This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via statedependent switching rules.Several attempts have been made to solve this problem via different types ...This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via statedependent switching rules.Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid.These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type.In some cases,the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function.In this paper,we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently.The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size.Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size,i.e.,a purely numerical method and a numerical-analytical one.In the former,both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities.The latter is established on a principle that the attractiveness of a set implies its invariance.Thus,the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension.Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.展开更多
This paper investigates the chaos synchronisation between two coupled chaotic Chua＇s circuits. The sufficient condition presented by linear matrix inequalities （LMIs） of global asymptotic synchronisation is attaine...This paper investigates the chaos synchronisation between two coupled chaotic Chua＇s circuits. The sufficient condition presented by linear matrix inequalities （LMIs） of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions （pwLDIs） model of synchronisation error dynamics, then we design a switching （piecewise-linear） feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results.展开更多
In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A（t）x on time scales. Moreover, for the nonlinear perturbed equation x△= A（t）x ...In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A（t）x on time scales. Moreover, for the nonlinear perturbed equation x△= A（t）x ＋ f（t, x） we give the instability of the zero solution when f is sufficiently small.展开更多
This study employs mathematical modeling to analyze the impact of active immigrants on Foot and Mouth Disease (FMD) transmission dynamics. We calculate the reproduction number (R<sub>0</sub>) using the nex...This study employs mathematical modeling to analyze the impact of active immigrants on Foot and Mouth Disease (FMD) transmission dynamics. We calculate the reproduction number (R<sub>0</sub>) using the next-generation matrix approach. Applying the Routh-Hurwitz Criterion, we establish that the Disease-Free Equilibrium (DFE) point achieves local asymptotic stability when R<sub>0</sub> α<sub>1</sub> and α<sub>2</sub>) are closely associated with reduced susceptibility in animal populations, underscoring the link between immigrants and susceptibility. Furthermore, our findings emphasize the interplay of disease introduction with population response and adaptation, particularly involving incoming infectious immigrants. Swift interventions are vital due to the limited potential for disease establishment and rapid susceptibility decline. This study offers crucial insights into the complexities of FMD transmission with active immigrants, informing effective disease management strategies.展开更多
In this paper, stability and disturbance attenuation issues for a class of Networked Control Systems （NCSs） under uncertain access delay and packet dropout effects are considered. Our aim is to find conditions on th...In this paper, stability and disturbance attenuation issues for a class of Networked Control Systems （NCSs） under uncertain access delay and packet dropout effects are considered. Our aim is to find conditions on the delay and packet dropout rate, under which the system stability and H∞ disturbance attenuation properties are preserved to a desired level. The basic idea in this paper is to formulate such Networked Control System as a discrete-time switched system. Then the NCSs＇ stability and performance problems can be reduced to the corresponding problems for switched systems, which have been studied for decades and for which a number of results are available in the literature. The techniques in this paper are based on recent progress in the discrete-time switched systems and piecewise Lyapunov functions.展开更多
This paper focuses on synchronization of fractionalorder complex dynamical networks with decentralized adaptive coupling.Based on local information among neighboring nodes,two fractional-order decentralized adaptive s...This paper focuses on synchronization of fractionalorder complex dynamical networks with decentralized adaptive coupling.Based on local information among neighboring nodes,two fractional-order decentralized adaptive strategies are designed to tune all or only a small fraction of the coupling gains respectively.By constructing quadratic Lyapunov functions and utilizing fractional inequality techniques,Mittag-Leffler function,and Laplace transform,two sufficient conditions are derived for reaching network synchronization by using the proposed adaptive laws.Finally,two numerical examples are given to verify the theoretical results.展开更多
The main purpose of this paper is to investigate the problem of quadratic stability and stabilization in switched linear systems using reducible Lie algebra. First, we investigate the structure of all real invariant s...The main purpose of this paper is to investigate the problem of quadratic stability and stabilization in switched linear systems using reducible Lie algebra. First, we investigate the structure of all real invariant subspaces for a given linear system. The result is then used to provide a comparable cascading form for switching models. Using the common cascading form, a common quadratic Lyapunov function is （QLFs） is explored by finding common QLFs of diagonal blocks. In addition, a cascading Quaker Lemma is proved. Combining it with stability results, the problem of feedback stabilization for a class of switched linear systems is solved.展开更多
In this paper, we discuss the problem of guaranteed cost control for uncertain linear systems subject to actuator saturation. Based on the quadratic and non-quadratic Lyapunov functions, sufficient conditions for the ...In this paper, we discuss the problem of guaranteed cost control for uncertain linear systems subject to actuator saturation. Based on the quadratic and non-quadratic Lyapunov functions, sufficient conditions for the robust stability and performance are derived. Moreover, all the conditions can be expressed as linear matrix inequalities （LMIs） or bilinear matrix inequalities （BMIs） in terms of the feedback gain. Thus, the static controller can be effectively synthesized via convex optimization. A numerical example illustrates the effectiveness of the method.展开更多
We consider two problems from stability theory of matrix polytopes: the existence of common quadratic Lyapunov functions and the existence of a stable member. We show the applicability of the gradient algorithm and gi...We consider two problems from stability theory of matrix polytopes: the existence of common quadratic Lyapunov functions and the existence of a stable member. We show the applicability of the gradient algorithm and give a new sufficient condition for the second problem. A number of examples are considered.展开更多
Asymptotic stability of linear systems is closely related to Hurwitz stability of the system matrices. For uncertain linear systems we consider stability problem through common quadratic Lyapunov functions (CQLF) and ...Asymptotic stability of linear systems is closely related to Hurwitz stability of the system matrices. For uncertain linear systems we consider stability problem through common quadratic Lyapunov functions (CQLF) and problem of stabilization by linear feedback.展开更多
In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated.When switched nonlinear systems have uniform normal form an...In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated.When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law.The results of this paper are also applied to switched linear systems.展开更多
This paper is concerned with the problem of H-infinity filtering for discrete-time switched linear systems under arbitrary switching laws.New sufficient conditions for the solvability of the problem are given via swit...This paper is concerned with the problem of H-infinity filtering for discrete-time switched linear systems under arbitrary switching laws.New sufficient conditions for the solvability of the problem are given via switched quadratic Lyapunov functions.Based on Finsler's lemma,two sets of slack variables with special structure are introduced to provide extra degrees of freedom in optimizing the guaranteed H-infinity performance.Compared to the existing methods,the proposed one has better performances and less conservatism.An example is given to illustrate its effectiveness.展开更多
The problem of finding stabilizing controllers for switched systems is an area of much research interest as conventional concepts from continuous time and discrete event dynamics do not hold true for these systems.Man...The problem of finding stabilizing controllers for switched systems is an area of much research interest as conventional concepts from continuous time and discrete event dynamics do not hold true for these systems.Many solutions have been proposed,most of which are based on finding the existence of a common Lyapunov function(CLF) or a multiple Lyapunov function(MLF) where the key is to formulate the problem into a set of linear matrix inequalities(LMIs).An alternative method for finding the existence of a CLF by solving two sets of linear inequalities(LIs) has previously been presented.This method is seen to be less computationally taxing compared to methods based on solving LMIs.To substantiate this,the computational ability of three numerical computational solvers,LMI solver,cvx,and Yalmip,as well as the symbolic computational program Maple were tested.A specific switched system comprising four second-order subsystems was used as a test case.From the obtained solutions,the validity of the controllers and the corresponding CLF was verified.It was found that all tested solvers were able to correctly solve the LIs.The issue of rounding-off error in numerical computation based software is discussed in detail.The test revealed that the guarantee of stability became uncertain when the rounding off was at a different decimal precision.The use of different external solvers led to the same conclusion in terms of the stability of switched systems.As a result,a shift from using a conventional numerical computation based program to using computer algebra is suggested.展开更多
文摘This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via statedependent switching rules.Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid.These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type.In some cases,the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function.In this paper,we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently.The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size.Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size,i.e.,a purely numerical method and a numerical-analytical one.In the former,both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities.The latter is established on a principle that the attractiveness of a set implies its invariance.Thus,the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension.Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.
基金Project partially supported by the grant from the Research Grants Council of the Hong Kong Special Administrative Region,China (Grant No. 101005)the National Natural Science Foundation of China (Grant No. 60904004)the Key Youth Science and Technology Foundation of University of Electronic Science and Technology of China (Grant No. L08010201JX0720)
文摘This paper investigates the chaos synchronisation between two coupled chaotic Chua＇s circuits. The sufficient condition presented by linear matrix inequalities （LMIs） of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions （pwLDIs） model of synchronisation error dynamics, then we design a switching （piecewise-linear） feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results.
文摘In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A（t）x on time scales. Moreover, for the nonlinear perturbed equation x△= A（t）x ＋ f（t, x） we give the instability of the zero solution when f is sufficiently small.
文摘This study employs mathematical modeling to analyze the impact of active immigrants on Foot and Mouth Disease (FMD) transmission dynamics. We calculate the reproduction number (R<sub>0</sub>) using the next-generation matrix approach. Applying the Routh-Hurwitz Criterion, we establish that the Disease-Free Equilibrium (DFE) point achieves local asymptotic stability when R<sub>0</sub> α<sub>1</sub> and α<sub>2</sub>) are closely associated with reduced susceptibility in animal populations, underscoring the link between immigrants and susceptibility. Furthermore, our findings emphasize the interplay of disease introduction with population response and adaptation, particularly involving incoming infectious immigrants. Swift interventions are vital due to the limited potential for disease establishment and rapid susceptibility decline. This study offers crucial insights into the complexities of FMD transmission with active immigrants, informing effective disease management strategies.
基金Supported by the State Key Program of National Natural Science of China （60534010）, National Basic Research Program of China （973 Program）（2009CB320604）, National Natural Science Foundation of China （60674021）, the Funds for Creative Research Groups of China （60521003）, the 111 Project（B08015）, and the Funds of Ph.D. Program of Ministry of Eduction, China （20060145019）.
文摘In this paper, stability and disturbance attenuation issues for a class of Networked Control Systems （NCSs） under uncertain access delay and packet dropout effects are considered. Our aim is to find conditions on the delay and packet dropout rate, under which the system stability and H∞ disturbance attenuation properties are preserved to a desired level. The basic idea in this paper is to formulate such Networked Control System as a discrete-time switched system. Then the NCSs＇ stability and performance problems can be reduced to the corresponding problems for switched systems, which have been studied for decades and for which a number of results are available in the literature. The techniques in this paper are based on recent progress in the discrete-time switched systems and piecewise Lyapunov functions.
基金supported by the"Chunhui Plan"Cooperative Research for Ministry of Education(Z2016133)the Open Research Fund of Key Laboratory of Automobile Engineering(Xihua University)+3 种基金Sichuan Province(szjj2016-017)the National Natural Science Foundation of China(51177137)the Scientific Research Foundation of the Education Department of Sichuan Province(16ZB0163)the China Scholarship Council
文摘This paper focuses on synchronization of fractionalorder complex dynamical networks with decentralized adaptive coupling.Based on local information among neighboring nodes,two fractional-order decentralized adaptive strategies are designed to tune all or only a small fraction of the coupling gains respectively.By constructing quadratic Lyapunov functions and utilizing fractional inequality techniques,Mittag-Leffler function,and Laplace transform,two sufficient conditions are derived for reaching network synchronization by using the proposed adaptive laws.Finally,two numerical examples are given to verify the theoretical results.
基金Supported partly by National Natural Science Foundation of PRC (No. 60343001, 60274010, 66221301 and 60334040)
文摘The main purpose of this paper is to investigate the problem of quadratic stability and stabilization in switched linear systems using reducible Lie algebra. First, we investigate the structure of all real invariant subspaces for a given linear system. The result is then used to provide a comparable cascading form for switching models. Using the common cascading form, a common quadratic Lyapunov function is （QLFs） is explored by finding common QLFs of diagonal blocks. In addition, a cascading Quaker Lemma is proved. Combining it with stability results, the problem of feedback stabilization for a class of switched linear systems is solved.
基金supported by the National Natural Science Foundation of China (No.60704004)
文摘In this paper, we discuss the problem of guaranteed cost control for uncertain linear systems subject to actuator saturation. Based on the quadratic and non-quadratic Lyapunov functions, sufficient conditions for the robust stability and performance are derived. Moreover, all the conditions can be expressed as linear matrix inequalities （LMIs） or bilinear matrix inequalities （BMIs） in terms of the feedback gain. Thus, the static controller can be effectively synthesized via convex optimization. A numerical example illustrates the effectiveness of the method.
文摘We consider two problems from stability theory of matrix polytopes: the existence of common quadratic Lyapunov functions and the existence of a stable member. We show the applicability of the gradient algorithm and give a new sufficient condition for the second problem. A number of examples are considered.
文摘Asymptotic stability of linear systems is closely related to Hurwitz stability of the system matrices. For uncertain linear systems we consider stability problem through common quadratic Lyapunov functions (CQLF) and problem of stabilization by linear feedback.
基金Supported partially by the National Natural Science Foundation of China (Grant No 50525721)
文摘In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated.When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law.The results of this paper are also applied to switched linear systems.
基金supported by the National Natural Science Foundation of China (No. 60974043,61074055)the Fundamental Research Funds for the Central Universities (No. N090604001,N090604002)+1 种基金the China Postdoctoral Science Foundation (No. 20100470203)the Fund of Beijing Excellent Talents Program (No. 2009D013001000016)
文摘This paper is concerned with the problem of H-infinity filtering for discrete-time switched linear systems under arbitrary switching laws.New sufficient conditions for the solvability of the problem are given via switched quadratic Lyapunov functions.Based on Finsler's lemma,two sets of slack variables with special structure are introduced to provide extra degrees of freedom in optimizing the guaranteed H-infinity performance.Compared to the existing methods,the proposed one has better performances and less conservatism.An example is given to illustrate its effectiveness.
文摘The problem of finding stabilizing controllers for switched systems is an area of much research interest as conventional concepts from continuous time and discrete event dynamics do not hold true for these systems.Many solutions have been proposed,most of which are based on finding the existence of a common Lyapunov function(CLF) or a multiple Lyapunov function(MLF) where the key is to formulate the problem into a set of linear matrix inequalities(LMIs).An alternative method for finding the existence of a CLF by solving two sets of linear inequalities(LIs) has previously been presented.This method is seen to be less computationally taxing compared to methods based on solving LMIs.To substantiate this,the computational ability of three numerical computational solvers,LMI solver,cvx,and Yalmip,as well as the symbolic computational program Maple were tested.A specific switched system comprising four second-order subsystems was used as a test case.From the obtained solutions,the validity of the controllers and the corresponding CLF was verified.It was found that all tested solvers were able to correctly solve the LIs.The issue of rounding-off error in numerical computation based software is discussed in detail.The test revealed that the guarantee of stability became uncertain when the rounding off was at a different decimal precision.The use of different external solvers led to the same conclusion in terms of the stability of switched systems.As a result,a shift from using a conventional numerical computation based program to using computer algebra is suggested.
