非线性系统辨识中的 turnpike 性质

中文摘要 AI 辅助

我们分析了使用仿真误差最小化（SEM）方法学习一般离散时间非线性状态空间模型的问题。在此设置中，模型参数通常通过最小化训练数据集或从中提取的较短子序列上的仿真输出与测量输出之间的失配来学习。具体而言，我们研究了基础SEM优化问题的累积输出turnpike性质，该性质要求从固定初始状态出发的最优输出序列逼近并保持接近具有自由初始状态的对应SEM问题的最优输出序列。在存在非唯一最优输出序列的情况下——例如在使用神经网络的全黑箱系统辨识中可能出现这种情况——该性质是针对最接近的此类序列表述的。Turnpike行为在实践中通常是理想的，因为它为采用计算上更易处理的固定初始状态SEM公式提供了理论依据，同时确保其最优输出序列保持接近无约束的最优序列。在温和的可达性条件下，我们建立了累积turnpike性质、值函数的强制性以及定制的严格耗散性概念之间的等价性。我们还引入了基数turnpike性质，并证明它严格弱于累积性质。最后，我们基于增量输出稳定性、阶段成本的凸性和合适的最优性条件建立了turnpike行为的充分条件，并通过数值示例说明了该理论。

英文摘要

We analyze the problem of learning general discrete-time nonlinear state-space models using the simulation error minimization (SEM) method. In this setting, model parameters are typically learned by minimizing the mismatch between simulated and measured outputs over a training dataset, or shorter subsequences extracted from it. Specifically, we study the cumulative output turnpike property of the underlying SEM optimization problem, which requires optimal output sequences emanating from a fixed initial state to approach and remain close to an optimal output sequence of the corresponding SEM problem with free initial state. In the presence of non-unique optimal output sequences---as may arise, for instance, in fully black-box system identification using neural networks---the property is formulated with respect to the closest such sequence. Turnpike behavior is generally desirable in practice, as it provides a theoretical justification for employing computationally more tractable SEM formulations with fixed initial states while ensuring that their optimal output sequences remain close to unconstrained optimal ones. Under a mild reachability condition, we establish equivalence between the cumulative turnpike property, coercivity of the value function, and a tailored notion of strict dissipativity. We additionally introduce a cardinality turnpike property and show that it is strictly weaker than the cumulative notion. Finally, we establish sufficient conditions for turnpike behavior based on incremental output stability, convexity of the stage cost, and a suitable optimality condition, and illustrate the theory by means of a numerical example.

发表机构

• Leibniz University Hannover, Institute of Automatic Control（汉诺威莱布尼茨大学，自动控制研究所）

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