A systematic review of discretisation methods for time-delay systems

Time delay is an inherent characteristic of real-world phenomena which may affect the system's characteristic. The systems including delay are known as time-delay systems, they are represented using delay differential equations. Modeling, discretisation, stability and control design for time-delay systems are still challenging in modern control theory. This paper systematically overviews available discretisation methods of linear and nonlinear time-delay systems. Emphasis is placed on illustrating fundamental results and recent progress on discretisation methods for delay systems. Numerous methods for the discretisation of linear and nonlinear systems considering input delays, state or output delays in the system's dynamics have been presented. A particular attention will be paid to illustrate effects of the discretisation process on the stability of discretised systems. Examples of mathematical descriptions, problems, and performance analysis for delay systems are presented. The presentation of discretisation methods is as easy as possible, focussing more on the main ideas and mathematical concepts by analogy. Finally, some possible future research directions to be tackled by researchers in this field are discussed.