Adaptive estimation in reproducing kernel Hilbert spaces
Adaptive estimation in reproducing kernel Hilbert spaces
复制标题
再生核希尔伯特空间中的自适应估计
DOI：
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发表时间：
2017
期刊：
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通讯作者：
J. Ferris
中科院分区：
文献类型：
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作者：
Parag Bobade;Suprotim Majumdar;Savio Pereira;A. Kurdila;J. Ferris
本文介绍了一种新的框架，研究的自适应或在线估计问题的一类常见的非线性系统的常微分方程（ODEs）上。与大多数常规的常微分方程的策略相比，这里的方法将出现在植物中的未知非线性函数的估计嵌入再生核希尔伯特空间（RKHS），H。然后将非线性自适应估计问题转化为未知函数的有限维状态估计和无穷维估计的乘积空间φ d × H中的时变估计问题。因此，自适应估计问题构成了一种类型的分布参数系统，即使原始系统是一个ODE的集合。RKHS中的未知函数是分布参数。本文给出了（1）解存在唯一的充分条件，（2）状态估计误差的稳定性和收敛性，（3）有限维近似解收敛于无限维状态空间上的解。新的配方提供了一个简洁和直接的方式，严格提出自适应估计问题，使用基地，适合分散近似。最后给出了道路或地形图自适应估计的一个数值例子，以说明本文所导出的函数估计的收敛性。
This paper introduces a novel framework for the study of adaptive or online estimation problems for a common class of nonlinear systems governed by ordinary differential equations (ODEs) on ℝd. In contrast to most conventional strategies for ODEs, the approach here embeds the estimate of the unknown nonlinear function appearing in the plant in a reproducing kernel Hilbert space (RKHS), H. The nonlinear adaptive estimation problem is then cast as a time-varying estimation problem in the product space ℝd × H of finite dimensional state estimates and infinite dimensional estimates of the unknown function. The adaptive estimation problem thereby constitutes a type of distributed parameter system, even though the original system is a collection of ODEs. The unknown function that lies in the RKHS is the distributed parameter. This paper derives (1) the sufficient conditions for the existence and uniqueness of solutions, (2) the stability and convergence of the state estimation error, and (3) the convergence of finite dimensional approximate solutions to the solution on the infinite dimensional state space. The new formulation provides a succinct and direct way of rigorously posing adaptive estimation problems using bases that are amenable to scattered approximation. A numerical example on adaptive estimation of road or terrain maps is presented to illustrate the convergence of the function estimates derived in this paper.