This paper provides stability analysis results for discretised time delay control(TDC)as implemented in a sampled data system with the standard form of zero-order hold.We first substantiate stability issues in discret...This paper provides stability analysis results for discretised time delay control(TDC)as implemented in a sampled data system with the standard form of zero-order hold.We first substantiate stability issues in discrete-time TDC using an example and propose sufficient stability criteria in the sense of Lyapunov.Important parameters significantly affecting the overall system stability are the sampling period,the desired trajectory and the selection of the reference model dynamics.展开更多
In the Lagrangian meshless(particle)methods,such as the smoothed particle hydrodynamics(SPH),moving particle semi-implicit(MPS)method and meshless local Petrov-Galerkin method based on Rankine source solution(MLPG_R),...In the Lagrangian meshless(particle)methods,such as the smoothed particle hydrodynamics(SPH),moving particle semi-implicit(MPS)method and meshless local Petrov-Galerkin method based on Rankine source solution(MLPG_R),the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities(such as the viscous stresses).In some meshless applications,the Laplacians are also needed as stabilisation operators to enhance the pressure calculation.The particles in the Lagrangian methods move following the material velocity,yielding a disordered(random)particle distribution even though they may be distributed uniformly in the initial state.Different schemes have been developed for a direct estimation of second derivatives using finite difference,kernel integrations and weighted/moving least square method.Some of the schemes suffer from a poor convergent rate.Some have a better convergent rate but require inversions of high order matrices,yielding high computational costs.This paper presents a quadric semi-analytical finite-difference interpolation(QSFDI)scheme,which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices,i.e.3×3 for 3D cases,compared with 6×6 or 10×10 in the schemes with the best convergent rate.Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations.The convergence,accuracy and robustness of the present schemes are compared with the existing schemes.It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures,particularly for estimating the Laplacian of given functions.展开更多
Time delay is an inherent characteristic of real-world phenomena which may affect the system's characteristic.The systems including delay are known as time-delay systems,they are represented using delay differenti...Time delay is an inherent characteristic of real-world phenomena which may affect the system's characteristic.The systems including delay are known as time-delay systems,they are represented using delay differential equations.Modeling,discretisation,stability and control design for time-delay systems are still challenging in modern control theory.This paper systematically overviews available discretisation methods of linear and nonlinear time-delay systems.Emphasis is placed on illustrating fundamental results and recent progress on discretisation methods for delay systems.Numerous methods for the discretisation of linear and nonlinear systems considering input delays,state or output delays in the system's dynamics have been presented.A particular attention will be paid to illustrate effects of the discretisation process on the stability of discretised systems.Examples of mathematical descriptions,problems,and performance analysis for delay systems are presented.The presentation of discretisation methods is as easy as possible,focussing more on the main ideas and mathematical concepts by analogy.Finally,some possible future research directions to be tackled by researchers in this field are discussed.展开更多
In this paper, we apply Hirota＇s discretisation to a three-dimensional integrable Lotka- Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is vol...In this paper, we apply Hirota＇s discretisation to a three-dimensional integrable Lotka- Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume- preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.展开更多
In this paper,three kinds of discrete formulae for the Caputo fractional derivative are studied,including the modified L1 discretisation forα∈(O,1),and L2 discretisation and L2C discretisation forα∈(1,2).The trunc...In this paper,three kinds of discrete formulae for the Caputo fractional derivative are studied,including the modified L1 discretisation forα∈(O,1),and L2 discretisation and L2C discretisation forα∈(1,2).The truncation error estimates and the properties of the coeffcients of all these discretisations are analysed in more detail.Finally,the theoretical analyses areverifiedby thenumerical examples.展开更多
The state equations of stochastic control problems,which are controlled stochastic differential equations,are proposed to be discretized by the weak midpoint rule and predictor-corrector methods for the Markov chain a...The state equations of stochastic control problems,which are controlled stochastic differential equations,are proposed to be discretized by the weak midpoint rule and predictor-corrector methods for the Markov chain approximation approach. Local consistency of the methods are proved.Numerical tests on a simplified Merton's portfolio model show better simulation to feedback control rules by these two methods, as compared with the weak Euler-Maruyama discretisation used by Krawczyk.This suggests a new approach of improving accuracy of approximating Markov chains for stochastic control problems.展开更多
A useful unified analysis framework is proposed for exploring the intriguing behaviors of a second-order switching control system. Complex discretization behaviors of the switching control system are explored in detai...A useful unified analysis framework is proposed for exploring the intriguing behaviors of a second-order switching control system. Complex discretization behaviors of the switching control system are explored in detail, and some intrinsic relationships between the system periodic behaviors and their associated symbolic sequences are studied. Keywords Switching control - Chaos - Discretisation - Periodicity This work was supported by the Australian Research Council and the Hong Kong Research Grants Council for their financial supports, under the CERG Grants CityU 1018/01E, 1004/02E, and 1115/03E.展开更多
The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional d...The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional derivatives are proposed,including the Fourier transform and the Laplace transform.Then,the L2 discretisation for the exponential Caputo derivative with a∈(1,2)is established.The estimation of the truncation error and the properties of the coefficients are discussed.In addition,a numerical example is given to verify the correctness of the derived L2 discrete formula.展开更多
It is well-known that physical laws for large chaotic dynamical systems are revealed statistically.Many times these statistical properties of the system must be approximated numerically.The main contribution of this m...It is well-known that physical laws for large chaotic dynamical systems are revealed statistically.Many times these statistical properties of the system must be approximated numerically.The main contribution of this manuscript is to provide simple and natural criterions on numerical methods (temporal and spatial discretization) that are able to capture the stationary statistical properties of the underlying dissipative chaotic dynamical systems asymptotically.The result on temporal approximation is a recent finding of the author,and the result on spatial approximation is a new one.Applications to the infinite Prandtl number model for convection and the barotropic quasi-geostrophic model are also discussed.展开更多
The approximation of problems with linear convection and degenerate nonlinear difFusion,which arise in the framework of the transport of energy in porous media with thermodynamic transitions,is done usingθ-scheme bas...The approximation of problems with linear convection and degenerate nonlinear difFusion,which arise in the framework of the transport of energy in porous media with thermodynamic transitions,is done usingθ-scheme based on the centred gradient discretisation method.The convergence of the numerical scheme is proved,although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field,owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting.Some numerical examples,using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme,show the role ofθfor stabilising the scheme.展开更多
A framework for parallel algebraic multilevel preconditioning methods presented for solving large sparse systems of linear equstions with symmetric positive definite coefficient matrices,which arise in suitable finite...A framework for parallel algebraic multilevel preconditioning methods presented for solving large sparse systems of linear equstions with symmetric positive definite coefficient matrices,which arise in suitable finite element discretizations of many second-order self-adjoint elliptic boundary value problems. This framework not only covers all known parallel algebraic multilevel preconditioning methods, but also yields new ones. It is shown that all preconditioners within this framework have optimal orders of complexities for problems in two-dimensional(2-D) and three-dimensional (3-D) problem domains, and their relative condition numbers are bounded uniformly with respect to the numbers of both levels and nodes.展开更多
This paper deals with the fault detection observer design problem for switched system with all modes unstable.First,the error system model is presented.With the consideration of the instability caused by the unobserva...This paper deals with the fault detection observer design problem for switched system with all modes unstable.First,the error system model is presented.With the consideration of the instability caused by the unobservable condition of(Ai,Ci),a switching signal is obtained via average dwell time method and discretised Lyapunov function.By means of linear matrix inequalities,sufficient conditions pledging exponential stability and H∞performance are proposed.Through transferring the observer design problem into H∞performance analysis,proper observer gain is obtained via solving these LMIs.In the end,the validity of the proposed Luenberger observer is verified by an example.展开更多
文摘This paper provides stability analysis results for discretised time delay control(TDC)as implemented in a sampled data system with the standard form of zero-order hold.We first substantiate stability issues in discrete-time TDC using an example and propose sufficient stability criteria in the sense of Lyapunov.Important parameters significantly affecting the overall system stability are the sampling period,the desired trajectory and the selection of the reference model dynamics.
文摘In the Lagrangian meshless(particle)methods,such as the smoothed particle hydrodynamics(SPH),moving particle semi-implicit(MPS)method and meshless local Petrov-Galerkin method based on Rankine source solution(MLPG_R),the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities(such as the viscous stresses).In some meshless applications,the Laplacians are also needed as stabilisation operators to enhance the pressure calculation.The particles in the Lagrangian methods move following the material velocity,yielding a disordered(random)particle distribution even though they may be distributed uniformly in the initial state.Different schemes have been developed for a direct estimation of second derivatives using finite difference,kernel integrations and weighted/moving least square method.Some of the schemes suffer from a poor convergent rate.Some have a better convergent rate but require inversions of high order matrices,yielding high computational costs.This paper presents a quadric semi-analytical finite-difference interpolation(QSFDI)scheme,which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices,i.e.3×3 for 3D cases,compared with 6×6 or 10×10 in the schemes with the best convergent rate.Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations.The convergence,accuracy and robustness of the present schemes are compared with the existing schemes.It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures,particularly for estimating the Laplacian of given functions.
文摘Time delay is an inherent characteristic of real-world phenomena which may affect the system's characteristic.The systems including delay are known as time-delay systems,they are represented using delay differential equations.Modeling,discretisation,stability and control design for time-delay systems are still challenging in modern control theory.This paper systematically overviews available discretisation methods of linear and nonlinear time-delay systems.Emphasis is placed on illustrating fundamental results and recent progress on discretisation methods for delay systems.Numerous methods for the discretisation of linear and nonlinear systems considering input delays,state or output delays in the system's dynamics have been presented.A particular attention will be paid to illustrate effects of the discretisation process on the stability of discretised systems.Examples of mathematical descriptions,problems,and performance analysis for delay systems are presented.The presentation of discretisation methods is as easy as possible,focussing more on the main ideas and mathematical concepts by analogy.Finally,some possible future research directions to be tackled by researchers in this field are discussed.
基金The authors would like to thank J. Niesen for his helpful suggestions in improving the presentation of the paper. The first author was supported by the Fundamental Research Funds for the Central Universities （WK2030040057）. The second author was sup- ported by the National Natural Science Foundation of China （11271357）, the Foundation for Innovative Research Groups of the NNSFC （11321061） and the ITER-China Program （2014G- B124005）. The third author was supported by the Foundation of the NNSFC （10990012） and the Marine Public Welfare Project of China （201105032）.
文摘In this paper, we apply Hirota＇s discretisation to a three-dimensional integrable Lotka- Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume- preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.
文摘In this paper,three kinds of discrete formulae for the Caputo fractional derivative are studied,including the modified L1 discretisation forα∈(O,1),and L2 discretisation and L2C discretisation forα∈(1,2).The truncation error estimates and the properties of the coeffcients of all these discretisations are analysed in more detail.Finally,the theoretical analyses areverifiedby thenumerical examples.
基金supported by the China Postdoctoral Science Foundation (No.20080430402).
文摘The state equations of stochastic control problems,which are controlled stochastic differential equations,are proposed to be discretized by the weak midpoint rule and predictor-corrector methods for the Markov chain approximation approach. Local consistency of the methods are proved.Numerical tests on a simplified Merton's portfolio model show better simulation to feedback control rules by these two methods, as compared with the weak Euler-Maruyama discretisation used by Krawczyk.This suggests a new approach of improving accuracy of approximating Markov chains for stochastic control problems.
文摘A useful unified analysis framework is proposed for exploring the intriguing behaviors of a second-order switching control system. Complex discretization behaviors of the switching control system are explored in detail, and some intrinsic relationships between the system periodic behaviors and their associated symbolic sequences are studied. Keywords Switching control - Chaos - Discretisation - Periodicity This work was supported by the Australian Research Council and the Hong Kong Research Grants Council for their financial supports, under the CERG Grants CityU 1018/01E, 1004/02E, and 1115/03E.
文摘The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional derivatives are proposed,including the Fourier transform and the Laplace transform.Then,the L2 discretisation for the exponential Caputo derivative with a∈(1,2)is established.The estimation of the truncation error and the properties of the coefficients are discussed.In addition,a numerical example is given to verify the correctness of the derived L2 discrete formula.
基金supported by the National Science Foundation (No.DMS0606671)a 111 project from the Chinese MOE
文摘It is well-known that physical laws for large chaotic dynamical systems are revealed statistically.Many times these statistical properties of the system must be approximated numerically.The main contribution of this manuscript is to provide simple and natural criterions on numerical methods (temporal and spatial discretization) that are able to capture the stationary statistical properties of the underlying dissipative chaotic dynamical systems asymptotically.The result on temporal approximation is a recent finding of the author,and the result on spatial approximation is a new one.Applications to the infinite Prandtl number model for convection and the barotropic quasi-geostrophic model are also discussed.
基金supported by the French Agence Nationale de la Recherche(CHARMS project,ANR-16-CE06-0009).
文摘The approximation of problems with linear convection and degenerate nonlinear difFusion,which arise in the framework of the transport of energy in porous media with thermodynamic transitions,is done usingθ-scheme based on the centred gradient discretisation method.The convergence of the numerical scheme is proved,although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field,owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting.Some numerical examples,using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme,show the role ofθfor stabilising the scheme.
文摘A framework for parallel algebraic multilevel preconditioning methods presented for solving large sparse systems of linear equstions with symmetric positive definite coefficient matrices,which arise in suitable finite element discretizations of many second-order self-adjoint elliptic boundary value problems. This framework not only covers all known parallel algebraic multilevel preconditioning methods, but also yields new ones. It is shown that all preconditioners within this framework have optimal orders of complexities for problems in two-dimensional(2-D) and three-dimensional (3-D) problem domains, and their relative condition numbers are bounded uniformly with respect to the numbers of both levels and nodes.
基金supported by the NationalNatural Science Foundation of China[61873057]Natural Science Foundation of Jilin Province[20180520211JH]Jilin City Science and Technology Bureau[201831727,201831731].
文摘This paper deals with the fault detection observer design problem for switched system with all modes unstable.First,the error system model is presented.With the consideration of the instability caused by the unobservable condition of(Ai,Ci),a switching signal is obtained via average dwell time method and discretised Lyapunov function.By means of linear matrix inequalities,sufficient conditions pledging exponential stability and H∞performance are proposed.Through transferring the observer design problem into H∞performance analysis,proper observer gain is obtained via solving these LMIs.In the end,the validity of the proposed Luenberger observer is verified by an example.
