Delayed feedback of sampled higher derivatives

Systems where the present rate of change of the state depends on the past values of the higher rates of change of the state are described by so-called advanced functional differential equations (AFDEs). In an AFDE, the highest derivative of the state-space coordinate appears with delayed argument only. The corresponding linearized equations are always unstable with infinitely many unstable poles, and are rarely related to practical applications due to their inherently implicit nature. In this paper, one of the simplest AFDEs, a linear scalar first-order system, is considered with the delayed feedback of the second derivative of the state in the presence of sampling in the feedback loop (i.e. in the case of digital control). It is shown that sampling of the feedback may stabilize the originally infinitely unstable system for certain parameter combinations. The result explains the stable behaviour of certain dynamical systems with feedback delay in the highest derivative.

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