Regularity Of Non-Square Descriptor Systems

In this dissertation, two nonregular descriptor systems for practical circuits are first introduced. The physical background of two examples is discussed. As one type of nonregular descriptor systems, rectangular descriptor systems can be resulted from modeling with insufficient knowledge of a plant, or just ignoring details about part of the plant. Furthermore, the introduction of unknown circuits which cause nonsquare descriptor systems can not affect the impulse phenomena happening before they are introduced. Nonsquare descriptor systems still have impulse solutions as the regular case.Then, the regularity of nonsquare descriptor systems is studied. The notion of generalized regularity is proposed for rectangular descriptor systems. Generalized regularizability of a rectangular descriptor system via different feedback forms, such as state feedback and output feedback, is considered. Necessary and sufficient conditions for generalized regularizability are obtained, which are only dependent upon the open-loop coefficient matrices. It is also shown that under these necessary and sufficient conditions, all the generalized regularizing feedback controllers form a Zariski open set. A numerical example demonstrates the proposed results.Based on the notion of generalized regularity, the generalized inverse form of a rectangular descriptor system is proposed. The relationship of C-controllability and C-observability between the generalized inverse form and the original system is studied. Unlike the relationship between the inverse form and the original form of a regular descriptor system, controllability and observability of the generalized inverse form are not equivalent to C-controllability and C-observability of the original rectangular system. C-controllability of the original system is sufficient for the controllability of its generalized inverse form, while C-observability of the original system is only a necessary condition for the observability of its generalized inverse form. Furthermore, the relationship of R-controllability and R-observability between the generalized inverse form and the original system is also discussed. At last, the conditions of I-controllability and I-observability for rectangular descriptor systems are introduced.