Exponential Stability Analysis Of Linear Uncertain Time-delay Systems

In the study of control theory,the system models are often highly abstracted from engineering objects,which leads to the appearance of time-delay and uncertainty in system models.It is well known that the performance of a control system will be greatly reduced,even to a large extent,affecting the stability,if it has both time-delay and uncertainties,simultaneously.However,in the actual industrial production process,the time-delay and uncertainty of the system are often inevitable,so it has important academic significance and application value to study stability of the systems with time delay and uncertainty.In this thesis,the necessary conditions and the necessary and sufficient conditions for robust exponential stability of linear uncertain time-delay systems are studied.The main contents of the research are divided into two parts:Firstly,by defining the system parameter-dependent fundamental matrix and the complete parameter-dependent Lyapunov–Krasovskii functional,and calculating the derivative of the parameter-dependent Lyapunov–Krasovskii functional along the system,furthermore,the concept of parameter-dependent Lyapunov matrix is given.When the system is robustly exponentially stable,the basic properties of the quadratic lower bound of the complete parameter-dependent functional and the parameter-dependent Lyapunov matrix are given,and the relationship between the parameter-dependent Lyapunov matrix and the parameter-dependent fundamental matrix is found.In addition,in order to calculate the parameter-dependent Lyapunov matrix,the differential equation method and Lagrange interpolation method are introduced.Furthermore,the necessary conditions for the robustness of the system are obtained by defining a new parameter-dependent bilinear functional.Numerical examples show that the necessary conditions can be used to estimate the stability region of linear uncertain time-delay systems.Secondly,by introducing the concept of Lyapunov condition and the related theorem,proving the existence and uniqueness of the parameter-dependent Lyapunov matrix of the system,and using the binary Lagrange interpolation method to calculate the Lyapunov matrix with two parameters,furthermore,the necessary and sufficient conditions for the robust exponential stability of linear uncertain time-delay systems are given.Finally,numerical examples verify the validity of the theoretical results.