Delay-dependent Admissibility And Robust Control For Singular Systems

Singular systems are extended from regular systems. singular system arises naturally as models for many dynamical processes, such as chemical processes, electric power systems, nuclear reactor, economic systems, etc. Moreover, due to information transmission, the natural properties of system elements, the computation of variables, and so forth, time delays usually exist in real-life systems. It is well known that time delay is one of the sources of instability and deterioration of system performances. The asymptotic stability and stabilization problem is one of the fundamental problems in the theory of singular systems with time-delays. The study of them is much more complicated than that of regular systems because the singular time-delay systems requires considering not only stability, but also regularity and impulse free. Therefore, the study on delay-dependent admissibility (regular, impulse-free and delay-dependent stability) and robust control for singular systems with time delays is important in theory and in practice.The delay-dependent admissibility and robust control problem for singular systems with time delays are studied in this dissertation. Based on integral equality, descriptor integral inequality and linear matrix inequality(LMI), this dissertation tackles the delay-dependent admissibility problem, delay-dependent and delay-rate-dependent admissibility problem, robust stabilization problem and robust H?control problem for linear singular systems with time delays via introducing some new Lyapunov-Krasovskii functional. The delay-dependent decentralized robust admissibility and stabilization problem for interconnected singular large-scale system with time delays are also studied. The major contributions of this dissertation are as follows:(1) Based on integral equality, the problem of delay-dependent admissibility, robust delay-dependent admissibility and robust delay-dependent H?control are discussed by LMI approach. Some delay-dependent admissibility conditions are obtained and the explicit expressions of the desired state feedback control laws are also given. We deal with the cross-term of the derivative of Lypunov-Krasovskii functional by using integral equality, which is different to the integral inequality method and preferably obtain the delay-dependent admissibility criteria for singular time delay systems.(2) Based on integral equality and double integral equality, the criteria of delay-dependent and delay-rate-dependent admissibility for linear singular systems with time-varying delay are obtained by constructing a new type of Lypunov-Krasovskii functional which contains some triple-integral terms. On this basis, the criteria are extended to singular systems with time-varying structured uncertainties. Furthermore, a new bounded real lemma with delay-dependent and delay-rate-dependent is also obtained by using the delay-dependent and delay-rate-dependent criteria. Based on this, the sufficient condition for the existence robust delay-dependent and delay-rate-dependent H?state feedback controller is obtained. The design methods of delay-dependent and delay-rate-dependent H?state feedback controller are also given by linear matrix inequality.(3) The condition when a singular system subject to multiple point delays is regular independent of time delays is given and it can be easily test with numerical or algebraic methods. For linear singular systems with both input and state multiple point delays, based on the Lyapunov-Krasovskii functional approach and the descriptor integral-inequality method, a sufficient condition for delay-dependent admissibility is obtained. The main idea is to design multiple memoryless state feedback control laws such that the resulting closed-loop system is regular independent of time delays, impulse free, and stable via solving matrix inequality problem. An explicit expression for the desired memoryless state feedback control law is also given. In terms of matrix inequalities, the sufficient condition for delay-dependent admissibility and the method to design multiple memoryless state feedback controller law are also given for linear singular systems only with input multiple point delays. Meanwhile, the memory state feedback controller is also designed for linear singular systems only with state multiple point delays. (4) Based on the Lyapunov stability theory and delay integral matrix inequality method The delay-dependent decentralized robust admissibility problem for interconnected singular large-scale system with uncertainties is investigated by constructing a special type of Lypunov-Krasovskii functional. The aim is to design a memoryless state feedback decentralized controllers such that the whole closed-loop system is regular, impulse-free and robust asymptotically stable. Sufficient conditions for the delay-dependent decentralized robust admissibility are obtained in terms of a set of matrix inequalities. The results depend on the size of the delays and are given in terms of matrix inequalities.