https://hal-mines-paristech.archives-ouvertes.fr/hal-01671724v3Barrau, AxelAxelBarrauSafran TechBonnabel, SilvèreSilvèreBonnabelCAOR - Centre de Robotique - MINES ParisTech - École nationale supérieure des mines de Paris - PSL - Université Paris sciences et lettresLinear observed systems on groupsHAL CCSD2019[SPI] Engineering Sciences [physics][SPI.AUTO] Engineering Sciences [physics]/AutomaticBonnabel, Silvere2019-05-15 05:04:162021-11-17 12:31:062019-05-23 08:41:55enJournal articleshttps://hal-mines-paristech.archives-ouvertes.fr/hal-01671724v2application/pdf3We 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.