Symmetries of differential delay systems with applications to observer design

In this paper we construct a general observer theory for differential delay systems based on different types of symmetries (exact symmetry or asymptotic symmetry), ending up with a certain number of semi-global and global observers, with bounded or unbounded system’s solutions. We introduce the notions of symmetry for a differential delay system, being inspired by well-known definitions of symmetry for an ordinary or partial differential system, and variational symmetry for the associated variational differential delay system. We illustrate observer design procedures in details, by proving that the existence of a (variational asymptotic) symmetrywith system’s detectability in the first approximation have a central role in the design of a state observer. The symmetry is a one-parameter group of transformationswhich maps the systeminto itself (exact symmetry) or into a different system (asymptotic symmetry), approximating the original one with better and better accuracy as the parameter of the symmetry is larger. The types of symmetries we consider here show an important contractive action on the state and input spaces for which the system’s solutions are mapped into arbitrarily small neighbourhoods of the origin in which the transformed system can be well-approximated by its linearization. The parameter of the symmetry may be constant (semiglobal observers) or updated on-line by a state norm estimator (global observers).