Editing Kalman filter (section)

== Sensitivity analysis ==
{{More citations needed section|date=December 2010}}

The Kalman filtering equations provide an estimate of the state <math>\hat{\mathbf{x}}_{k\mid k}</math> and its error covariance <math>\mathbf{P}_{k\mid k}</math> 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.<ref name=anderson>{{cite book|last1= Anderson|first1= Brian D. O.|last2=Moore|first2=John B.|year=1979 |title=Optimal Filtering |publisher=[[Prentice Hall]]|place=New York|pages=129–133|isbn= 978-0-13-638122-8}}</ref> In the absence of reliable statistics or the true values of noise covariance matrices <math>\mathbf{Q}_{k}</math> and <math>\mathbf{R}_k</math>, the expression
:<math>\mathbf{P}_{k\mid k} = \left(\mathbf{I} - \mathbf{K}_k\mathbf{H}_k\right)\mathbf{P}_{k\mid k-1}\left(\mathbf{I} - \mathbf{K}_k\mathbf{H}_k\right)^\textsf{T} + \mathbf{K}_k\mathbf{R}_k\mathbf{K}_k^\textsf{T}</math>

no longer provides the actual error covariance. In other words, <math>\mathbf{P}_{k \mid k} \neq E\left[\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k}\right)\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}\right)^\textsf{T}\right]</math>.  In most real-time applications, the covariance matrices that are used in designing the Kalman filter are different from the actual (true) noise covariances matrices.{{citation needed|date=December 2010}} This sensitivity analysis describes the behavior of the estimation error covariance when the noise covariances as well as the system matrices <math>\mathbf{F}_k</math> and <math>\mathbf{H}_k</math> 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 <math>\mathbf{Q}^a_k</math> and <math>\mathbf{R}^a_k</math> respectively, whereas the design values used in the estimator are <math>\mathbf{Q}_k</math> and <math>\mathbf{R}_k</math> respectively. The actual error covariance is denoted by <math>\mathbf{P}_{k \mid k}^a</math> and <math>\mathbf{P}_{k \mid k}</math> as computed by the Kalman filter is referred to as the Riccati variable. When <math>\mathbf{Q}_k \equiv \mathbf{Q}^a_k</math> and <math>\mathbf{R}_k \equiv \mathbf{R}^a_k</math>, this means that <math>\mathbf{P}_{k \mid k} = \mathbf{P}_{k \mid k}^a</math>. While computing the actual error covariance using <math>\mathbf{P}_{k \mid k}^a = E\left[\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}\right)\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}\right)^\textsf{T}\right] </math>, substituting for <math>\widehat{\mathbf{x}}_{k \mid k}</math> and using the fact that <math>E\left[\mathbf{w}_k\mathbf{w}_k^\textsf{T}\right] = \mathbf{Q}_k^a</math> and <math>E\left[\mathbf{v}_k \mathbf{v}_k^\textsf{T}\right] = \mathbf{R}_k^a</math>, results in the following recursive equations for <math>\mathbf{P}_{k \mid k}^a</math> :
:<math>\mathbf{P}_{k \mid k-1}^a = \mathbf{F}_k\mathbf{P}_{k-1 \mid k-1}^a \mathbf{F}_k^\textsf{T} + \mathbf{Q}_k^a </math>

and
:<math>\mathbf{P}_{k \mid k}^a = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)\mathbf{P}_{k \mid k-1}^a \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)^\textsf{T} + \mathbf{K}_k \mathbf{R}_k^a \mathbf{K}_k^\textsf{T}</math>

While computing <math>\mathbf{P}_{k \mid k}</math>, by design the filter implicitly assumes that <math>E\left[\mathbf{w}_k \mathbf{w}_k^\textsf{T}\right] = \mathbf{Q}_k</math> and <math>E\left[\mathbf{v}_k \mathbf{v}_k^\textsf{T}\right] = \mathbf{R}_k</math>. The recursive expressions for <math>\mathbf{P}_{k \mid k}^a</math> and <math>\mathbf{P}_{k \mid k}</math> are identical except for the presence of <math>\mathbf{Q}_k^a</math> and <math>\mathbf{R}_k^a</math> in place of the design values <math>\mathbf{Q}_k</math> and <math>\mathbf{R}_k</math> respectively. Researches have been done to analyze Kalman filter system's robustness.<ref>Jingyang Lu.  [https://web.archive.org/web/20200224230739/https://ieeexplore.ieee.org/document/6916211, "False information injection attack on dynamic state estimation in multi-sensor systems"], Fusion 2014</ref>