## 1 Introduction  
Time delay systems are often found in dynamic processes. It is well known that when the delay is small enough in comparison with the dynamics of the process, it can be neglected. However, if this is not the case, a careful design of the control stage becomes necessary. Systems with a delay are common in different fields of application, for example, in chemical processes [1], satellite communications [2], hydraulic systems [3], economic systems [4] and technological processes, such as remote operation [5, 6]. From the control point of view, linear delayed systems pose a challenge because when closing the control loop, the delay term is located in the denominator of the transfer function. As a consequence, the characteristic equation has an infinite number of roots. Numerous works have addressed the control of delayed systems in continuous time; for example, some authors propose representing the delay as a rational function by using Taylor, Bessel, or Pade approximations [7, 8, 9]. Others propose strategies based on classical controllers such as Proportional (P), Proportional-Derivative (PD), Proportional-Integral (PI), and Proportional-Integral-Derivative (PID); see, for example, [10, 11, 12, 13, 1]. On the other hand, when a digital control is implemented, the process to be controlled should be discretized. In the particular case of a discretization by using a zero-order hold (ZOH) of the delayed system, the delay term yields \(n\) poles at the origin, where \(n\) is an integer and must satisfy \(T=\tau/n\), where \(T\) is the sampling period and \(\tau\) is the time delay. This approach allows designing a digital controller using any classical technique in the discrete domain (for instance, P, PD, PI, or PID controllers). However, some robustness problems can occur when the delay of the system differs from the model. Another way to use a discrete model of the system, as mentioned above, is to perform a discrete analysis of the transfer function together with the controller by leading the sampling time \(T\to 0\) in the analysis (or equivalently \(n\rightarrow\infty\)) to get stability conditions in continuous time [14, 15]. From a different perspective, the control of the delayed system has been treated by using the well-known Smith Predictor (PS) [16]. This strategy estimates a signal before being delayed, and later this estimated signal is used to apply the control to the delay-free system. However, the main limitation of this strategy is that it only allows the treatment of stable systems. Therefore, numerous modifications have been developed in the structure of the PS. Some modifications of the PS for unstable systems with delay are presented by [17; 18; 19; 20; 21; 22; 23]. Recently, in [24], an observer-predictor is presented for the case of one, two, and even three unstable poles with time delay. In particular, the stabilization for a system with one unstable pole, several stable poles, and time delay, the same kind of plant as that analyzed in the present work, is discussed. Stability conditions with respect to the time delay are given based on the use of an observer-predictor scheme and a PID-type controller. In this work, an observer-predictor control scheme for high-order delayed systems with an unstable pole and several stable poles is proposed. For the convergence of the observer-predictor, a special injection scheme is considered, which is discretized, and later the stability condition of the system in continuous time is obtained under the consideration of a sampling period \(T=\frac{\tau}{n}\), with \(T\to 0\) (or equivalently when \(n\rightarrow\infty\)). Special attention is given to the case of a delayed unstable second order (one stable and one unstable pole) because it is a system commonly treated in the literature [21; 25; 26; 27]. Later, the proposed control strategy is generalized to high-order unstable systems with delay.