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

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