Tutor: DiDing

School: Northeastern University

Course: Operational Research and Cybernetics

Keywords: Lur’e-type discrete-time descriptor systems strongly absolute stability circlecriterion Popov criterion generalized Lyapunov function linear matrix inequality (LMI) integral mean value theorem

CLC: N941.1

Type: Master's thesis

Year: 2011

Downloads: 2

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Descriptor systems are described by a set of differential equation and algebraic equations, which are more general and can be applied more widely than normal systems. During the past forty years, descriptor systems have been investigated by many scholars and many results on controllability and observability, stability, controller and observer design, H2and H∞control, robust control and optimal control have been reported.Stability is a fundamental requirement for a control system. Thus it’s important to study the stability of control systems. Lur’e-type systems are nonlinear systems with typical structure, whose feed-forward path is a linear time-invariant system and feedback path contains sector restricted nonlinearities. Absolute stability, which is focused on Lur’e-type systems, is an important branch of control system stability. In recent years, research on absolute stability has attracted much attention and many valuable absolute stability criteria have been proposed.Lur’e-type descriptor systems whose feed-forward path is a linear time-invariant descriptor system and feedback path contains the sector restricted nonlinearities are typical nonlinear descriptor systems. There have been many results on Lur’e-type continuous-time descriptor systems. However, the research on Lur’e-type discrete-time descriptor systems is still premature. Therefore, it’s necessary to investigate Lur’e-type discrete-time descriptor systems. This thesis mainly studies the strongly absolute stability problem of Lur’e-type discrete-time descriptor systems and the main results are summarized as follows:(1) The notion of strongly absolute stability of Lur’e-type discrete-time descriptor systems are defined by combining the concept of absolute stability and the property of index one together. This concept is a generalization of both the absolute stability for normal systems and the admissibility for linear time-invariant descriptor systems. According this definition, we have that the admissibility of (E, A) is a necessary condition for Lur’e-type discrete-time descriptor systems to be strongly absolutely stable. This is a solid theoretical foundation for driving the main results. (2) The circle criterion for strongly absolute stability of Lur’e-type discrete-time descriptor systems is proposed. By using generalized Lyapunov function, S-procedure and linear matrix inequality (LMI), the circle criterion basing on strict LMI is established. A numerical example is given to illustrate the effectiveness and feasibility of the proposed results.(3) The Popov criterion for strongly absolute stability of Lur’e-type discrete-time descriptor systems is proposed. By using generalized Lyapunov function, S-procedure and LMI, Popov criterion expressed by non-strict LMIs is obtained. Then, a strict LMI based algorithm is proposed. Finally, a numerical example is given to illustrate the effectiveness and feasibility of the result.(4) A criterion is proposed for strongly absolute stability of Lur’e-type discrete-time descriptor systems, whose nonlinearities are assumed to be sector and slope-restricted. By using the generalized Lyapunov function, S-procedure, LMI and integral mean value theorem, two stability criteria are given. Furthermore, we reduce the non-strict conditions to strict LMI based algorithms without any conservatism. Finally, a numerical example is given to illustrate the effectiveness of the proposed methods. |

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