Delay-Dependent Stability Of Linear Delay Differential-Algebraic System And Its Numerical Solutions

Delay differential-algebraic systems,which are widely applied in engineering fields,are an important branch of modern control theory.Compared with classical delay system,delay differential-algebraic system endows many special features such as impulse terms and input derivatives in the state response,non-causality between input and state(or output),consistent initial conditions,etc.Hence,the study of delay differential-algebraic system and its numerical solutions more sophisticated than classical delay system,and this study has important theoretical value and practical significance.The main research object of this PhD thesis is linear delay differential-algebraic sys-tem.Based on the argument principle of complex variable function theory,delay-dependent stability of linear delay differential-algebraic system and its numerical solutions is studied in this thesis.And numerical examples are given to illustrate the effectiveness of the pro-posed results.This thesis is organized as follows:The first chapter mainly outlines the research background and significance of the sta-bility of delay differential-algebraic system and its numerical solutions,introduces the main research work and chapter arrangement.The second chapter introduces the basic knowledge of mathematics and some lemmas.The third chapter studies the delay-dependent stability of linear delay differential-algebraic system.First,using the matrix theory,the bounded semicircle region where all the unstable eigenvalues of the system are located is determined.Then,based on the ar-gument principle,the delay-dependent stability criteria of the system are established,and the number of unstable characteristic roots of the system is discussed.The new stability criteria need only to estimate the characteristic function and the argument on the boundary of the bounded semicircle region.Numerical examples are proposed to illustrate the less conservatism of the obtained results.The fourth chapter investigates the delay-dependent stability of Runge-Kutta methods for linear delay differential-algebraic system.Based on the argument principle,the delay-dependent stability criteria of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods for linear delay differential-algebraic system are established respectively.By the given algorithm,several numerical examples are shown to illustrate the effective-ness of the proposed results.Similar to the fourth chapter,the fifth chapter assesses the delay-dependent stability of linear multi-step methods for linear delay differential-algebraic system.Also based on the argument principle,the delay-dependent stability criteria of implicit linear multi-step methods for linear delay differential-algebraic system are obtained.By the proposed algo-rithm,several numerical examples are presented to illustrate the theoretical results,which indicates that our main results work well in the practical computations.In sixth chapter,the main work of this paper is summarized and further research topics are listed.