Jianming She

Date of Graduation


Document Type



From engineering point of view, a complete accomplishment should be supported by rigorous mathematical analyses, feasible computational algorithms, and concrete practical applications. The research work of this dissertation is done with an attempt of trying to approach this principle and to understand some basic facts of singular systems in these three aspects. In Part I, a new complementary solution formulation is derived from both the time domain and the frequency domain for singular systems. The solution expressions of regular systems and singular systems are unified. Further, the dynamic characteristics of singular systems are studied. We discussed in detail on the memory and memoryless properties of singular systems and we gave a new concept defined as infinitesimal memory which is more proper to describe the pure singular system dynamic behaviors. We also gave a necessary and sufficient condition under which the resultant system of two connectable regular systems by feedback connection is again a regular system. As a bridge from the mathematical theory to practical applications, we have developed some feasible computational algorithm for singular system in Part II. The first goal attacked in Part II is the algorithm for Jordan canonical form which is mainly motivated by the computation of Kronecker decomposition of matrix pencil of singular systems. We developed an algorithm for identifying multiple eigenvalues with nonlinear elementary factors. On the basis of the Jordan canonical computation algorithm, we gave a practical algorithm for computing Kronecker form of matrix pencil which is the key point of many singular system theories and applications. We also clarified that the popular difference algorithm for singular systems is not distributional convergent. In Part III, we investigate in detail for switched networks as the concrete application of singular systems. We expand the setting of singular systems to t {dollar}<{dollar} 0 and analyzed the role of pre-system and initial conditions to the system transit behaviors. Transmission line fault studies and power system voltage instability investigations are conducted based on the theories and algorithms developed in Part I and Part II. From these applications, the advantages of the singular system models are exhibited.