Linear Descriptor System Analysis Based On Distribution Theory

Descriptor system is a dynamic system which has more extensive forms than normal system. It exists in various fields of social production. Compared with the regular differential system, linear descriptor system has many unique properties, which also add a lot of characteristics on linear descriptor system. Particularly, linear descriptor system may contain the impulse items in its solution structure. Its controllability and observability problems have already attracted the attention of scientists for decades.Distribution is a part of the functional analysis knowledge. It provides simple and rigorous justifications for a number of formal manipulations often found in the technical literature. The knowledge of distribution includes the definition of test function and distribution, the differentiation, the convergence, the convolution, Fourier transform of tempered distribution and Laplace transform. This thesis briefly introduces the definition of test function and distribution, the differentiation, the convolution, and Laplace transform. In many areas of applied mathematics, especially ones involving differential equations, it supplies powerful tools that greatly simplify the analysis.Base on the theoretical knowledge, the main contribution of the thesis can be summarized as follows:first, we can get the distributional solution to the linear time invarying descriptor system in Jordan form, which simplifies the process of solution. Then, we can obtain the controllability and observability criterion of the linear time varying descriptor system in standard canonical form. Furthermore, based on the distributional solution to the linear time varying descriptor system in a new form, we can analyze the inconsistency between initial value and control input excites impulse clearly. In the end, impulse controllability and impulse observability criterion for linear time varying descriptor system are defined in this thesis.