RKHS嵌入式自适应估计中激励的部分持续性
Partial Persistence of Excitation in RKHS Embedded Adaptive Estimation
摘要：
本文提出了一种自适应非参数方法来估计由常微分方程（ODEs）控制的不确定性系统中的标量值非线性函数。通过采用无限维再生核希尔伯特空间（RKHS）作为假设空间，将有限维欧几里得空间中的非线性估计问题转化为在无限维RKHS中构造线性观测器的问题.通过引入部分持续激励（部分PE）的新条件，这是定义为RKHS的子空间的收敛性分析。利用这个条件，我们证明了函数估计误差在PE子空间上的投影依范数渐近收敛于零。虽然这是一个抽象的收敛概念，隐式地依赖于用于定义RKHS的内核，我们推导出的条件，确保逐点收敛的PE子集上的函数估计。本文还介绍了一个较弱的，但几何直观的概念，部分PE条件，一个类似PE条件，因为他们已经制定了历史上在欧几里得空间。充分条件推导出描述时，这两个条件是等价的。最后，通过数值模拟说明了文中给出的收敛性证明的定性性质.
英文摘要：
In this paper, an adaptive non-parametric method is proposed to estimate the scalar-valued nonlinear function that appears in uncertain systems governed by ordinary differential equations (ODEs). By employing an infinite-dimensional reproducing kernel Hilbert space (RKHS) as the hypothesis space, the nonlinear estimation problem in finite-dimensional Euclidean space is recast into that of constructing a linear observer in the infinite-dimensional RKHS. The analysis of convergence is facilitated by the introduction of a novel condition of partial persistent excitation (partial PE), which is defined for a subspace of the RKHS. Using this condition, we prove that the projection of the function estimation error onto the PE subspace converges in norm asymptotically to zero. While this is an abstract notion of convergence that depends implicitly on the kernel used to define the RKHS, we derive conditions that ensure the pointwise convergence of the function estimates over the PE subset. This paper additionally introduces a weaker but geometrically intuitive notion of a partial PE condition, one that resembles PE conditions as they have been formulated historically in Euclidean spaces. Sufficient conditions are derived that describe when the two conditions are equivalent. Finally, qualitative properties of the convergence proofs derived in the paper are illustrated with numerical simulations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
