奇异系统又称广义系统、广义状态空间系统、隐式系统、微分-代数系统。它是一种由微分方程和代数方程共同构成的大规模复杂系统，广泛应用于机械系统、电路系统、电力系统、航空航天系统、生物工程等实际系统。由于其不满秩的半状态动态特性，使得奇异系统的稳定性和性能分析问题变得非常困难，因而奇异系统得到了许多学者的广泛关注。另一方面，时滞现象的普遍存在可能导致系统响应发生改变，从而使系统性能下降，甚至使系统的稳定性遭到破坏。因此，研究奇异时滞系统的性能分析以及控制器设计问题具有重要意义。 近二十多年来，基于Lyapunov-Krasovskii泛函(LK 泛函)方法，奇异时滞系统的研究已经取得了许多有价值的成果。然而，受到奇异矩阵的影响，利用LK泛函研究奇异时滞系统相比一般时滞系统要困难得多，所获得的结果无论是在保守性方面还是计算量方面都存在较大的改进空间。围绕这两方面的问题，本文针对奇异时滞系统的稳定性、耗散性以及H〓控制开展研究。主要工作和研究成果如下: (1)基于中立型模型变换的奇异时滞系统稳定性分析 针对具有时变时滞的奇异系统稳定性问题，利用中立型模型变换将奇异时滞系统转化为中立型系统，再针对变换后的中立型系统，通过增加积分项构造含有更多交叉关系的增广LK泛函，并利用自由矩阵积分不等式和松弛积分不等式对LK泛函导数进行估计，建立中立型系统的时滞相关稳定性判定准则，从而提出一种基于中立型模型变换的奇异时滞系统稳定性分析方法，相比于现有方法具有更低的保守性。 (2)奇异时滞系统稳定性分析的状态分解方法 针对奇异时滞系统稳定性问题，利用受限等价变换将奇异时滞系统分解为微分方程和代数方程的形式，将系统状态分解为两个部分，利用分解后的系统状态构造与奇异矩阵不相关的LK泛函，提出一种奇异时滞系统稳定性分析的状态分解方法，为奇异时滞系统的研究提供一种新的思路，在此基础上，建立保守性低且计算量小的稳定性判定准则。 (3)基于状态分解增广LK泛函的奇异时滞系统耗散性分析 针对奇异时滞系统耗散性问题，首先，构造与奇异矩阵相关的增广LK泛函，建立基于增广LK泛函的耗散性判定准则，降低保守性。其次，在系统状态分解的基础上，构造状态分解增广LK泛函，建立基于状态分解增广LK泛函的时滞相关耗散性判定准则，在降低保守性的前提下，减少计算量。 针对具有时变时滞的奇异系统耗散性问题，利用状态分解方法，构造时变时滞的状态分解增广LK泛函，提出一种基于状态分解增广LK泛函的具有时变时滞奇异系统耗散性分析方法，降低保守性，减少计算量。 (4)基于状态分解增广LK泛函的奇异时滞系统H〓控制器设计 针对奇异时滞系统H〓控制问题，首先,构造和奇异矩阵相关的增广LK泛函，建立基于增广LK泛函的奇异时滞系统H〓性能分析条件，提出一种基于状态反馈的奇异时滞系统H〓控制器设计方法。其次，利用状态分解方法，构造状态分解增广LK泛函，建立基于状态分解增广LK泛函的H〓性能分析条件，提出一种基于记忆状态反馈的奇异时滞系统H〓控制器设计方法，相比于传统的奇异时滞系统H〓控制器设计方法，能够实现更好的控制效果。 关键词：奇异时滞系统；Lyapunov-Krasovskii泛函；稳定性；耗散性；H〓控制；状态分解方法

 Singular systems are called generalized systems, generalized state space systems, descriptor systems, implicit systems and differential-algebraic systems. They are large-scale complex systems composed of differential equations and algebraic equations, and widely used in mechanical systems, electrical systems, power systems, aerospace systems, biological engineering and other practical applications. The stability and performance analysis of singular systems are very difficult because of the semi-state dynamic feature. Thus, the research of singular systems has attracted much attention. On the other hand, the time delays are common phenomenon, which may degrade the performance of the systems and even destroy the stability of singular systems. Thus, the research on the performance analysis and controller design of singular time-delay systems has important significance. Based on the Lyapunov-Krasovskii functional (LKF) method, most results for singular systems with time delays have been reported in the past two decades. However, due to the influence of singular matrix, it is more difficult to study singular systems with time delays than regular systems with time delays, and the obtained results have much room for improvement in both conservatism and computational complexity. In view of above problems, this dissertation studies the stability, dissipativity and H〓 control of singular systems with time delays. The main results and contributions are presented as follows. (1)Stability analysis of singular systems with time delays based on the neutral model transformation For the stability analysis of singular systems with time-varying delays, singular systems with time delays are transformed to neutral systems by the neutral model transformation. For the neutral systems, the integral terms are added to construct an augmented LKF to take more cross relationships into consideration. The derivative of the LKF is estimated by the free-matrix-based integral inequality and the relaxed integral inequality respectively, and thus two stability criteria of the neutral systems are derived, which shows a stability analysis method based on the neutral model transformation. Compared with the existing methods, this method is less conservative. (2)State-decomposition-based method for stability analysis of singular systems with time delays For the stability of singular systems with time delays, singular systems with time delays are transformed into differential equations and algebraic equations by using the constrained equivalent transformation, and the system state is decomposed into two parts. These parts are used to construct LKFs which do not involve with the singular matrix. Then, a state-decomposition-based method for the stability is proposed, which can provide a new idea for investigating the singular systems. On the basis of this, a stability criterion, which has less conservatism and less computational complexity, is derived. (3)The dissipativity analysis of singular systems with time delays based on state-decomposition-based augmented LKFs For the dissipativity analysis of singular systems with time delays, an augmented LKF involved with the singular matrix is firstly constructed to derive a less conservative disspativity criterion. Then, the other disspativity criterion is presented by constructing a state-decomposition-based augmented LKF, which has less conservatism and computational complexity. For the dissipativity analysis of singular systems with time-varying delays, by constructing a state-decomposition-based augmented LKF, a disspativity analysis method is presented, which is dependent upon the criterion with less conservatism and computational complexity. (4) H〓 controller design of singular systems with time delays based on state-decomposition-based augmented LKFs For H〓 control design of singular systems with time delays, an H〓 performance criterion is firstly derived by constructing an augmented LKF involved with the singular matrix. On this basis, an H〓 state feedback controller design method is proposed. Then, a state-decomposition-based augmented LKF is constructed to present the other H〓 performance criterion, and then an H〓 memory state feedback controller design method is proposed, which has better control performance compared with other traditional control methods. Keywords: Singular systems with time delays; Lyapunov-Krasovskii functional; Stability; Dissipativity; H〓 control; State-decomposition-based method