In this paper, we propose adaptive algorithms of gain tuning for Kalman filters and switching Kalman and $\mathcal {H}_{\infty }$ filters for discrete systems. Both of the gain tuning and switching rely on square means of innovations. This paper also provides stability analyses on time-varying Kalman filters and derives a sufficient condition for their asymptotic stability, on which our gain-tuning algorithm is based. It should be noted that the stability condition is available for even nonstabilizable systems having their uncontrollable poles on the unit circle. To illustrate those algorithms, we perform simulations using a harmonic oscillator model that is nonstabilizable and has its poles on the unit circle. Furthermore, we apply the algorithms to estimation of a ship's oscillation, particularly, with time-varying frequencies by simulations and model experiments. Consequently, all the results of stability analyses, simulations, and experiments have convinced that the algorithms are solid and effective.

中文翻译：

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调谐和切换卡尔曼滤波器和 H∞ 滤波器的自适应算法及其在时变频率船舶振荡估计中的应用

在本文中，我们提出了卡尔曼滤波器的增益调谐自适应算法和离散系统的切换卡尔曼和 $\mathcal {H}_{\infty }$ 滤波器。增益调整和切换都依赖于创新的平方手段。本文还提供了时变卡尔曼滤波器的稳定性分析，并推导出了它们渐近稳定性的充分条件，我们的增益调整算法基于该条件。应该注意的是，即使是在单位圆上具有不可控极点的不稳定系统，稳定条件也是可用的。为了说明这些算法，我们使用不可稳定且极点在单位圆上的谐振子模型进行仿真。此外，我们将算法应用于船舶振荡的估计，特别是，通过模拟和模型实验具有随时间变化的频率。因此，稳定性分析、模拟和实验的所有结果都确信这些算法是可靠且有效的。