Sensitivity Analysis
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The Kalman filtering equations provide an estimate of the state and its error covariance recursively. The estimate and its quality depend on the system parameters and the noise statistics fed as inputs to the estimator. This section analyzes the effect of uncertainties in the statistical inputs to the filter. In the absence of reliable statistics or the true values of noise covariance matrices and, the expression
no longer provides the actual error covariance. In other words, . In most real time applications the covariance matrices that are used in designing the Kalman filter are different from the actual noise covariances matrices. This sensitivity analysis describes the behavior of the estimation error covariance when the noise covariances as well as the system matrices and that are fed as inputs to the filter are incorrect. Thus, the sensitivity analysis describes the robustness (or sensitivity) of the estimator to misspecified statistical and parametric inputs to the estimator.
This discussion is limited to the error sensitivity analysis for the case of statistical uncertainties. Here the actual noise covariances are denoted by and respectively, whereas the design values used in the estimator are and respectively. The actual error covariance is denoted by and as computed by the Kalman filter is referred to as the Riccati variable. When and, this means that . While computing the actual error covariance using, substituting for and using the fact that and, results in the following recursive equations for :
- While computing, by design the filter implicitly assumes that and .
- The recursive expressions for and are identical except for the presence of and in place of the design values and respectively.
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