A Fusion Kalman Filter and UFIR Estimator Using the Influence Function Method Wei Xue, Member, IEEE, Xiaoli Luan, Shunyi Zhao, Senior Member, IEEE, and Fei Liu Abstract—In this paper, the Kalman filter (KF) and the unbiased finite impulse response (UFIR) filter are fused in the discrete-time state-space to improve robustness against uncertainties. To avoid the problem where fusion filters may give up some advantages of UFIR filters by fusing based on noise statistics, we attempt to find a way to fuse without using noise statistics. The fusion filtering algorithm is derived using the influence function that provides a quantified measure for disturbances on the resulting filtering outputs and is termed as an influence finite impulse response (IFIR) filter. The main advantage of the proposed method is that the noise statistics of process noise and measurement noise are no longer required in the fusion process, showing that a critical feature of the UFIR filter is inherited. One numerical example and a practice-oriented case are given to illustrate the effectiveness of the proposed method. It is shown that the IFIR filter has adaptive performance and can automatically switch from the Kalman estimate to the UFIR estimates according to operating conditions. Moreover, the proposed method can reduce the effects of optimal horizon length on the UFIR estimate and can give the state estimates of best accuracy among all the compared methods. Index Terms—Fusion filter, influence function, Kalman filter (KF), robustness, unbiased finite impulse response (FIR).    I. Introduction T O estimate the states of industrial systems, including power electronic systems, large-scale systems, cyber- physical systems, static neural networks and motion control systems, state estimators are considered to be a fundamental tool [1]–[5]. Kalman and Bucy proposed the famous Kalman filter (KF) in the 1970s [6], which is a simple and globally optimal state estimator for linear Gaussian processes [7]. Up until now, it has been widely used in numerous areas with great success. Given an accurate linear model, the KF can theoretically reach optimal estimates [8]–[10], while its errors will rise dramatically once the underlying model is slightly mismatched or there is colored noise. Due to the complexity industrial processes, it is difficult and time-consuming to find an accurate filtering model, and more importantly, the random external interference barely obeys the Gaussian and white statistics. Therefore, many efforts have been made during the last two decades to improve KF performance under different environments [11]–[14]. As a type of finite impulse response (FIR) filters, the unbiased finite impulse response (UFIR) filtering algorithm is constructed and analyzed in [15]. This algorithm ignores the statistical characteristics of noise sources and initial distribution and uses an optimal estimation interval to drive estimation accuracy to approach its optima in the minimum mean square error sense [15]–[17]. Unlike the KF, which recursively computes state estimates, the UFIR filter operates with a finite number of most recent data either in a batch form or in an iterative structure. Therefore, the UFIR filter accumulates estimation errors only within a limited horizon [18]–[21]. Under harsh industrial operating conditions, it is expected that the UFIR filter exhibits better robustness against uncertainties and will be insensitive to changes in the noisy environment. A detailed comparison of the UFIR filter and the KF is provided in [22], [23] with practical examples. As discussed, each filtering algorithm has its features. Specifically, the KF provides the best linear estimates (or almost the best) when the underlying linear model is accurate or nearly accurate, while the UFIR estimator shows impressive robustness against uncertainties. With the boom in the development of both filters comes a variety of fusion strategies. There are self-fusion strategies for the same filter to make better use of the characteristic of the filter [24], [25]. If one wishes to design a filter to achieve the optimality of the KF and the robustness of the UFIR filter simultaneously, a common practice is to find an appropriate strategy to fuse them. For example, an infinite impulse response (IIR)-type filter and an FIR-type filter are merged in [26] using the mixing probability calculated based on the residuals and their covariances. Later, in [27] the KF and UFIR filters are fused by assigning probabilistic weights to achieve smaller errors. With the same motivations, the weighted UFIR filter is derived using the Frobenius norm in [28], [29]. Reference [30] uses measurement differencing and by de-correlating noise vectors to fuse the two filters. A unified fusion framework employed in these approaches is demonstrated in Fig. 1, implying that the IIR and FIR filters give respective estimates, and a fusing procedure then achieves the overall output. Although this structure is clear and pellucid, fusing the UFIR estimate and the Kalman estimate mathematically may not be as intuitive as Fig. 1. The main difficulty is that the Manuscript received January 26, 2021; revised June 10, 2021; accepted August 16, 2021. This work was supported in part by the National Natural Science Foundation of China (61973136, 61991402, 61833007) and the Natural Science Foundation of Jiangsu Province (BK20211528). Recommended by Associate Editor Qinglai Wei. (Corresponding author: Xiaoli Luan and Shunyi Zhao.) Citation: W. Xue, X. L. Luan, S. Y. Zhao, and F. Liu, “A fusion Kalman filter and UFIR estimator using the influence function method,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 4, pp. 709–718, Apr. 2022. The authors are with the Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Jiangsu 214000, China (e-mail: 6191905041@stu.jiangnan. edu.cn; shunyi@jiangnan.edu.cn; xlluan@jiangnan.edu.cn; fliu@jiangnan. edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2021.1004389 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 9, NO. 4, APRIL 2022 709