# Linear observed systems on groups

Abstract : We propose a unifying and versatile framework for a class of discrete time systems whose state is an element of a general group $G$, that we call linear observed systems on groups. Those systems strictly mimic linear systems in the sense that + is replaced with group multiplication, and linear maps with automorphisms. We argue they are the true generalization of linear systems of the form X_{n+1}=F_n X_n+B_n u_n in the context of state estimation, since 1- when G is the Euclidean space R^N the latter systems are recovered, 2- they are proved to possess the preintegration'' property, a characteristic property of linear systems that relates continuous time to discrete time, and has recently proved extremely useful in robotics applications, and 3- we can build observers that ensure the evolution between the true state and estimated state does not depend on the followed trajectory, a characteristic feature of Luenberger (and invariant) observers. The theory is applied to a 3D inertial navigation example. Interestingly, this example cannot be put in the form of an invariant system and the proposed generalization is required.
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Journal articles
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Cited literature [21 references]

https://hal-mines-paristech.archives-ouvertes.fr/hal-01671724
Contributor : Silvere Bonnabel Connect in order to contact the contributor
Submitted on : Wednesday, May 15, 2019 - 5:04:16 AM
Last modification on : Wednesday, November 17, 2021 - 12:31:06 PM

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Linear_Observed_Systems.pdf
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• HAL Id : hal-01671724, version 3

### Citation

Axel Barrau, Silvère Bonnabel. Linear observed systems on groups. Systems and Control Letters, Elsevier, 2019, 129, pp.36-42. ⟨hal-01671724v3⟩

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