A new class of symmetry-preserving observers

This paper introduces the concept of $\theta$-invariant systems, which can be regarded as a generalization of symmetry-preserving systems. This new property enables the re-scaling of time coordinates while dilating the solutions. We discuss some of the implications of this concept for observer design and adapt the results about the existence of a canonical representation for an observer obtained in the context of invariant systems. Furthermore, we demonstrate how our generalization can be used to construct an observer design framework for homogeneous systems with arbitrary degrees.