Prescribed-time stabilisation of linear systems with matched unknown external disturbance by smooth periodic delayed feedback

This paper considers the prescribed-time stabilisation of the linear system with matched unknown external disturbance. The disturbance is supposed to be generated by an exogenous system. In the case only the output signal of the original system is available for feedback, a prescribed-time linear observer for the augmented system consisting of the original system and the exogenous system is designed. Then the prescribed-time stabilisation problem can be achieved by using the state-and-disturbance-observerbased periodic delay feedback. Furthermore, in the case the state of the original system is measurable, three classes of prescribed-time disturbance observers for the exogenous system are designed. The prescribedtime stabilisation problem can also be solved by applying the disturbance-observer-based periodic delay feedback. A numerical example is posted to demonstrate the effectiveness of the proposed approaches.