Designing asymptotic observers for linear completely regular differential algebraic systems with delay

We study the problem of generating a solution estimate based on the observed output signal data for linear autonomous completely regular differential algebraic systems with commensurate delays. To obtain an estimate for the solution, two types of observers are proposed: an asymptotic observer and an asymptotic observer with bounded error. An asymptotic observer is characterized by the fact that its error asymptotically approaches zero. In this case, if the original system has the property of final observability, then the rate of convergence of the estimation error to zero can be set in advance by choosing the characteristic quasi-polynomial of the homogeneous system that describes the behavior of the error. Otherwise, the estimation error is described by an inhomogeneous system, and the rate of its convergence to zero depends not only on the choice of the characteristic quasi-polynomial of the homogeneous system, but also on the behavior of the inhomogeneous part, the dynamics of which depends on the matrices that determine the structure of the output signal. A distinctive feature of an asymptotic observer with bounded error is that its estimation error remains bounded by some constant depending on the observer's initial condition. In this case, the conditions for the existence of such an observer impose weaker requirements on the parameters of the original system in comparison with the conditions for the existence of an asymptotic observer.