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This article addresses the boundary control problem of the transport equation. Namely, we propose a control method, which is merely a delayed output feedback relying on a partial pole placement idea, that consists in assigning an appropriate exponential decay rate to the closed‐loop system's solution. The proposed control structure appearing in the transport boundary, which has proven its effectiveness in controlling finite dimensional systems, consists of an autoregressive relation linking the transport equation's input and output. The obtained result provides an analytical lower bound for the solution's exponential decay. The contribution is concluded with a numeric discussion on the concrete implementation of the method with a special emphasis on the delay seen as a control parameter. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we study the behavior of characteristic roots of a class of strictly proper LTI SISO systems that are subject to a PD controller for which the derivative action has been implemented by means of a delay‐difference approximation scheme that uses multiple delays. To such an end, the improperly‐posed case is discussed in depth, that is, the situations where "small delays" induce instability in the closed‐loop system. The effect of adding more delays on the improperly‐posed case is analyzed and a methodology for determining the set of parameters that produces such a situation for a given number of delays is derived. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

This article focuses on the root behavior of the characteristic function of some LTI SISO systems controlled by a PD‐control when a delay‐difference operator is used to approximate the derivative action. We will focus on the cases when the corresponding stability problem is improperly posed for "small" delay values. In this context, we will express the solutions of the corresponding characteristic function as power series and exploit their structure in deriving the asymptotic behavior of the characteristic roots. This analysis will allow detecting and explicitly characterizing the cases when delay‐difference approximations lead to improperly posed stability problems. Furthermore, in the case when the derivative approximation guarantees the properly posedness of the closed‐loop system, the robustness of the scheme with respect to the delay margin of the delay‐difference approximation and control gains parameters are explicitly addressed. More precisely, upper bounds on the delays as well as the robustness of the corresponding controller are computed. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]