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-observer-based 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 prescribed-time 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.