An efficient identification algorithm is given for commensurate order linear time-invariant fractional systems. This algorithm can identify not only model coefficients of the system, but also its differential order at...An efficient identification algorithm is given for commensurate order linear time-invariant fractional systems. This algorithm can identify not only model coefficients of the system, but also its differential order at the same time. The basic idea is to change the system matrix into a diagonal one through basis transformation. This makes it possible to turn the system’s input-output relationships into the summation of several simple subsystems, and after the identification of these subsystems, the whole identification system is obtained which is algebraically equivalent to the former system. Finally an identification example verifies the effectiveness of the method previously mentioned.展开更多
This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the ...This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the sake of overcoming the singularity, sequences of approximate solutions to the boundary value problem are obtained by applying the fixed point index theory on the cone. Next, it is demonstrated that these sequences of approximate solutions are uniformly bounded and equicontinuous. The main results are then established through the Ascoli-Arzelà theorem. Ultimately, an instance is worked out to test and verify the validity of the main results.展开更多
Methods based on numerical optimization are useful and effective in the design of control systems. This paper describes the design of retarded fractional delay differential systems (RFDDSs) by the method of inequali...Methods based on numerical optimization are useful and effective in the design of control systems. This paper describes the design of retarded fractional delay differential systems (RFDDSs) by the method of inequalities, in which the design problem is formulated so that it is suitable for solution by numerical methods. Zakian's original formulation, which was first proposed in connection with rational systems, is extended to the case of RFDDSs. In making the use of this formulation possible for RFDDSs, the associated stability problems are resolved by using the stability test and the numerical algorithm for computing the abscissa of stability recently developed by the authors. During the design process, the time responses are obtained by a known method for the numerical inversion of Laplace transforms. Two numerical examples are given, where fractional controllers are designed for a time-delay and a heat-conduction plants.展开更多
The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is estab- lished, and the fractional Lagrange equations are obtained by virtue of the d'Alembert-Lagrange princi...The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is estab- lished, and the fractional Lagrange equations are obtained by virtue of the d'Alembert-Lagrange principle with fractional derivatives. As the Lie symmetry theorem is based on the invariance of differential equations under infinitesimal trans- formations, by introducing the differential operator of infinitesimal generators, the determining equations are obtained. Furthermore, the limit equations, the additional restriction equations, the structural equations, and the conserved quantity of Lie symmetry are acquired. An example is presented to illustrate the application of results.展开更多
By using power mapping(s =v^m),stability analysis of fractional order polynomials was simplified to the stability analysis of expanded degree integer order polynomials in the first Riemann sheet.However,more investiga...By using power mapping(s =v^m),stability analysis of fractional order polynomials was simplified to the stability analysis of expanded degree integer order polynomials in the first Riemann sheet.However,more investigation is needed for revealing properties of power mapping and demonstration of conformity of Hurwitz stability under power mapping of fractional order characteristic polynomials.Contributions of this study have two folds: Firstly,this paper demonstrates conservation of root argument and magnitude relations under power mapping of characteristic polynomials and thus substantiates validity of Hurwitz stability under power mapping of fractional order characteristic polynomials.This also ensures implications of edge theorem for fractional order interval systems.Secondly,in control engineering point of view,numerical robust stability analysis approaches based on the consideration of minimum argument roots of edge and vertex polynomials are presented.For the computer-aided design of fractional order interval control systems,the minimum argument root principle is applied for a finite set of edge and vertex polynomials,which are sampled from parametric uncertainty box.Several illustrative examples are presented to discuss effectiveness of these approaches.展开更多
In this paper, a modified impulsive control scheme is proposed to realize the complete synchronization of fractional order hyperchaotic systems. By constructing a suitable response system, an integral order synchroniz...In this paper, a modified impulsive control scheme is proposed to realize the complete synchronization of fractional order hyperchaotic systems. By constructing a suitable response system, an integral order synchronization error system is obtained. Based on the theory of Lyapunov stability and the impulsive differential equations, some effective sufficient conditions are derived to guarantee the asymptotical stability of the synchronization error system. In particular, some simpler and more convenient conditions are derived by taking the fixed impulsive distances and control gains. Compared with the existing results, the main results in this paper are practical and rigorous. Simulation results show the effectiveness and the feasibility of the proposed impulsive control method.展开更多
Existence of periodic solutions and stability of fractional order dynamic systems are two important and difficult issues in fractional order systems(FOS) field. In this paper, the relationship between integer order sy...Existence of periodic solutions and stability of fractional order dynamic systems are two important and difficult issues in fractional order systems(FOS) field. In this paper, the relationship between integer order systems(IOS) and fractional order systems is discussed. A new proof method based on the above involved relationship for the non existence of periodic solutions of rational fractional order linear time invariant systems is derived. Rational fractional order linear time invariant autonomous system is proved to be equivalent to an integer order linear time invariant non-autonomous system. It is further proved that stability of a fractional order linear time invariant autonomous system is equivalent to the stability of another corresponding integer order linear time invariant autonomous system. The examples and state figures are given to illustrate the effects of conclusion derived.展开更多
Leader-following consensus of fractional order multi-agent systems is investigated. The agents are considered as discrete-time fractional order integrators or fractional order double-integrators. Moreover, the interac...Leader-following consensus of fractional order multi-agent systems is investigated. The agents are considered as discrete-time fractional order integrators or fractional order double-integrators. Moreover, the interaction between the agents is described with an undirected communication graph with a fixed topology. It is shown that the leader-following consensus problem for the considered agents could be converted to the asymptotic stability analysis of a discrete-time fractional order system. Based on this idea, sufficient conditions to reach the leader-following consensus in terms of the controller parameters are extracted. This leads to an appropriate region in the controller parameters space. Numerical simulations are provided to show the performance of the proposed leader-following consensus approach.展开更多
Practical stabilities for linear fractional impulsive hybrid systems are investigated in detail.The transformation from a linear fractional differential system to a fractional impulsive hybrid system is interpreted.Wi...Practical stabilities for linear fractional impulsive hybrid systems are investigated in detail.The transformation from a linear fractional differential system to a fractional impulsive hybrid system is interpreted.With the help of the Mittag-Leffler functions for matrix-type,several practical stability criteria for fractional impulsive hybrid systems are derived.Finally,a numerical example is provided to illustrate the effectiveness of the results.展开更多
I.INTRODUCTION FRACTIONAL calculus has been applied in all MAD(modeling,analysis and design)aspects of control systems engineering since Shunji Manabe’s pioneering work in early 1960s.The 2016 International Conferenc...I.INTRODUCTION FRACTIONAL calculus has been applied in all MAD(modeling,analysis and design)aspects of control systems engineering since Shunji Manabe’s pioneering work in early 1960s.The 2016 International Conference on Fractional Differentiation and Its Applications(ICFDA)was held in Novi Sad,Serbia,July 18-20.Quoting from the展开更多
Under some local superquadratic conditions on <em>W</em> (<em>t</em>, <em>u</em>) with respect to <em>u</em>, the existence of infinitely many solutions is obtained for ...Under some local superquadratic conditions on <em>W</em> (<em>t</em>, <em>u</em>) with respect to <em>u</em>, the existence of infinitely many solutions is obtained for the nonperiodic fractional Hamiltonian systems<img src="Edit_b2a2ac0a-6dde-474f-8c75-e9f5fc7b9918.bmp" alt="" />, where <em>L</em> (<em>t</em>) is unnecessarily coercive.展开更多
I.INTRODUCTION FRACTIONAL calculus is about differentiation and integration of non-integer orders.Using integer-order models and controllers for complex natural or man-made systems is simply for our own convenience wh...I.INTRODUCTION FRACTIONAL calculus is about differentiation and integration of non-integer orders.Using integer-order models and controllers for complex natural or man-made systems is simply for our own convenience while the nature runs in a fractional order dynamical way.Using integer order traditiona tools for modelling and control of dynamic systems may resul in suboptimum performance,that is,using fractional order calculus tools,we could be'more optimal'as already doc-展开更多
As one of secure communication means, chaotic communication systems has been well-developed during the past three decades. Technical papers, both for theoretical and practical investigations, have reached a huge amoun...As one of secure communication means, chaotic communication systems has been well-developed during the past three decades. Technical papers, both for theoretical and practical investigations, have reached a huge amount in number. On the other hand, fractional chaos, as a parallel ongoing research topic, also attracts many researchers to investigate. As far as the IT field is concerned, the research on control systems by using fractional chaos known as FOC (fractional order control) has been a hot issue for quite a long time. As a comparison, interesting enough, up to now we have not found any research result related to Fractional Chaos Communi- cation (FCC) system, i.e., a system based on fractional chaos. The motivation of the present article is to reveal the feasibility of realizing communication systems based upon FCC and their superiority over the conventional integer chaotic communication systems. Principles of FCC and its advantages over integer chaotic communication systems are also discussed.展开更多
The state-space representation of linear time-invariant (LTI) fractional order systems is introduced, and a proof of their stability theory is also given. Then an efficient identification algorithm is proposed for tho...The state-space representation of linear time-invariant (LTI) fractional order systems is introduced, and a proof of their stability theory is also given. Then an efficient identification algorithm is proposed for those fractional order systems. The basic idea of the algorithm is to compute fractional derivatives and the filter simultaneously, i.e., the filtered fractional derivatives can be obtained by computing them in one step, and then system identification can be fulfilled by the least square method. The instrumental variable method is also used in the identification of fractional order systems. In this way, even if there is colored noise in the systems, the unbiased estimation of the parameters can still be obtained. Finally an example of identifying a viscoelastic system is given to show the effectiveness of the aforementioned method.展开更多
In this paper,we consider the following perturbed fractional Hamiltonian systems{tD_(∞)^(α)(_(-∞)D_(t)^(α)u(t))+L(t)u(t)=■_(u)W(t,u(t))+■(u)G(t,u(t)),t∈R,u∈H^(α)(R,R^(N)),whereα∈(1/2,1],L∈C(R,R^(N×N))...In this paper,we consider the following perturbed fractional Hamiltonian systems{tD_(∞)^(α)(_(-∞)D_(t)^(α)u(t))+L(t)u(t)=■_(u)W(t,u(t))+■(u)G(t,u(t)),t∈R,u∈H^(α)(R,R^(N)),whereα∈(1/2,1],L∈C(R,R^(N×N))is symmetric and not necessarily required to be positive definite,W∈C1(R×R^(N,R))is locally subquadratic and locally even near the origin,and perturbed term G∈C1(R×R^(N,R))maybe has no parity in u.Utilizing the perturbed method improved by the authors,a sequence of nontrivial homo clinic solutions is obtained,which generalizes previous results.展开更多
In this article, we study the existence, uniqueness, stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional differential systems by relax...In this article, we study the existence, uniqueness, stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional differential systems by relaxing the linear growth conditions. Finally, we give examples to illustrate its applications.展开更多
This paper focuses on a new approach to design(possibly fractional) set-point filters for fractional control systems.After designing a smooth and monotonic desired output signal,the necessary command signal is obtaine...This paper focuses on a new approach to design(possibly fractional) set-point filters for fractional control systems.After designing a smooth and monotonic desired output signal,the necessary command signal is obtained via fractional input-output inversion.Then,a set-point filter is determined based on the synthesized command signal.The filter is computed by minimizing the 2-norm of the difference between the command signal and the filter step response.The proposed methodology allows the designer to synthesize both integer and fractional setpoint filters.The pros and cons of both solutions are discussed in details.This approach is suitable for the design of two degreeof-freedom controllers capable to make the set-point tracking performance almost independent from the feedback part of the controller.Simulation results show the effectiveness of the proposed methodology.展开更多
Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities.Moiréphotonic and optical lattices—two-dimen...Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities.Moiréphotonic and optical lattices—two-dimensional twisted patterns lie somewhere in between perfect periodic structures and aperiodic ones—are a new emerging investigative tool for studying nonlinear localized waves of diverse types.Herein,a theory of two-dimensional spatial localization in nonlinear periodic systems with fractional-order diffraction(linear nonlocality)and moiréoptical lattices is investigated.Specifically,the flat-band feature is well preserved in shallow moiréoptical lattices which,interact with the defocusing nonlinearity of the media,can support fundamental gap solitons,bound states composed of several fundamental solitons,and topological states(gap vortices)with vortex charge s=1 and 2,all populated inside the finite gaps of the linear Bloch-wave spectrum.Employing the linear-stability analysis and direct perturbed simulations,the stability and instability properties of all the localized gap modes are surveyed,highlighting a wide stability region within the first gap and a limited one(to the central part)for the third gap.The findings enable insightful studies of highly localized gap modes in linear nonlocality(fractional)physical systems with shallow moirépatterns that exhibit extremely flat bands.展开更多
This paper investigates the problem of stability analysis for a class of incommensurate nabla fractional order systems.In particular,both Caputo definition and Riemann-Liouville definition are under consideration.With...This paper investigates the problem of stability analysis for a class of incommensurate nabla fractional order systems.In particular,both Caputo definition and Riemann-Liouville definition are under consideration.With the convex assumption,several elementary fractional difference inequalities on Lyapunov functions are developed.According to the essential features of nabla fractional calculus,the sufficient conditions are given first to guarantee the asymptotic stability for the incommensurate system by using the direct Lyapunov method.To substantiate the efficacy and effectiveness of the theoretical results,four examples are elaborated.展开更多
The consensus problem for fractional multi-agent systems(MASs)with time delay is considered.The distributed fractional proportional-integral(PI)-type controller is designed so that the consensus of the proposed system...The consensus problem for fractional multi-agent systems(MASs)with time delay is considered.The distributed fractional proportional-integral(PI)-type controller is designed so that the consensus of the proposed systems is achieved.Moreover,explicit condition to determine the crossing directions is developed.The results show that with the increase of time delay,the closed-loop system has two different dynamic characteristics:From consensus to nonconsensus and consensus switching.Furthermore,delay margin within which consensus of MASs will always hold is determined.The results should provide useful guidelines in the consensus analysis and in the analytical design of the distributed controllers.展开更多
基金Sponsored by 863 Project (Grant No.2002AA517020) Developing Fund of Shanghai Science Committee (Grant No.011607033).
文摘An efficient identification algorithm is given for commensurate order linear time-invariant fractional systems. This algorithm can identify not only model coefficients of the system, but also its differential order at the same time. The basic idea is to change the system matrix into a diagonal one through basis transformation. This makes it possible to turn the system’s input-output relationships into the summation of several simple subsystems, and after the identification of these subsystems, the whole identification system is obtained which is algebraically equivalent to the former system. Finally an identification example verifies the effectiveness of the method previously mentioned.
文摘This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the sake of overcoming the singularity, sequences of approximate solutions to the boundary value problem are obtained by applying the fixed point index theory on the cone. Next, it is demonstrated that these sequences of approximate solutions are uniformly bounded and equicontinuous. The main results are then established through the Ascoli-Arzelà theorem. Ultimately, an instance is worked out to test and verify the validity of the main results.
基金supported by the AUN/SEED-Net collaborative research program.
文摘Methods based on numerical optimization are useful and effective in the design of control systems. This paper describes the design of retarded fractional delay differential systems (RFDDSs) by the method of inequalities, in which the design problem is formulated so that it is suitable for solution by numerical methods. Zakian's original formulation, which was first proposed in connection with rational systems, is extended to the case of RFDDSs. In making the use of this formulation possible for RFDDSs, the associated stability problems are resolved by using the stability test and the numerical algorithm for computing the abscissa of stability recently developed by the authors. During the design process, the time responses are obtained by a known method for the numerical inversion of Laplace transforms. Two numerical examples are given, where fractional controllers are designed for a time-delay and a heat-conduction plants.
基金supported by the National Natural Science Foundation of China(Grant Nos.11272287 and 11472247)the Program for Changjiang Scholars and Innovative Research Team in University of China(Grant No.IRT13097)
文摘The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is estab- lished, and the fractional Lagrange equations are obtained by virtue of the d'Alembert-Lagrange principle with fractional derivatives. As the Lie symmetry theorem is based on the invariance of differential equations under infinitesimal trans- formations, by introducing the differential operator of infinitesimal generators, the determining equations are obtained. Furthermore, the limit equations, the additional restriction equations, the structural equations, and the conserved quantity of Lie symmetry are acquired. An example is presented to illustrate the application of results.
文摘By using power mapping(s =v^m),stability analysis of fractional order polynomials was simplified to the stability analysis of expanded degree integer order polynomials in the first Riemann sheet.However,more investigation is needed for revealing properties of power mapping and demonstration of conformity of Hurwitz stability under power mapping of fractional order characteristic polynomials.Contributions of this study have two folds: Firstly,this paper demonstrates conservation of root argument and magnitude relations under power mapping of characteristic polynomials and thus substantiates validity of Hurwitz stability under power mapping of fractional order characteristic polynomials.This also ensures implications of edge theorem for fractional order interval systems.Secondly,in control engineering point of view,numerical robust stability analysis approaches based on the consideration of minimum argument roots of edge and vertex polynomials are presented.For the computer-aided design of fractional order interval control systems,the minimum argument root principle is applied for a finite set of edge and vertex polynomials,which are sampled from parametric uncertainty box.Several illustrative examples are presented to discuss effectiveness of these approaches.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 50830202 and 51073179)the Natural Science Foundation of Chongqing,China (Grant No. CSTC 2010BB2238)+2 种基金the Doctoral Program of Higher Education Foundation of Institutions of China (Grant Nos. 20090191110011 and 20100191120025)the Natural Science Foundation for Postdoctoral Scientists of China (Grant Nos. 20100470813 and 20100480043)the Fundamental Research Funds for the Central Universities(Grant Nos. CDJZR11 12 00 03 and CDJZR11 12 88 01)
文摘In this paper, a modified impulsive control scheme is proposed to realize the complete synchronization of fractional order hyperchaotic systems. By constructing a suitable response system, an integral order synchronization error system is obtained. Based on the theory of Lyapunov stability and the impulsive differential equations, some effective sufficient conditions are derived to guarantee the asymptotical stability of the synchronization error system. In particular, some simpler and more convenient conditions are derived by taking the fixed impulsive distances and control gains. Compared with the existing results, the main results in this paper are practical and rigorous. Simulation results show the effectiveness and the feasibility of the proposed impulsive control method.
文摘Existence of periodic solutions and stability of fractional order dynamic systems are two important and difficult issues in fractional order systems(FOS) field. In this paper, the relationship between integer order systems(IOS) and fractional order systems is discussed. A new proof method based on the above involved relationship for the non existence of periodic solutions of rational fractional order linear time invariant systems is derived. Rational fractional order linear time invariant autonomous system is proved to be equivalent to an integer order linear time invariant non-autonomous system. It is further proved that stability of a fractional order linear time invariant autonomous system is equivalent to the stability of another corresponding integer order linear time invariant autonomous system. The examples and state figures are given to illustrate the effects of conclusion derived.
文摘Leader-following consensus of fractional order multi-agent systems is investigated. The agents are considered as discrete-time fractional order integrators or fractional order double-integrators. Moreover, the interaction between the agents is described with an undirected communication graph with a fixed topology. It is shown that the leader-following consensus problem for the considered agents could be converted to the asymptotic stability analysis of a discrete-time fractional order system. Based on this idea, sufficient conditions to reach the leader-following consensus in terms of the controller parameters are extracted. This leads to an appropriate region in the controller parameters space. Numerical simulations are provided to show the performance of the proposed leader-following consensus approach.
基金Natural Science Foundation of Shanghai China (No. 10ZR1400100)
文摘Practical stabilities for linear fractional impulsive hybrid systems are investigated in detail.The transformation from a linear fractional differential system to a fractional impulsive hybrid system is interpreted.With the help of the Mittag-Leffler functions for matrix-type,several practical stability criteria for fractional impulsive hybrid systems are derived.Finally,a numerical example is provided to illustrate the effectiveness of the results.
文摘I.INTRODUCTION FRACTIONAL calculus has been applied in all MAD(modeling,analysis and design)aspects of control systems engineering since Shunji Manabe’s pioneering work in early 1960s.The 2016 International Conference on Fractional Differentiation and Its Applications(ICFDA)was held in Novi Sad,Serbia,July 18-20.Quoting from the
文摘Under some local superquadratic conditions on <em>W</em> (<em>t</em>, <em>u</em>) with respect to <em>u</em>, the existence of infinitely many solutions is obtained for the nonperiodic fractional Hamiltonian systems<img src="Edit_b2a2ac0a-6dde-474f-8c75-e9f5fc7b9918.bmp" alt="" />, where <em>L</em> (<em>t</em>) is unnecessarily coercive.
文摘I.INTRODUCTION FRACTIONAL calculus is about differentiation and integration of non-integer orders.Using integer-order models and controllers for complex natural or man-made systems is simply for our own convenience while the nature runs in a fractional order dynamical way.Using integer order traditiona tools for modelling and control of dynamic systems may resul in suboptimum performance,that is,using fractional order calculus tools,we could be'more optimal'as already doc-
文摘As one of secure communication means, chaotic communication systems has been well-developed during the past three decades. Technical papers, both for theoretical and practical investigations, have reached a huge amount in number. On the other hand, fractional chaos, as a parallel ongoing research topic, also attracts many researchers to investigate. As far as the IT field is concerned, the research on control systems by using fractional chaos known as FOC (fractional order control) has been a hot issue for quite a long time. As a comparison, interesting enough, up to now we have not found any research result related to Fractional Chaos Communi- cation (FCC) system, i.e., a system based on fractional chaos. The motivation of the present article is to reveal the feasibility of realizing communication systems based upon FCC and their superiority over the conventional integer chaotic communication systems. Principles of FCC and its advantages over integer chaotic communication systems are also discussed.
文摘The state-space representation of linear time-invariant (LTI) fractional order systems is introduced, and a proof of their stability theory is also given. Then an efficient identification algorithm is proposed for those fractional order systems. The basic idea of the algorithm is to compute fractional derivatives and the filter simultaneously, i.e., the filtered fractional derivatives can be obtained by computing them in one step, and then system identification can be fulfilled by the least square method. The instrumental variable method is also used in the identification of fractional order systems. In this way, even if there is colored noise in the systems, the unbiased estimation of the parameters can still be obtained. Finally an example of identifying a viscoelastic system is given to show the effectiveness of the aforementioned method.
基金Supported by National Natural Science Foundation of China(Grant No.12171355)Elite Scholar Program in Tianjin University,P.R.China。
文摘In this paper,we consider the following perturbed fractional Hamiltonian systems{tD_(∞)^(α)(_(-∞)D_(t)^(α)u(t))+L(t)u(t)=■_(u)W(t,u(t))+■(u)G(t,u(t)),t∈R,u∈H^(α)(R,R^(N)),whereα∈(1/2,1],L∈C(R,R^(N×N))is symmetric and not necessarily required to be positive definite,W∈C1(R×R^(N,R))is locally subquadratic and locally even near the origin,and perturbed term G∈C1(R×R^(N,R))maybe has no parity in u.Utilizing the perturbed method improved by the authors,a sequence of nontrivial homo clinic solutions is obtained,which generalizes previous results.
文摘In this article, we study the existence, uniqueness, stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional differential systems by relaxing the linear growth conditions. Finally, we give examples to illustrate its applications.
基金supported by the Australian Research Council(DP160104994)
文摘This paper focuses on a new approach to design(possibly fractional) set-point filters for fractional control systems.After designing a smooth and monotonic desired output signal,the necessary command signal is obtained via fractional input-output inversion.Then,a set-point filter is determined based on the synthesized command signal.The filter is computed by minimizing the 2-norm of the difference between the command signal and the filter step response.The proposed methodology allows the designer to synthesize both integer and fractional setpoint filters.The pros and cons of both solutions are discussed in details.This approach is suitable for the design of two degreeof-freedom controllers capable to make the set-point tracking performance almost independent from the feedback part of the controller.Simulation results show the effectiveness of the proposed methodology.
基金This work was supported by the National Natural Science Foundation of China(NSFC)(No.12074423)Young Scholar of Chinese Academy of Sciences in Western China(No.XAB2021YN18)China Postdoctoral Science Foundation(No.2023M733722).
文摘Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities.Moiréphotonic and optical lattices—two-dimensional twisted patterns lie somewhere in between perfect periodic structures and aperiodic ones—are a new emerging investigative tool for studying nonlinear localized waves of diverse types.Herein,a theory of two-dimensional spatial localization in nonlinear periodic systems with fractional-order diffraction(linear nonlocality)and moiréoptical lattices is investigated.Specifically,the flat-band feature is well preserved in shallow moiréoptical lattices which,interact with the defocusing nonlinearity of the media,can support fundamental gap solitons,bound states composed of several fundamental solitons,and topological states(gap vortices)with vortex charge s=1 and 2,all populated inside the finite gaps of the linear Bloch-wave spectrum.Employing the linear-stability analysis and direct perturbed simulations,the stability and instability properties of all the localized gap modes are surveyed,highlighting a wide stability region within the first gap and a limited one(to the central part)for the third gap.The findings enable insightful studies of highly localized gap modes in linear nonlocality(fractional)physical systems with shallow moirépatterns that exhibit extremely flat bands.
基金supported by the National Natural Science Foundation of China under Grant No.62273092the Science Climbing Project under Grant No.4307012166+3 种基金the Anhui Provincial Natural Science Foundation under Grant No.1708085QF141the Fundamental Research Funds for the Central Universities under Grant No.WK2100100028the General Financial Grant from the China Postdoctoral Science Foundation under Grant No.2016M602032the fund of China Scholarship Council under Grant No.201806345002。
文摘This paper investigates the problem of stability analysis for a class of incommensurate nabla fractional order systems.In particular,both Caputo definition and Riemann-Liouville definition are under consideration.With the convex assumption,several elementary fractional difference inequalities on Lyapunov functions are developed.According to the essential features of nabla fractional calculus,the sufficient conditions are given first to guarantee the asymptotic stability for the incommensurate system by using the direct Lyapunov method.To substantiate the efficacy and effectiveness of the theoretical results,four examples are elaborated.
基金supported in part by the National Natural Science Foundation of China under Grant Nos.61773207,62173183the Natural Science Fund for Distinguished Young Scholars of Jiangsu Province under Grant No.BK20190020.
文摘The consensus problem for fractional multi-agent systems(MASs)with time delay is considered.The distributed fractional proportional-integral(PI)-type controller is designed so that the consensus of the proposed systems is achieved.Moreover,explicit condition to determine the crossing directions is developed.The results show that with the increase of time delay,the closed-loop system has two different dynamic characteristics:From consensus to nonconsensus and consensus switching.Furthermore,delay margin within which consensus of MASs will always hold is determined.The results should provide useful guidelines in the consensus analysis and in the analytical design of the distributed controllers.