He Kong, Mao Shan, Daobilige Su, Yongliang Qiao, Abdullah Al-Azzawi, Salah Sukkarieh, Filtering for systems subject to unknown inputs without a priori initial information, . Automatica, Volume 120, 2020 DOI: 10.1016/j.automatica.2020.109122.
The last few decades have witnessed much development in filtering of systems with Gaussian noises and arbitrary unknown inputs. Nonetheless, there are still some important design questions that warrant thorough discussions. Especially, the existing literature has shown that for unbiased and minimum variance estimation of the state and the unknown input, the initial guess of the state has to be unbiased. This clearly raises the question of whether and under what conditions one can design an unbiased and minimum variance filter, without making such a stringent assumption. The above-mentioned question will be investigated systematically in this paper, i.e., design of the filter is sought to be independent of a priori information about the initial conditions. In particular, for both cases with and without direct feedthrough, we establish necessary and sufficient conditions for unbiased and minimum variance estimation of the state/unknown input, independently of a priori initial conditions, respectively. When the former conditions do not hold, we carry out a thorough analysis of all possible scenarios. For each scenario, we present detailed discussions regarding whether and what can be achieved in terms of unbiased estimation, independently of a priori initial conditions. Extensions to the case with time-delays, conceptually like Kalman smoothing where future measurements are allowed in estimation, will also be presented, amongst others.
Shunyi Zhao, Biao Huang, Trial-and-error or avoiding a guess? Initialization of the Kalman filter, . Automatica, Volume 121, 2020 DOI: 10.1016/j.automatica.2020.109184.
As a recursive state estimation algorithm, the Kalman filter (KF) assumes initial state distribution is known a priori, while in practice the initial distribution is commonly treated as design parameters. In this paper, we will answer three questions concerning initialization: (1) At each time step, how does the KF respond to measurements, control signals, and more importantly, initial states? (2) What is the price (in terms of accuracy) one has to pay if inaccurate initial states are used? and (3) Can we find a better strategy rather than through guessing to improve the performance of KF in the initial estimation phase when the initial condition is unknown? To these ends, the classical recursive KF is first transformed into an equivalent but batch form, from which the responses of the KF to measurements, control signal, and initial state can be clearly separated and observed. Based on this, we isolate the initial distribution by dividing the original state into two parts and reconstructing a new state-space model. An initialization algorithm is then proposed by employing the Bayesian inference technique to estimate all the unknown variables simultaneously. By analyzing its performance, an improved version is further developed. Two simulation examples demonstrate that the proposed initialization approaches can be considered as competitive alternatives of various existing initialization methods when initial condition is unknown.