Stability And Feedback Stabilization Of Neutral Delay Systems

Time-delays are frequently encountered in the behavior of many physical processes and very often are the main cause for poor performance and instability of control systems. In view of this, the control issue of time-delay systems is a topic of great practical importance which has attracted a great deal of interest for recent years, but the development of neutral systems is relatively slow and the references are fewer. The main reason is that difference operator D of neutral systems is so difficult to deal with that the properties of the solutions to neutral systems are more complicated than those normal delay systems. In this paper, we consider the feedback control problem of neutral systems.In chapter 1, some examples show applied background, theoretical and applied value of neutral systems. Reviewing development history and study actuality of delay systems, we point out that the work of neutral systems is weak in the control theory.In chapter2, we describe briefly the basic problem of neutral systems: initial value problem, local existence and uniqueness of solutions, continuous dependence of solutions on initial values, and some properties of the eigenvalues equation of linear neutral systems, Razumikhi-type theorems and neutral model transformations.In chapter 3, the stabilization problem of linear neutral system is discussed. Firstly, we reduce the neutral system to a delay-free system using a linear transformation. Under the assumption of spectral stabilizability, based on the solution vectors of the linear algebraic equations corresponding to a given eigenvalues, it is shown how to obtain a stabilizing feedback control law on the delay-free system. Secondly, under the condition of state determination, a design method of state feedback control is given by the solution of a linear matrix inequality (LMI). Finally, we focus on the design of dynamic output feedback controller for neutral linear systems that its state cannot be determined, a design program that the design of observer and controller is separated, is proposed. The delay-independent feedback controller, which satisfied the design demands, is obtained by the solutions of 3-LMIs, and the design method is easily realized.In chapter 4, the robustness of a class of neutral systems with nonlinear perturbations is considered. By Lyapunov-Krasovskii functional and Lyapunov stability theory, the delay-independent stability conditions are derived, which are based on the asymptotically stability of the normal neutral systems. Some analytical methods are employed to investigate the bound on the perturbations so that the systems remain stable, and its computation is transformed to solve a linear matrix inequality leading to the free parameters in the coupling Riccati equations do not need to be readjusted. Furthermore, the stabilization of the neutral systems with multiple time-delays is discussed. A state feedback control law via compound memory andmemoryless feedback is derived, which stabilized the described systems, and by construct Lyapunov functional, delay-independent stability criteria are proposed that are sufficient to ensure a uniform asymptotic stability property. Finally, delay-dependent guaranteed cost control problem is concerned for the neutral-type system with norm bounded time-varying nonlinear uncertainty and a given quadratic cost function. New delay-dependent conditions for the existence of the guaranteed cost controller are presented in term of LMIs. An algorithm involving optimization is proposed to design a controller achieving a suboptimal guaranteed cost such that the system can be stabilized for all admissible uncertainties.In chapter 5, the Hm stabilization problem of the neutral systems with delay-varying is firstly investigated. A memory feedback //? controller is derived based on the solutions of a linear matrix inequality, which is the delayed-dependent condition to stabilize the neutral systems. Secondly, the robust Hx performance of neutral delay systems involving an unknown time-delay and time-varying parameter uncertainties is discussed. Based on LMI, the sufficient criteria which assure the quadratic stability as well as an //ao-norm bound are proposed. Meanwhile, the results show that the condition is necessary to stability of the neutral systems, which assures the delay systems without delayed derivative term is Lyapunov stable or pseudo-quadratically stable with //oo-norm bound. Furthermore, an equivalent system with //^norm bound is obtained in the sense of Lyapunov stability or pseudo-quadratically stability. Finally, the robust H^ control problem of uncertain neutral systems is considered, and the design robust H^ controller can be achieved by the solutions of a LMI.In chapter 6, the robust absolute stability of neutral Lurie control systems with multiple delays is discussed. The sufficient conditions of delay-independent for uncertain neutral systems are obtained via the method of Lyapunov functional, which are equivalent to the solvability of a LMI on the free parameters. The Lyapunov functional constructed by the solution of the LMI is adopted to guarantee the robust stability of systems, and the parameters do not need to be readjusted. Meanwhile, some concise examples are provided to illustrate the effectiveness of the presented conclusions, and the relation between the size of the sector and the robustness is analyzed.