Abstract: The term "Kalman-like" or "Kalman-type" is commonly used whenever the standard linear Kalman filtering algorithm is modified to estimate state of the nonlinear model, under unknown initial conditions, in the presence of nonwhite or multiplicative noise sources, etc. In such improper applications for the Kalman filter, the Kalman-like one is designed to save the recursive structure, while connecting the algorithm components with the model in different ways. Because there can be found an infinity of the Kalman-like solutions depending on applications, we encounter a number of propositions suggesting some new qualities while saving (or not deteriorating substantially) the advantages of the Kalman filter: accuracy, fast computation, and small memory. The extended and unscented algorithms are among the widely recognized Kalman-like ones suitable for nonlinear problems. Nahi proposed a modification for uncertain observations by including the multiplicative noise component to the measurement matrix. For hidden Markov trees, the efficient restoration Kalman-like algorithm was discussed by Basseville et al. Implying nonlinear modeling for hidden Markov chains, the Kalman filter was modified by Baccarelli and Cusani to have the gain dependent on the observations. Most recently, Ait-El-Fquih and Desbouvries applied the Kalman-like approach to triple Markov chains. We also meet a new Kalman-like tracking algorithm applied to the autoregressive channel process estimation with fading by Stefanatos and Katsaggelos. Different kinds of the Kalman-like algorithms can also be found in the area of control. Becis-Aubrya et al, discussed the two-step one with a switching gain matrix. Carli et al. employed the concept of the centralized Kalman filter for state estimation in the complex sensor networks, and the list of the developments can be extended. There were also proposed the iterative Kalman-like forms for the finite impulse response (FIR) time invariant filters. Han, Kwon, and Kim suggested a relevant algorithm for deterministic control systems and Shmaliy derived an algorithm for the p-shift unbiased FIR estimator.
This lecture introduces readers to the recently developed p-shift general iterative linear Kalman-like FIR estimation algorithm intended for filtering (p = 0), prediction (p > 0), and smoothing (p < 0) of linear discrete time-varying state-space models. The algorithm is designed to have no requirements for noise and initial conditions and thus has strong engineering features. A solution is first found in a batch form and then represented in the computationally efficient iterative Kalman-like one with the following advantages peculiar to FIR structures: guarantied bounded input/bounded output (BIBO) stability, better robustness against temporary model uncertainties and round-off errors, and low sensitivity to noise and initial conditions. It is shown that the estimator proposed overperforms the Kalman one when 1) the noise covariances and initial conditions are not known exactly, 2) the noise constituents are not white sequences, and 3) both the system and measurement noise components need to be filtered out. Otherwise, the Kalman-like and Kalman estimators produce similar errors. A payment for this is a larger operation time featured to averaging. These conclusions are supported by extensive numerical investigations and comparisons with the Kalman smoothing, filtering, and predictive estimates of multistate space models. Examples of applications are taken from signal and image processing, clock synchronization, and control. If one still wonders why the Kalman-like FIR algorithm ignoring noise and initial conditions is able to provide errors similar or even lower than in the Kalman one, then there is no magic. Just recall the rule of thumb of averaging: the estimate variance diminishes as a reciprocal of the averaging interval irrespective of the model.

Brief Biography of the Speaker:
Dr. Yuriy S. Shmaliy is Full Professor of Electrical and Electronics Engineering of the University of Guanajuato (DICIS campus in Salamanca), Mexico. He received the B.S., M.S., and Ph.D. degrees in 1974, 1976 and 1982, respectively, from the Kharkiv Aviation Institute, Ukraine, all in Electrical Engineering. In 1992 he received the Doctor of Technical Sc. degree from the Kharkiv Railroad Institute. In March 1985, he joined the Kharkiv Military University. He serves as Full Professor beginning in 1986 and has a certificate of Professor from the Ukrainian Government in 1993. Since 1993 to 1999, he has been a director-collaborator of the Scientific Center "Sichron" (Kharkiv, Ukraine) working in the field of precise time and frequency. Since 1999 to 2009, he has been with the Kharkiv National University of Radio Electronics, and, since November 1999, he has been with the Guanajuato University of Mexico. His books Continuous-Time Signals (2006) and Continuous-Time Systems (2007) were published by Springer, New York. His book GPS-based Optimal FIR Filtering of Clock Models was published by Nova Science Publ., New York. He also contributed to several books with invited chapters. Dr. Shmaliy has 262 Journal and Conference papers and 80 patents. He was rewarded a title, Honorary Radio Engineer of the USSR, in 1991; was listed in Marquis Who's Who in the World in 1998; was listed in Outstanding People of the 20th Century, Cambridge, England in 1999; and was listed in The Contemporary Who’s Who, American Bibliographical Institute, 2003. He is Senior Member of IEEE and belongs to several other professional Societies. He is currently an Associate Editor of Recent Patents on Space Technology. He is a member of the Organizing and Program Committees of various Int. Symposia. He is a founder and organizer of the Int. Symposium on Precision Oscillators in Electronics and Optics. He was multiply invited to give tutorial, plenary, and seminar lectures. His current interests include statistical signal processing, optimal estimation, and stochastic system theory.