Unstable time-delay systems and recycling systems are challenging problems for control analysis and design. When an unstable time-delay system has a recycle, its control problem becomes even more difficult. A control methodology for this class of systems is proposed in this paper. The considered strategy is based on the fact that if some internal system signals are available for measurement, then it will be possible to decouple the backward dynamics of the system and then a feedback controller could be designed for the forward dynamics. The key point for this strategy to be carried out is an asymptotic observer-predictor proposed to estimate these required internal signals. Necessary and sufficient conditions to assure convergence of this observer are given. After proving that the proposed control scheme tracks a step input signal and at the same time reject step disturbances, a procedure summarizing the methodology is provided. Robustness with respect to delay uncertainty and model parameters are also analyzed.

This work considers the stabilization problem of n-dimensional unstable linear systems with time delay, specifically with one unstable pole and n-1 stable poles, which could be complex conjugate. Necessary and sufficient conditions are stated to guarantee the stability of the system in closed loop with a P, PI, PD, or PID controller. The performance of the control strategies are evaluated by considering the linear approximation of an unstable continuously stirred tank reactor and a high-order academic system. Figure 5. PI controller, example 1: (a) Nyquist diagram and (b) output signal evolution. This figure is available in colour online at www.apjChemEng.com. Asia-Pacific Journal of Chemical Engineering PID CONTROL TO COMPLEX CONJUGATE POLES 693

This work considers the problem of stabilization of a class of unstable first order linear systems subject to a large input-output time delay. Necessary and sufficient conditions are stated to guarantee the stability of the closed loop delayed system by means of a compensation scheme based on two static gains and an induced delay term. The proof of the main result is derived by considering a discrete time approach that, under adequate assumptions, allows to conclude the stability condition for the continuous time case.

No abstract

Recycling systems are a challenging problem from the control viewpoint due to their detrimental effects. This paper proposes a simple way to stabilize and control a class of high-order recycling systems with time delays, internal instability, and possible complex conjugate poles. The strategy proposes P∕PI∕PD∕PID-like delayed controllers. Conditions to guarantee the stability of the controlled closed-loop system are stated. The performance of the proposed delayed controllers is illustrated via numerical simulations of a high-order unstable recycling system and an industrial chemical process.