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A generalization of the graph Laplacian with application to a distributed consensus algorithm

100%
Zhai G.
International Journal of Applied Mathematics and Computer Science
|
2015
|
tom 25
|
nr 2
353-360
EN
In order to describe the interconnection among agents with multi-dimensional states, we generalize the notion of a graph Laplacian by extending the adjacency weights (or weighted interconnection coefficients) from scalars to matrices. More precisely, we use positive definite matrices to denote full multi-dimensional interconnections, while using nonnegative definite matrices to denote partial multi-dimensional interconnections. We prove that the generalized graph Laplacian inherits the spectral properties of the graph Laplacian. As an application, we use the generalized graph Laplacian to establish a distributed consensus algorithm for agents described by multi-dimensional integrators.
2
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A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching

64%
Zhai G., Xu X.
International Journal of Applied Mathematics and Computer Science
|
2010
|
tom 20
|
nr 2
249-259
EN
We establish a unified approach to stability analysis for switched linear descriptor systems under arbitrary switching in both continuous-time and discrete-time domains. The approach is based on common quadratic Lyapunov functions incorporated with linear matrix inequalities (LMIs). We show that if there is a common quadratic Lyapunov function for the stability of all subsystems, then the switched system is stable under arbitrary switching. The analysis results are natural extensions of the existing results for switched linear state space systems.
3
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Generalized practical stability analysis of discontinuous dynamical systems

64%
Zhai G., Michel A. N.
International Journal of Applied Mathematics and Computer Science
|
2004
|
tom 14
|
nr 1
5-12
EN
In practice, one is not only interested in the qualitative characterizations provided by the Lyapunov stability, but also in quantitative information concerning the system behavior, including estimates of trajectory bounds, possibly over finite time intervals. This type of information has been ascertained in the past in a systematic manner using the concept of practical stability. In the present paper, we give a new definition of generalized practical stability (abbreviated as GP-stability) and establish some sufficient conditions concerning GP-stability for a wide class of discontinuous dynamical systems. As in the classical Lyapunov theory, our results constitute a Direct Method, making use of auxiliary scalar-valued Lyapunov-like functions. These functions, however, have properties that differ significantly from the usual Lyapunov functions. We demonstrate the applicability of our results by means of several specific examples.
4
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Extended lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems

51%
Zhai G., Xu X., Lin H., Liu D.
International Journal of Applied Mathematics and Computer Science
|
2007
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tom 17
|
nr 4
447-454
EN
We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.
5
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Decentralized design of interconnected $H_∞$ feedback control systems with quantized signals

51%
Zhai G., Chen N., Gui W.
International Journal of Applied Mathematics and Computer Science
|
2013
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tom 23
|
nr 2
317-325
EN
In this paper, we consider the design of interconnected $H_∞$ feedback control systems with quantized signals. We assume that a decentralized dynamic output feedback has been designed for an interconnected continuous-time LTI system so that the closed-loop system is stable and a desired $H_∞$ disturbance attenuation level is achieved, and that the subsystem measurement outputs are quantized before they are passed to the local controllers. We propose a local-output-dependent strategy for updating the parameters of the quantizers, so that the overall closed-loop system is asymptotically stable and achieves the same $H_∞$ disturbance attenuation level. Both the pre-designed controllers and the parameters of the quantizers are constructed in a decentralized manner, depending on local measurement outputs.
6
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Hybrid stabilization of discrete-time LTI systems with two quantized signals

45%
Zhai G., Matsumoto Y., Chen X., Imae J., Kobayashi T.
International Journal of Applied Mathematics and Computer Science
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2005
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tom 15
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nr 4
509-516
EN
We consider stabilizing a discrete-time LTI (linear time-invariant) system via state feedback where both the quantized state and control input signals are involved. The system under consideration is stabilizable and stabilizing state feedback has been designed without considering quantization, but the system's stability is not guaranteed due to the quantization effect. For this reason, we propose a hybrid quantized state feedback strategy asymptotically stabilizing the system, where the values of the quantizer parameters are updated at discrete time instants. We also extend the result to the case of static output feedback.
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