This paper presents a novel method of estimating the optimal steady state Kalman filter gain of a linear discrete time-invariant system from a non-optimal Kalman filter residual sequence. The relation between the optimal residual sequence and a signal derived from the non-optimal residual sequence is described by a moving average (MA) model whose coefficients are expressed in terms of the state space parameters and the optimal steady state Kalman filter gain. In order to identify the MA model, a whitening filter of the derived signal, which corresponds to an autoregressive (AR) model of the signal, is first identified using the least- squares method. Then the inverse filter of the whitening filter, which corresponds to the MA model, is calculated. From the coefficients of the identified MA model, the optimal steady state Kalman filter gain can be obtained. Numerical example is provided to illustrate the feasibility of this approach.
KEYWORDS: Filtering (signal processing), Digital filtering, Electronic filtering, Acquisition tracking and pointing, Data processing, Error analysis, Systems modeling, Stochastic processes, Data modeling, Signal to noise ratio
This paper proposes a new approach of obtaining adaptive state estimation of a system in the presence of unknown system disturbances and measurement noise. In the beginning, a non-optimal Kalman filter with arbitrary initial guess for the process and measurement noises is implemented. At the same time, an adaptive transversal predictor (ATP) based on the recursive least-squares (ilLS) algorithm is used to yield optimal one- to p- stepahead output predictions using the previous input/output data. Referring to these optimal predictions the Kalman filter gain is updated and the performance of the state estimation is thus improved. If forgetting factor is implemented in the recursive least-squares algorithm, this method is also capable of dealing with the situation when the noise statistics are slowly time-varying. This feature makes this new approach especially suitable for the control of flexible structures. A numerical example demonstrates the feasibility of this real time adaptive state estimation method.
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