Random Coefficient Matrix Of The Kalman Filter And Its Application

Kalman Filtering is an efficient method for the data processing of dynamic system, which uses the measurement vector to estimate the time variant state vector and is widely used in various dynamic systems. The common filtering is classic Kalman filtering for the linear model that requires restrictive conditions for the system. If the conditions can not be met in the practice, the filtering results could be unsatisfactory. The inaccurate targeting of the state transition and measurement matrices may cause the variation of the state estimation. In practice, due to many random factors, the state transition and measurement matrices might be uncertain. If we apply the classic Kalman filtering on those systems, the filtering results are distorted.Discrete-time systems with stochastic parameters arise in many areas such as digital control of chemical processes, economic systems and stochastically sampled digital control systems. The above system can be converted to a linear dynamic system with deterministic parameter matrices but state-dependent process and measurement noises. Therefore, the conditions of standard Kalman filtering is violated and the recursive formulae given by W.L.De Koning can not be derived directly from the Kalman filtering theory. A rigorous analysis in this paper shows under mild conditions, the converted system still satisfies the three conditions of the standard Kalman filtering; therefore, the recursive estimation of the system is still in the form of the modified Kalman filtering. More importantly, this result can be applied to many practical problems related to Kalman filtering such as the system with uncertain observation and Multi-Model dynamic processes. In the last session of this paper, we consider the estimation fusion of multi-sensor. If the distributed dynamic systems are linear and the measurement noise of each sensor is not cross-correlated, the performance of the distributed Kalman filtering fusion is the same as that of the centralized fusion. We show the result is still true for the distributed systems with random parameter matrices.