In this paper a new control methodology is proposed for unstable linear time-delay systems with recycle. For this kind of systems time delays are present in the forward and backward paths increasing control difficulty. The strategy is based on the observation that if some internal system signals were known then it would be possible to remove the backward dynamics. In this way, a controller feedback could be designed by considering only the dynamics of the forward loop. To carry out this strategy an asymptotic observer is proposed to estimate the internal signals needed. Necessary and sufficient conditions to assure convergence of the proposed observerpredictor are given. An overall procedure for the proposed methodology is provided and numerical simulations to illustrate its performance are presented.

Unstable time delay system and recycling system pose a challenge problem in their own. When unstable time delay system have recycle the control problem becomes even more difficult. A control methodology for these systems is proposed in this paper. The strategy is based on the observation that if some internal system were available, then it would be possible to decouple the backward dynamics and then a feedback controller could be designed for the forward dynamics. The key for this strategy to be carried out is the asymptotic observerpredictor proposed to estimate the required internal signals. Necessary and sufficient conditions to assure convergence of this observer are given. Then, the strategy is complemented by using a PID controller with two degree of freedom in order to track step input references. Robustness to delay uncertainty is also discussed.

Abstract:In this work the problem of stabilization and control for recycling system is considered. Such class of systems is characterized by possessing two main paths named through this work as the direct (feedforward) and the recycling feedback) paths. This work considers recycling systems composed by a system of order n with one unstable pole at the direct path and a stable system of order m in the recycling path, both with different time delays. Two different dynamic delayed controllers are proposed in order to achieve a stable behavior of the closed-loop system. Stability conditions for the existence of these controllers are stated. The problems of step tracking and reject step disturbances are also addressed.

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