AI 中文总结
针对高维离散时间非线性系统自适应控制的维度相关性能局限，提出带非欧几里得Polyak型步长的镜像下降型自适应律，其后悔界对环境维度至多呈对数依赖，数值实验验证了方案有效性。
AI 中文摘要
受实时控制问题中现代高容量模型应用的驱动，本文研究具有未知矩阵值参数的高维离散时间非线性系统的自适应控制。我们关注未知参数元素数量大，但参数矩阵具有可利用结构的场景，如元素级稀疏性、组稀疏性、低秩、行随机或密度矩阵结构。为量化暂态性能，我们考虑相对于完全知晓真实参数的标称控制器的后悔准则。我们证明，标准欧几里得更新方案（包括递归最小二乘和梯度下降）不适用于该场景：即使真实参数具有低内在复杂度，其暂态性能也会随维度增加而恶化。为解决此局限，我们提出一类新型镜像下降型自适应律，配备利用参数结构诱导几何的非欧几里得Polyak型步长。对于所提出的更新律，我们建立渐近状态收敛性，并推导出对环境维度至多呈对数依赖的后悔界。数值实验验证了所提方案的有效性。
英文摘要
Motivated by the use of modern high-capacity models in real-time control problems, this paper studies the adaptive control of high-dimensional discrete-time nonlinear systems with an unknown matrix-valued parameter. We focus on regimes where the number of unknown parameter entries is large, but the parameter matrix possesses exploitable structure, such as entrywise sparsity, group sparsity, low rank, and row-stochastic or density-matrix structure. To quantify transient performance, we consider a regret criterion relative to a nominal controller with full knowledge of the true parameter. We show that standard Euclidean update schemes, including recursive least squares and gradient descent, are ill-suited to this setting: their transient performance deteriorates as the dimension increases, even when the true parameter has low intrinsic complexity. To address this limitation, we propose a novel class of mirror-descent-type adaptive laws equipped with a non-Euclidean Polyak-type step size that exploit the geometry induced by the parameter structure. For the proposed update laws, we establish asymptotic state convergence and derive regret bounds with at most logarithmic dependence on the ambient dimension. Numerical experiments demonstrate the effectiveness of the proposed schemes.
