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 Xn+1=FnXn+Bnun in the context of state estimation, since 1—when G=RN 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.