Identification of unknown noise statistics for non-stationary state space systems, страница 3

,                                                                                                       (11)

.                                                                                                       (12)

The expressions (9 –12) determine the consistent estimations of time-invariant noise variances in conditions of unequally-spaced observations when time intervals between measurements are bounded.

The Figure 1 depicts the behavior of mean square errors of estimations  (Figure 1a) and  (Figure 1b) obtained on the basis of statistical modeling.

Figure 2. The behavior of mean square errors of estimations: a); b) . True values ; .               Solid curve: , probability of measurement omission ; dash curve:  , .

As Figure shows, the accuracy of estimation is high despite the fact that half of measurements were missed.

The expressions (11), (12) use every new pseudo-measurement  and  as equivalent additional information to already available information. When the variances  and  are time-varying, the current pseudo-measurements are more important than previous, so the previous measurements have to be used with smaller weight. Therefore, the method of exponential smoothing with smoothing constants and can be used for the identification of time-varying noise variances in conditions of unequally-spaced observations.

,                                                                                    (13)

                                                                                             (14)

To increase the prediction accuracy of non-stationary processes, it is rational to suppose about existence of non-zero expectation  of noise  and to build adaptive Kalman filter on the basis of identification of both the noise variances and expectation.

The following 1-depentent sequence can be considered as pseudo-measurements of noise expectation  in conditions of unequally-spaced observations

                                                                                                                             (13)

The consistent estimate of time-invariant expectation is giving by                .

Time-varying expectation in conditions of unequally-spaced observations is defined in the following way

,                                                                                                (14)

in which  - smoothing constant.

In conditions of non-zero expectation of noise  in the expressions (9) and (10) instead of residuals  and  centralized values  and  are used.

So, the identification algorithm in conditions of unequally-spaced observations includes next steps:

1. Identification of expectation  in accordance with expressions (13), (14).

2. Identification of variance in accordance with expression (9), (13).

3. Estimation of variance in accordance with expression (14).

To estimate effectiveness of proposed algorithm, the modeling of adaptive Kalman filter for non-stationary trajectory of random walk was performed. The Figure 2a shows the modeling behavior of true expectation of noise  (solid curve) and its estimation in conditions of measurements omission (dotted curve). The Figure 2b depicts the modeling trajectory of random walk and its extrapolated values for one step on the basis of the identification of noise statistics , , .