Synchronous Models and Fundamental Systems in Observer Design

This paper introduces the concept of asynchronous modelas an extension of the internal model concept used in observer design for dynamical systems. A system is said to contain a synchronous model of another if there is a suitable error function between the two systems that remains stationary for all of the trajectories of the two systems. For any given system, we provide necessary and sufficient conditions that there exists another system that contains a synchronous model of the first, and a method to construct a (there may be many) model should one exist. We characterise the class of all systems for which synchronous models can be constructed by showing that they consist of fundamental vector fields induced by a Lie group action, a class of systems we termfundamental systems. For fundamental systems we propose a simple synchronous observer design methodology, for which we show how correction terms can be discretised and combined easily, facilitating global characterisation of convergence and performance. Finally, we provide three examples to demonstrate the key concepts of synchrony, symmetry construction, and observer design for a fundamental system.