Observer Design for a Class of Delay Systems Using a Fredholm Transform

In this paper, we design an observer for a system represented by a general class of Integral Delay Equations (IDE). This class of equations encompasses various systems, from chemical or biological processes to the propagation of electric pulses on excitable media. The available measurement corresponds to a discrete and distributed-delayed value of the state. Under an appropriate spectral observability assumption, an implementable observer is proposed. Inspired by the fact that first-order hyperbolic Partial Derivative Equations (PDE) and timedelay systems are closely related, we use a PDE formulation and the well-known backstepping methodology for the design. However, due to integral terms in the dynamics, the use of a Fredholm integral transform is required. We prove its existence and invertibility by using an operator framework. Both proofs derive from the spectral observability assumption. Some test case simulations end the paper.