Recent Results on Analysis of Systems with Time-Delay

Abstract

It is well known that time delays as a source of the generation of oscillation and a source of instability are frequently encountered in various engineering systems such as long transmission lines in pneumatic systems, nuclear reactors, rolling mills, hydraulic systems and manufacturing processes. Therefore, the problem of stability analysis and control of time-delay systems has attracted much attention, and considerable effort has been applied to different aspects of linear time-delay systems during recent years [153], [128], [141], [64], [112], [140], [206], [66], [65], [204], [87], [39], [214], [199], [125], [157], [155], [156], [158], [159], [193], [192]. It has been proven that the linear time delay system \(\dot x(t)=-bx(t-\tau), b>0, \tau>0\) is stable if \(\tau<\frac{\pi}{2b}\), and the system is unstable if τ is too large. Existing criteria for asymptotic stability of time-delay systems can be classified into two types, that is, delay-independent stability and delay-dependent stability; the former does not include any information on the size of delay while the latter employs such information. It is known that delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the size of the delay is small.