I'm studying the topic from Probabilistic Robotics by Thrun Burgard and Fox.
In the Extended Kalman Filter algorithm, we linearized the action model in the following way.
𝑔(𝑢(𝑡),𝑥(𝑡-1)) = 𝑔(𝑢(𝑡),𝜇(𝑡−1)) + 𝐺(𝑡)⋅(𝑥(𝑡−1)−𝜇(𝑡−1))
𝑔(𝑢(𝑡),𝑥(𝑡-1)) is the action model and 𝐺(𝑡) is its Jacobian matrix with respect to the state 𝑥(𝑡−1).
I don't see how this guarantees linearity because 𝑔 could be nonlinear in 𝑢(𝑡). The authors don't mention anything about why this is the case.
In other words, I imagined that the multivariate Taylor expansion for this where we get a linear function in both 𝑢(𝑡) and 𝑥(𝑡−1)
Hi,
It should be noted that the Kalman filter is part of the theory of estimation of states of a system represented by a linear model. It is therefore an algorithm that provides estimates of some unknown variables from observed measurements over time. The filter uses as input the command u(t) and the output of the model y(t) and as output it provides an estimate of the output, in other words an estimate of the states of the system. For non-linear systems there is an extension that can deal with these cases: the extended Kalman filter. In my opinion there is no particular requirement on the control when applying this algorithm.
Also please take a look at the links.
https://www.sciencedirect.com/topics/engineering/kalman-filter
https://www.sciencedirect.com/topics/earth-and-planetary-sciences/kalman-filter
https://engineeringmedia.com/controlblog/the-kalman-filter
https://www.cs.unc.edu/~welch/kalman/media/pdf/Julier1997_SPIE_KF.pdf
Best regards
Hi,
It should be noted that the Kalman filter is part of the theory of estimation of states of a system represented by a linear model. It is therefore an algorithm that provides estimates of some unknown variables from observed measurements over time. The filter uses as input the command u(t) and the output of the model y(t) and as output it provides an estimate of the output, in other words an estimate of the states of the system. For non-linear systems there is an extension that can deal with these cases: the extended Kalman filter. In my opinion there is no particular requirement on the control when applying this algorithm.
Also please take a look at the links.
https://www.sciencedirect.com/topics/engineering/kalman-filter
https://www.sciencedirect.com/topics/earth-and-planetary-sciences/kalman-filter
https://engineeringmedia.com/controlblog/the-kalman-filter
https://www.cs.unc.edu/~welch/kalman/media/pdf/Julier1997_SPIE_KF.pdf
Best regards
Dear,
The Kalman filter is naturally linear. When you use this estimator in the structure of the linear state-space control system, the stabilization of complete feedback loop is straitforward. However, you are not forced to use Kalman only for linear systems. You might use the Kalman filter in the nonlinear systems. In such applications, you should create theorems for stability of overall feedback loop.
Dear Essam,
Conceptually, in a stochastic linear system the functionality of a basic Kalman filter is to produce an optimal filtering of the current noisy system state x(t), x^(t|t), based on the noisy measurements up to current time instant t, y(t’), t’
No. You can use it in the structure of nonlinear control systems as well. But Laplace transform is not applicable for robust stability proof.
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