Delay-dependent Robust Stability And Stabilization Based On Free-weighting Matrices

Systems with time delays are frequently encountered in areas as diverse as engineering, biology, and economics, and have been attracting a great deal of attention over the past few decades because delays are often a source of instability. There are two types of stability criteria for such systems: delay-dependent and delay-independent. Since delay-dependent criteria make use of information on the length of delays, they are less conservative than delay-independent ones. The main method employed to derive delay-dependent criteria involves a fixed model transformation, which expresses the delay term in terms of an integral. And there are four basic model transformations that are commonly used. Among them, the descriptor model transformation combined with Park's or Moon et al.'s inequalities is the most efficient and least conservative. However, it uses the Leibniz-Newton formula in the derivative of the Lyapunov functional, and the delay term x(t — h) is replaced by x(t) — integral from n=(t-h) to t(x|.(s)ds) in some places but retained in others in order to make the Lyapunov functional easier to handle. While this method is equivalent to using preselected fixed weighting matrices to express the relationships between the terms in the Leibniz-Newton formula, there must exist optimal weighting matrices for those terms. So, there is room to reduce the conservatism of the method.In this dissertation, the limitations of the main methods used to handle delay-dependent problems are first examined, especially those involving a fixed model transformation. A free-weighting-matrix method is proposed to relax the limitations. In this method, free weighting matrices express the relationships between the terms in the Leibniz-Newton formula so that a solution can be obtained by solving linear matrix inequalities (LMIs), which makes the method less conservative than those employing fixed weighting matrices.For systems with time-varying delays, the new method employs free weighting matrices to express the relationships between the terms in the Leibniz-Newton formula, and to derive delay-dependent stability conditions. Moreover, the criteria are easy to extend to the delay-dependent/rate-independent stability conditions. Two criteria are obtained, depending on how the term x{t) in the derivative of the Lyapunov functional is handled; and their equivalence is proved. On this basis, the criteria are extended to systems with time-varying structured uncertainties. Furthermore, since the new way of handling the term x{t) separates the Lyapunov matrices and the system matrices in an easy way, this treatment is easily extended to deal with the delay-dependent stability of systems with polytopic uncertainties, based on a parameter-dependent Lyapunov functional.Based on the delay-dependent stability criteria thus derived, an iterative method of solving a nonlinear minimization problem subject to LMI conditions and a method of adjusting the parameters are employed to design a static state feedback controller to stabilize the system. In addition, a delay-dependent/rate-independent stabilization condition based on LMI conditions is obtained.For a system with two time delays, delay-dependent criteria are derived by taking the relationship between the delays into account in the free-weighting-matrix method. When the delays are equal, the criteria are equivalent to those for a system with a single time delay-a result that has been obtained for the first time. In addition, this idea is extended to the derivation of delay-dependent stability criteria for a system with multiple time delays.The free-weighting-matrix method is also employed to analyze the discrete-delay-dependent and neutral-delay-independent stability of a neutral system with a time-varying discrete-delay. This method, and also this method in combination with a parameterized model transformation, are used to derive delay-dependent stability criteria for neutral systems with identical discrete- and neutral-delays. Based on this method, discrete- and neutral-delay-dependent stability criteria are obtained for a neutral system with different discrete- and neutral- delays. It is shown that these criteria include those for identical discrete- and neutral- delays as a special case.Necessary and sufficient conditions are devised for the existence of an extended Lyapunov functional in the Lurie form that guarantees the absolute stability of a Lurie control system with a delay and multiple nonlinearities. It simplifies the existence problem to one of solving a set of LMIs. These conditions are then extended to a Lurie control system with a delay, multiple nonlinearities, and time-varying structured uncertainties. Moreover, the free-weighting-matrix method is employed to obtain delay-dependent absolute stability criteria for Lurie control systems with a time-varying delay as well.For discrete-time systems with an interval-like time-varying delay, the free-weighting- matrix method is used to study the delay-dependent/range-dependent stability and stabilization problem, and the results are extended to a system with time-varying structured uncertainties. Then, based on the stability analysis, a method of designing a delay-dependent memoryless state feedback controller is derived. Numerical examples show that both the upper bound and the range of the time-varying delay affect the results.Finally, how the free-weighting-matrix method can be used tosolve the delay-dependent robust stability and stabilization problem for linear systems, neutral systems, Lurie control systems, and discrete-time systems with delays is outlined. Possible applications to other areas are also suggested. In addition, problems with delay-dependent conditions and the direction of future studies are mentioned.