The article revises properties of two identification/adaptation algorithms proposed by Lion and Kreisselmeier more than 40 years ago to accelerate parametric convergence under regressor persistency of excitation (PE) condition. First, motivated by the paper we demonstrate that these algorithms can provide asymptotic (not exponential) parametric convergence under a simple condition which is weaker than the requirement of PE. Second, it is shown that for some special choice of adaptation gain the algorithms can provide finite time parametric convergence if the regressor satisfies the interval excitation (IE) condition that is even weaker than the condition of asymptotic convergence. Third, it is shown that for some certain structural requirements these algorithms can generate the high-order time derivatives of the adjustable parameters. This property can be used for solution of a wide range of problems of adaptive control including, in particular, model reference adaptive control and modular backstepping design with high-order tuners.

中文翻译：

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关于 Lion 和 Kreisselmeier 适应算法的关键特性

本文修改了 Lion 和 Kreisselmeier 40 多年前提出的两种识别/适应算法的特性，以加速回归量激励持续性 (PE) 条件下的参数收敛。首先，受论文的启发，我们证明了这些算法可以在比 PE 要求弱的简单条件下提供渐近（非指数）参数收敛。其次，表明对于自适应增益的一些特殊选择，如果回归器满足区间激励，算法可以提供有限时间参数收敛。(IE) 条件比渐近收敛条件更弱。第三，表明对于某些特定的结构要求，这些算法可以生成可调参数的高阶时间导数。该属性可用于解决自适应控制的广泛问题，特别是模型参考自适应控制和具有高阶调谐器的模块化反推设计。