The philosophy and the historical development of Kalman filter from ancient times to the present is followed by the connection between randomness, probability, statistics, random process, estimation theory, and the Kalman filter. A brief derivation of the filter is followed by its appreciation, aesthetics, beauty, truth, perspectives, competence, and variants. The menacing and notorious problem of specifying the filter initial state, measurement, and process noise covariances and the unknown parameters remains in the filter even after more than five decades of enormous applications in science and technology. Manual approaches are not general and the adaptive ones are difficult. The proposed reference recursive recipe (RRR) is simple and general. The initial state covariance is the probability matching prior between the Frequentist approach via optimization and the Bayesian filtering. The filter updates the above statistics after every pass through the data to reach statistical equilibrium within a few passes without any optimization. Further many proposed cost functions help to compare the present and earlier approaches. The efficacy of the present RRR is demonstrated by its application to a simulated spring, mass, and damper system and a real airplane flight data having a larger number of unknown parameters and statistics.
Part of the book: Kalman Filters
For designing an optimal Kalman filter, it is necessary to specify the statistics, namely the initial state, its covariance and the process and measurement noise covariances. These can be chosen by minimising some suitable cost function J . This has been very difficult till recently when a near optimal Recurrence Reference Recipe (RRR) was proposed without any optimisation but only filtering. In many filter applications after the initial transients, the gain matrix K tends to a constant during the steady state, which points to design the filter based on constant gains alone. Such a constant gain Kalman filter (CGKF) can be designed by minimising any suitable cost function. Since there are no covariances in CGKF, only the state equations need to be propagated and updated at a measurement, thus enormously reducing the computational load. Though CGKF results may not be too close to those of RRR, they are acceptable. It accepts extremely simple models and the gains are robust in handling similar scenarios. In this chapter, we provide examples of applying the CGKF by ancient Indian astronomers, parameter estimation of spring, mass and damper system, airplane real flight test data, ballistic rocket, re-entry of space object and the evolution of space debris.
Part of the book: Introduction and Implementations of the Kalman Filter