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Article Dans Une Revue Systems and Control Letters Année : 2019

Linear observed systems on groups

Axel Barrau
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Silvère Bonnabel


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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Dates et versions

hal-01671724 , version 1 (22-12-2017)
hal-01671724 , version 2 (08-02-2018)
hal-01671724 , version 3 (15-05-2019)


  • HAL Id : hal-01671724 , version 3


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