Linear observed systems on groups

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 Xn+1 = FnXn + BnUn in the context of state estimation, since 1-when G = 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. (C) 2019 Elsevier B.V. All rights reserved.