Adaptive Algorithms of Tuning and Switching Kalman and $\mathcal {H}_{\infty }$ Filters and Their Application to Estimation of Ship Oscillation With Time-Varying Frequencies

In this paper, we propose adaptive algorithms of gain tuning for Kalman filters and switching Kalman and H
_∞
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.