Innovation sequence

Influence of initial values of the diagonal values of the variance-covariance matrix P(0) and the variance of the experimental error on the gain vector and the Innovation sequence (see Table 41.1 for the experimental values, > )... 

Failure detection methods are based on successive monitoring of the innovation sequence and statistical tests. Basically, the standard filter calculations are performed until some form of aberrant behavior is detected. A test was suggested first by Wilsky and Jones (1976) and is based on the following. 

Under normal behavior of the filter, the innovation sequence is as follows ... 

During the standard Kalman filter calculations, the matrix M has been evaluated from Eq. (8.38). When the test gives an alarm of malfunction, one or more elements of the innovation sequence vector is supposed to be at fault. In order to satisfy the abnormal situation, if one element is assumed to be at fault, the corresponding term in the matrix M would be greater than under normal circumstances. Thus,... 

The behavior of the detection algorithm is illustrated by adding a bias to some of the measurements. Curves A, B, C, and D of Fig. 3 illustrate the absolute values of the innovation sequences, showing the simulated error at different times and for different measurements. These errors can be easily recognized in curve E when the chi-square test is applied to the whole innovation vector (n = 4 and a = 0.01). Finally, curves F,G,H, and I display the ratio between the critical value of the test statistic, r, and the chi-value that arises from the source when the variance of the ith innovation (suspected to be at fault) has been substantially increased. This ratio, which is approximately equal to 1 under no-fault conditions, rises sharply when the discarded innovation is the one at fault. 

The proposed technique is based on an extension to time-varying systems of Wiener s optimal filtering method (l-3). The estimation of the corrected chromato gram is optimal in the sense of minimizing the estimation error variance. A test for verifying the results is proposed, which is based on a comparison between the "innovations" sequence and its corresponding expected standard deviation. The technique is tested on both synthetic and experimental examples, and compared with an available recursive algorithm based on the Kalman filter ( ). 

GUGLIOTTA ET AL. Instrumental Broadening in SEC The innovations sequence is defined by ... 

The proposed check consists in matching the innovations sequence obtained from Equation 25 with the corresponding expected time-varying variance provided ty Equation 28. If the innovations sequence is assumed zero-mean Gaussian white, then e2 (k) should be within the bounds for approximately two thirds of the time, (og (k) re-preients the standard deviation of e2(k), found by square rooming the diagonal elements of Eg ). 

Figure 2 Example 1 a) Comparison between the "true" input u(k), the estimation of that input through the present technique u(k) and the same estimation through the method described in ( ) uj (k) b) Innovations sequence and Og (k) bounds corresponding to u(k).

In which M=100 represents the estimation window size. However, it is noted that (5) gives a valid result when the innovation sequence is stationary and ergodic over the M sample steps. 

Discrimination between performance degradation due to increases in unmeasured disturbances and changes in process parameters is a question of model validation. Consider an idealized case where disturbances can be regarded as white noise. If the model is perfect, the innovation sequence is white noise as well [2]. Imperfect models change the color of the innovation sequence that can be detected using various methods. 

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