Plenary lecture 3: an iterative Kalman-like algorithm with no requirements for noise and initial conditions
Yuriy S. Shmaliy · International Conference on Electronics, Hardware, Wireless and Optical Communications · 2011
The term 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 statespace 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.