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Stability Analysis and Control for Lurie Descriptor Systems

Author: ZhouZheng
Tutor: ZhangQingLing
School: Northeastern University
Course: Operational Research and Cybernetics
Keywords: Lurie descriptor systems strongly absolute stability input-to-statestability observer controller Lyapunov function linear matrix inequality (LMI)
CLC: N941.1
Type: Master's thesis
Year: 2011
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Descriptor systems are more general than normal systems and widely applied in circuits, economics and mechanical systems. Since1970s, the theory of descriptor systems has become an independent branch of modern control theory. The stability problem is one of the fundamental problems in the theory of descriptor systems and many stability results for normal systems have been generalized to descriptor systems. Recently, investigations on nonlinear descriptor systems have attracted much attention due to their wide applications.Lurie descriptor systems (LDS) whose feed forward path is a linear time-invariant descriptor system and feedback path contains the sector restricted nonlinearities are typical nonlinear descriptor systems. Therefore, investigations on Lurie descriptor systems can help us reveal the characteristics of general nonlinear descriptor systems. The existing results on Lurie descriptor systems are all based on the admissibility of the linear subsystems, which restricts the applications of these methods. This thesis studies the problems of input-to-state stability, strongly absolute stability and observer based control for Lurie descriptor systems whose linear subsystems are allowed to be not admissible. The main results are summarized as follows:(1) Input-to-state stability of Lurie descriptor systems with disturbances is studied. By using Lyapunov stability theory and linear matrix inequality (LMI), an LMI-based input-to-state stability criterion is derived. Compared with the existing methods, the newly developed one removes the restriction on the admissibility of the linear subsystems. Furthermore, the trajectory boundedness of descriptor systems is discussed by the newly developed result. Finally, an example is given to show the effectiveness of the proposed methods.(2) Strong absolute stability for Lurie descriptor systems whose linear subsystems have impulse models is considered. The system under consideration is transformed into an LDS with uncertainty in the derivative matrix. Then using the existing results on strong absolute stability for LDS and linear matrix inequality technology, a strong absolute stability criterion for LDS with impulse models in the linear subsystems is given. Finally, an example is given to show the effectiveness of the proposed method.(3) Observer based controller design for Lurie descriptor systems is investigated. By the existing results on strong absolute stability for LDS, the state-feedback controller and observer are designed independently. Then by establishing a separation principle, an observer based controller design method is proposed. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

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