Abstract
As discussed in the earlier chapters, the system state contains vital information for control system analysis and design, but it is not always directly measurable. Control engineers will have to obtain the information of system states based on accessible information of the known system dynamics, inputs and outputs (measurements).
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Notes
- 1.
The Kalman’s original derivation did not use the Bayes’ rule and does not require the exploitation of any specific error distribution information. The Kalman filter is the minimum variance estimator if the noise is Gaussian, and it is the linear minimum variance estimator for linear systems with non-Gaussian noises (Kalman 1960; Simon 2006; Julier and Uhlmann 2004; Van Der Merwe 2004).
- 2.$$\begin{aligned} (\mathbf P _{k|k-1}\mathbf H _k^T\mathbf K _k^T)^T= & {} \mathbf K _k\mathbf H _k\mathbf P _{k|k-1} \\ Tr(\mathbf P _{k|k-1}\mathbf H _k^T\mathbf K _k^T)^T= & {} Tr(\mathbf K _k\mathbf H _k\mathbf P _{k|k-1}) \end{aligned}$$
- 3.
For any matrix, M, and a symmetric matrix, N,
$$\begin{aligned} \frac{\partial Tr({MNM^T})}{\partial M}= 2MN. \end{aligned}$$ - 4.
\(\mathcal{H}_{\infty }\) filters minimises the worst-case energy gain from the noise input to the estimation error, which is equivalent to minimising the corresponding \(\mathcal{H}_{\infty }\) norm.
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Chandra, K.P.B., Gu, DW. (2019). Kalman Filter and Linear State Estimations. In: Nonlinear Filtering. Springer, Cham. https://doi.org/10.1007/978-3-030-01797-2_3
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