作为静态控制器的隐式神经网络：证书与性能分离
Implicit Neural Networks as Static Controllers: Certificates and Performance Separation
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中文总结 AI 辅助
研究将隐式神经网络用作静态控制器，提出其隐式表示，为有限维线性时不变系统开发分析理论及综合方法，经训练和检查得到控制器，还建立约束控制分离结果，显示INC在特定系统中成本更小。
中文摘要 AI 辅助
隐式神经控制器（INCs）是通过代数不动点方程进行评估的静态反馈定律，其中包括神经网络控制器作为特殊情况。我们提出一种所谓的神经网络隐式表示作为关键支持工具，将控制器展现为通过已知静态激活映射闭合的可训练线性互连，使适定性和李雅普诺夫/IQC分析在数学上易于处理。对于有限维线性时不变系统，我们首先为给定的INC开发严格分析理论，包括适定性的Perron - Frobenius和范数条件、指数稳定性的LMI/IQC证书以及折扣无限时域二次性能的LMI。然后将综合表述为与认证兼容的启发式搜索：在明确的适定性约束下进行训练，隐式微分公式提供梯度，只有在独立的训练后LMI或区域可容许性检查可行后才接受所得控制器。最后，我们建立约束控制分离结果：对于具有硬执行器界限的特定标量不稳定系统，INC实现的折扣无限时域成本比任何可容许的有限阶动态线性控制器都严格更小。其他结果涵盖二次状态 - 输入成本、与线性静态输出反馈的比较以及可计算的上下界证书。数值示例说明了该机制和所得的认证性能。
英文摘要
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.
