On The Feedback Stabilizable Delay Margin For Linear Delay Systems

Over the past decades,time-delay systems have been extensively studied in the research community,due in part to their ubiquity in engineering and in part to their bad effect on the performance of control systems.Enormous amount of research has been devoted to the stability analysis and control design of the system.However,little attention is paid on the feedback stabilizable delay margin,nor relevant quan-titative analysis on system performance,essential properties of system,and input delay.The feedback stabilizability,especially the study of time delay limitation,is particularly important.The dissertation investigate linear systems with input delay,for the feedback stabilizability for linear systems subject to the time delay constraints.We focus on the calculation of upper bounds of the delay margin,aiming to reveal the intrinsically relationship between the feedback stabilizability,the internal structure of the system,the input delay and the control design.The delay margin problem is one of the fundamental limitations of time-delay systems,and is one of the open problems in mathematical systems and control theory.The main contribution of this dissertation is stated as follows.1.For linear systems with multiple unstable poles and constant input delay,based on a novel bilinear transform technique,the original problem is translated into a finite dimensional optimization problem.Some upper bounds of the delay margin to maintain stability of the closed system,by using a linear time invariant controller,are obtained by exploring the frequency domain method and optimization theory.The results presented herein shown that the feedback stabilizability is affected by the unstable poles of a given plant and the input delay,and the larger is the poles,the lower the delay margin.The qualitative expression of their relationship provide theoretical guidance for control system design.In particular,the results both unify and generalize some existing results.2.We extended the delay margin problem from constant delay to time-varying delay.For linear systems with time-varying input delay,we first draw upon alge-braic Riccati equation and Halanay-type inequality to study the condition on the time-varying delay for the system to be stabilized by a static feedback control.The condition is related to an integral of the time delay squared,allowing the delay to be discontinuous and without limit on the upper bound of the time delay,so it can be arbitrarily large.In addition,we obtain another delay margin by exploring model transformation technique and small gain theorem.It is shown that the feedback sta-bilizability of the system is determined by the transfer function of the closed system and the input delay.Several numerical examples demonstrate the effectiveness of our results.3.For multiple-input multiple-output linear time-delay systems,by using simi-lar method as above,we obtain some feedback stabilzable delay margin.It is shown that the feedback stabilizability is affected by the unstable poles,pole direction of the given plant,and the input delay.The results unify many existing results,and some of them are special cases of ours.4.We extended the consensusable delay margin problem from one input delay to two input delay.For linear multi-agent systems with two input delays,the bounds of time delay margin to obtain consensus is addressed via the frequency domain method and algebraic graph theory.The results show that the consensusability of the multi-agent depends not only on the inherent characteristics of the controlled plant,such as unstable poles,but also the network topology and the input delay.The results quantitatively reveal how the input delay impact on the consensusability.