In recursive least squares (RLS), a persistently exciting (PE) regressor guarantees global asymptotic stability (GAS) of the estimation error relative to the zero equilibrium, and hence convergence of the parameter estimates to their true values. It is known, however, that PE is sufficient but not necessary for GAS, and thus GAS might be achieved even if the regressor is not PE. Since analyses based on PE are ubiquitous in the recursive estimation literature, the existence of non-PE regressors that ensure GAS raises the question of how PE can be directly generalized into a necessary and sufficient condition for the GAS of RLS, and whether or not such a generalization would provide a simple characterization of non-PE regressors for which RLS is GAS. In this paper, we introduce WPE, a direct generalization of PE that is necessary and sufficient for GAS, and explain how it can be understood as a “non-uniform” extension of PE to specific classes of summation windows and lower bound sequences. Next, we show that WPE is equivalent to a condition that emerges from extending certain proofs of non-negative series divergence, such as Oresme’s divergence proof for the harmonic series, to sequences of real symmetric positive-semidefinite matrices.

在递归最小二乘法 (RLS) 中，持续激励 (PE) 回归器保证估计误差相对于零平衡的全局渐近稳定性 (GAS)，从而保证参数估计收敛到其真实值。然而，众所周知，对于 GAS 来说 PE 是足够的但不是必需的，因此即使回归器不是 PE，也可能实现 GAS。由于基于 PE 的分析在递归估计文献中无处不在，确保 GAS 的非 PE 回归量的存在提出了一个问题，即 PE 如何直接推广为 RLS 的 GAS 的必要和充分条件，以及是否这样概括将提供 RLS 为 GAS 的非 PE 回归量的简单表征。在本文中，我们介绍了 WPE，对 GAS 必要且充分的 PE 的直接概括，并解释了如何将其理解为 PE 对特定类别的求和窗口和下界序列的“非均匀”扩展。接下来，我们证明 WPE 等价于将非负级数散度的某些证明（例如 Oresme 对调和级数的散度证明）扩展到实对称正半定矩阵序列所产生的条件。