论文标题
使用LTI过滤器的动态回归器扩展和混合参数估计器收敛的条件
Conditions for Convergence of Dynamic Regressor Extension and Mixing Parameter Estimator Using LTI Filters
论文作者
论文摘要
在本说明中，我们研究了最近引入的动态回归延伸和混合（DREM）参数估计器的收敛条件，当时使用LTI过滤器生成扩展回归器。 In particular, we are interested in relating these conditions with the ones required for convergence of the classical gradient (or least squares), namely the well-known persistent excitation (PE) requirement on the original regressor vector, $ϕ(t) \in \mathbb{R}^q$, with $q \in \mathbb{N}$ the number of unknown parameters.此外，我们研究了只有间隔激发（IE）的情况，在此情况下，DREM，并发和复合学习方案确保了全局收敛，这是DREM在有限时间内的收敛。关于PE，我们证明，在某些温和的技术假设下，如果$ ϕ（t）$是pe，则DREM的标量回归器，$δ（t）\ in \ Mathbb {r} $，也是PE，可以确保指数收敛。关于IE，我们证明，如果$ ϕ（t）$是IE，则$δ（t）$也是IE。所有这些结果都是在几乎确定的意义上确定的，即证明了该索赔未持有的过滤器参数的集合为零。我们证明中使用的主要技术工具的灵感来自于最近在文献中报道的非自主非线性系统的Luenberger观察者的研究。
In this note we study the conditions for convergence of recently introduced dynamic regressor extension and mixing (DREM) parameter estimator when the extended regressor is generated using LTI filters. In particular, we are interested in relating these conditions with the ones required for convergence of the classical gradient (or least squares), namely the well-known persistent excitation (PE) requirement on the original regressor vector, $ϕ(t) \in \mathbb{R}^q$, with $q \in \mathbb{N}$ the number of unknown parameters. Moreover, we study the case when only interval excitation (IE) is available, under which DREM, concurrent and composite learning schemes ensure global convergence, being the convergence for DREM in finite time. Regarding PE we prove that, under some mild technical assumptions, if $ϕ(t)$ is PE then the scalar regressor of DREM, $Δ(t) \in \mathbb{R}$, is also PE, ensuring exponential convergence. Concerning IE we prove that if $ϕ(t)$ is IE then $Δ(t)$ is also IE. All these results are established in the almost sure sense, namely proving that the set of filter parameters for which the claims do not hold is of zero measure. The main technical tool used in our proof is inspired by a study of Luenberger observers for nonautonomous nonlinear systems recently reported in the literature.
