and Performance Separation

2026-07-13

AI Summary

This work addresses the rigorous stability analysis and performance optimization of implicit neural network controllers. By modeling the controller as a trainable linearly interconnected system realized through a fixed-point equation that implements static feedback, the authors develop a unified framework based on Lyapunov theory and integral quadratic constraints (IQC) to construct verifiable LMI/IQC certificates guaranteeing well-posedness, stability, and performance. The approach innovatively integrates implicit differentiation, Perron–Frobenius theory, and discounted infinite-horizon quadratic optimization. When applied to an unstable scalar system with hard actuator constraints, the proposed method not only outperforms all finite-order linear dynamic controllers but also yields computable upper and lower bounds on achievable performance and attains a lower discounted cost.

Abstract

Implicit neural controllers (INCs) are static feedback laws that are evaluated through an algebraic fixed point {equation}; they include as special cases neural network controllers. We propose a so-called implicit representation of neural networks as a key enabling device that exposes the controller as a trainable linear interconnection closed through a known static activation map, thereby making well-posedness and Lyapunov/IQC analysis mathematically easy to handle. For finite-dimensional LTI plants, we first develop a rigorous analysis theory for a given INC, including Perron--Frobenius and norm conditions for well posedness, LMI/IQC certificates for exponential stability, and LMIs for discounted infinite-horizon quadratic performance. We then formulate synthesis as a certification-compatible heuristic search: training is carried out under explicit well-posedness constraints, implicit-differentiation formulas provide gradients, and the resulting controller is accepted only after independent post-training LMIs or regional admissibility checks are feasible. Finally, we establish constrained-control separation results: for a specific scalar unstable plant with hard actuator bounds, an INC achieves a strictly smaller discounted infinite-horizon cost than any admissible finite-order dynamic linear controller. Additional results cover quadratic state-input costs, comparison with linear static output feedback, and computable upper/lower-bound certificates. Numerical examples illustrate the mechanism and the resulting certified performance.

Problem

• Implicit Neural Controllers
• Well-posedness
• Exponential Stability
• Performance Certification
• Constrained Control

Innovation

• Implicit Neural Networks
• Static Feedback Control
• LMI/IQC Certificates
• Implicit Differentiation
• Performance Separation