Stabilisation of open-loop unstable plants under feedback control with distributed delays

A general class of linear time-invariant feedback control systems with a distributed delay in the feedback line is treated. Differently from present literature, both limits of the delay are taken as non-zero, which brings an interesting new perspective to the problem. The authors start with a theorem stating the equivalence of a general class of such distributed-delay systems to a system with multiple discrete and independent delays. Although this connection has been recognised earlier, what is interesting and novel in this study is the first deployment of a paradigm called the cluster treatment of characteristic roots (CTCR). CTCR declares the stability outlook of the system in the space of the delays non-conservatively and exhaustively. This capability forms a very important foundation over which the authors build further contributions. Firstly, the authors describe a systematic procedure to use distributed-delayed feedback logic in order to stabilise an unstable plant. Secondly, they present an intriguing but very simple control strategy which is called sign inverting control. This seemingly paradoxical proposition (i.e. the inversion of the control’s polarity) imparts considerably enhanced robustness of the control system against the variations in delay bounds. Example case studies are provided to validate these features. 1 Motivation and introduction Dynamic systems with time delays have been in the spotlight of research for over six decades with ever-increasing intensity. Along with the cases where some discrete delays appear in the dynamics [1–5], there has been a considerable effort invested into the general class of dynamics where ‘distributed delays’ exist [6–8]. Primary focus in these studies is the question of stability of the system [8]. From these prior investigations one may pose a simple benchmark control problem as follows: consider a linear time-invariant time-delayed system (LTI-TDS) with distributed delays as