Analysis And Synthesis Of Time-Delay Systems Based On Finsler’s Lemma

Time-delays are frequently encountered in various engineering systems. It has been shown that time-delay is often a source of the performance degradation and instability in control systems. Therefore, the problem of stability analysis and control synthesis of time-delay systems has attracted much attention during the past years, which is of both practical and theoretical importance. Since the conditions obtained by using the simple-LKF methods are rarely sufficient, reducing the conservatism and the computation burden are important for the practical implement of control algorithm.The problem of analysis and synthesis for the time-delay systems is studied based on simple-LKF method, and a unified framework and a series of improved results are proposed.The main contents of this thesis are as follows:(1) This paper investigates the problems of delay-dependent stability and stabilization of time-delay systems based on Finsler’s lemma. It is found that the existing four methods, including model transformation method, free weighting matrices method, integral inequality method and Jensen inequality method, are special cases under the proposed unified framework and free matrices are multiplier matrices essentially, the underlying effect of which is decoupling. From these, it is illustrated that the necessity of introducing multiplier matrices for stability analysis of time-varying delay system and the unnecessity for stability analysis of constant delay system. How the construction of the multiplier matrices affects the conservatism of the results is discussed. Some improved stability criteria are obtained by combining triple-integral method, singular model transformation method and time-varying time-delaydecomposition method under the proposed unified framework based on Finsler’s lemma.(2) For the NLMI problems in the controller designing for time-delay systems, based on the analysis of the existing methods, three improved controlled designing algorithms are proposed:stochastically indirect iterative algorithm, direct controller designing algorithm, controller designing algorithm based on BMI. The proposed algorithms are used to design controller for TCP/AQM systems and networked AGC systems, which illustrates its practical value and availability.(3) The delay-dependent exponential stability for neutral systems time-varying delay is investigated based on the exponential model transformation. The exponential parameter-dependent LMI is reduced to the multi-affine parameter dependent LMI by polytopic covering technique. The absolute stability of neutral systems with a time-varying delay and sector-bounded nonlinearity is investigated. By constructing a novel Lyapunov-Krasovskii functional based on the ideal of the delay decomposition, less conservative delay-dependent stability criteria are obtained and formulated in terms of linear matrix inequalities.(4) Under the unified framework for time-delay system based on Finsler’s lemma, by combining the constant time-delay decomposition and the time-varying time-delay decomposition, at the same time, constructing a novel Lyapunov-Krasovskii functional, the delay-dependent stochastic stability and BRL are studied for Markovian jump linear systems with interval time-varying delays. The problem of delay-dependent robust stability for a class of discrete-time uncertain singular Markovian jump systems with interval time-varying delay is studied. By using a new bounding inequality, a novel stability criterion is obtained, which involves only Lyapunov variables and has less computation burden. Under the unified study framework for time-delay system based on Finsler’s lemma, another novel stability criterion is proposed, which introduces some free matrix variables and is less conservative.(5) Delay-dependent robust H∞control for discrete-time Takagi-Sugeno (T-S) fuzzy systems with interval time-varying input delay is considered under the unified study framework based on Finsler’s lemma, by constructing a new Lyapunov-Krasovskii functional. An iterative algorithm based on the modified SLPM algorithm is proposed to solve the fuzzy H∞controller.(6) The problems of analysis and synthesis for the networked systems are studied based on time-delay system approach. For the data packet dropout and networked induced delay in closed-loop networked control systems, under the unified framework based on Finsler’s lemma, a novel delay-dependent stability criterion is proposed. By introducing some inequalities, the controller gain matrices are solved in terms of LMIs. Considering with delay, packet-dropout and quantization, by using the Lyapunov-Krasovskii functional method and the section bound method, the controller designing approach is proposed for the systems with two quantizers in both sides of sensors and controller. Finally, by using two characters of the networked control systems-the sawtooth time-varying delays and the impulsive control inputs, a less conservative stability criterion is obtained.