In this paper, we study the superconvergence of the error for the local discontinuous Galerkin （LDG） finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove t...In this paper, we study the superconvergence of the error for the local discontinuous Galerkin （LDG） finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise k-th degree polynomials, the error between the LDG solution and the exact solution is （k ＋ 2）-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is （k ＋ 2）-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for P^k polynomials with arbitrary k ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.展开更多
To solve the receding horizon control （RHC） problem in an online manner, a novel numerical method called the indirect Radau pseudospectral method （IRPM） is proposed in this paper. Based on calculus of variations a...To solve the receding horizon control （RHC） problem in an online manner, a novel numerical method called the indirect Radau pseudospectral method （IRPM） is proposed in this paper. Based on calculus of variations and the first-order necessary optimality condition, the RHC problem for linear time-varying （LTV） system is transformed into the two-point boundary value problem （TPBVP）. The Radau pseudospectral approximation is employed to discretize the TPBVP into well-posed linear algebraic equations. The resulting linear algebraic equations are solved via a matrix partitioning approach afterwards to obtain the optimal feedback control law. For the nonlinear system, the linearization method or the quasi linearization method is employed to approximate the RHC problem with successive linear approximations. Subsequently, each linear problem is solved via the similar method which is used to solve the RHC problem for LTV system. Simulation results of three examples show that the IRPM is of high accuracy and of high compu- tation efficiency to solve the RHC problem and the stability of closed-loop systems is guaranteed.展开更多
Aiming at increasing the calculation efficiency of the pseudospectral methods, a multiple- interval Radau pseudospectral method （RPM） is presented to generate a reusable launch vehicle （RLV） ＇s optimal re-entry t...Aiming at increasing the calculation efficiency of the pseudospectral methods, a multiple- interval Radau pseudospectral method （RPM） is presented to generate a reusable launch vehicle （RLV） ＇s optimal re-entry trajectory. After dividing the optimal control problem into many intervals, the state and control variables are approximated using many fixed- and low-degree Lagrange polyno- mials in each interval. Convergence of the numerical discretization is then achieved by increasing the number of intervals. With the application of the proposed method, the normal nonlinear program- ming （NLP） problem transcribed from the optimal control problem can avoid being dense because of the low-degree approximation polynomials in each interval. Thus, the NLP solver can easily compute a solution. Finally, simulation results show that the optimized re-entry trajectories satisfy the path constraints and the boundary constraints successfully. Compared with the single interval RPM, the multiple-interval RPM is significantly faster and has higher calculation efficiency. The results indicate that the multiple-interval RPM can be applied for real-time trajectory generation due to its high effi- ciency and high precision.展开更多
The study of series–parallel plug-in hybrid electric vehicles(PHEVs)has become a research hotspot in new energy vehicles.The global optimal Pareto solutions of energy management strategy(EMS)play a crucial role in th...The study of series–parallel plug-in hybrid electric vehicles(PHEVs)has become a research hotspot in new energy vehicles.The global optimal Pareto solutions of energy management strategy(EMS)play a crucial role in the development of PHEVs.This paper presents a multi-objective global optimization algorithm for the EMS of PHEVs.The algorithm combines the Radau Pseudospectral Knotting Method(RPKM)and the Nondominated Sorting Genetic Algorithm(NSGA)-II to optimize both energy conservation and battery lifespan under the suburban driving conditions of the New European Driving Cycle.The driving conditions are divided into stages at evident mode switching points and the optimal objectives are computed using RPKM.The RPKM results serve as the fitness values in iteration through the NSGA-II method.The results of the algorithm applied to a PHEV simulation show a 26.74%–53.87%improvement in both objectives after 20 iterations compared to the solutions obtained using only RPKM.The proposed algorithm is evaluated against the weighting dynamic programming and is found to be close to the global optimality,with the added benefits of faster and more uniform solutions.展开更多
In this paper, we derive optimal order a posteriori error estimates for the local dis- continuous Galerkin （LDC） method for linear convection-diffusion problems in one space dimension. One of the key ingredients in ...In this paper, we derive optimal order a posteriori error estimates for the local dis- continuous Galerkin （LDC） method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver- gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 （2015）, pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to （p ＋ 1）-degree right and left Radau interpolating polynomials under mesh re- finement. The order of convergence is proved to be p ＋ 2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to （p ＋ 1）-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at （.9（hp＋2） rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at （9（h） rate. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be p＋3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using PP polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.展开更多
In this study,the problem of time-optimal reconnaissance trajectory design for the aeroassisted vehicle is considered.Different from most works reported previously,we explore the feasibility of applying a high-order a...In this study,the problem of time-optimal reconnaissance trajectory design for the aeroassisted vehicle is considered.Different from most works reported previously,we explore the feasibility of applying a high-order aeroassisted vehicle dynamic model to plan the optimal flight trajectory such that the gap between the simulated model and the real system can be narrowed.A highly-constrained optimal control model containing six-degree-of-freedom vehicle dynamics is established.To solve the formulated high-order trajectory planning model,a pipelined optimization strategy is illustrated.This approach is based on the variable order Radau pseudospectral method,indicating that the mesh grid used for discretizing the continuous system experiences several adaption iterations.Utilization of such a strategy can potentially smooth the flight trajectory and improve the algorithm convergence ability.Numerical simulations are reported to demonstrate some key features of the optimized flight trajectory.A number of comparative studies are also provided to verify the effectiveness of the applied method as well as the high-order trajectory planning model.展开更多
Optimal trajectory planning of high-speed trains(HSTs)aims to obtain such speed curves that guarantee safety,punctuality,comfort and energy-saving of the train.In this paper,a new shrinking horizon model predictive co...Optimal trajectory planning of high-speed trains(HSTs)aims to obtain such speed curves that guarantee safety,punctuality,comfort and energy-saving of the train.In this paper,a new shrinking horizon model predictive control(MPC)algorithm is proposed to plan the optimal trajectories of HSTs using real-time traffic information.The nonlinear longitudinal dynamics of HSTs are used to predict the future behaviors of the train and describe variable slopes and variable speed limitations based on real-time traffic information.Then optimal trajectory planning of HSTs is formulated as the shrinking horizon optimal control problem with the consideration of safety,punctuality,comfort and energy consumption.According to the real-time position and running time of the train,the shrinking horizon is updated to ensure the recursive feasibility of the optimization problem.The optimal speed curve of the train is computed by online solving the optimization problem with the Radau Pseudo-spectral method(RPM).Simulation results demonstrate that the proposed method can satisfy the requirements of energy efficiency and punctuality of the train.展开更多
This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations.By the technique of constru...This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations.By the technique of constructing some special correction functions,we prove the(2k+1)-th-order superconvergence for the cell averages,and the numerical traces in the discrete L^(2) norm.In addition,superconvergence of orders k+2 and k+1 is obtained for the error and its derivative at generalized Radau points.All the theoretical findings are confirmed by numerical experiments.展开更多
文摘In this paper, we study the superconvergence of the error for the local discontinuous Galerkin （LDG） finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise k-th degree polynomials, the error between the LDG solution and the exact solution is （k ＋ 2）-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is （k ＋ 2）-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for P^k polynomials with arbitrary k ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.
基金supported by the National Natural Science Foundation of China(Nos.61174221 and 61402039)
文摘To solve the receding horizon control （RHC） problem in an online manner, a novel numerical method called the indirect Radau pseudospectral method （IRPM） is proposed in this paper. Based on calculus of variations and the first-order necessary optimality condition, the RHC problem for linear time-varying （LTV） system is transformed into the two-point boundary value problem （TPBVP）. The Radau pseudospectral approximation is employed to discretize the TPBVP into well-posed linear algebraic equations. The resulting linear algebraic equations are solved via a matrix partitioning approach afterwards to obtain the optimal feedback control law. For the nonlinear system, the linearization method or the quasi linearization method is employed to approximate the RHC problem with successive linear approximations. Subsequently, each linear problem is solved via the similar method which is used to solve the RHC problem for LTV system. Simulation results of three examples show that the IRPM is of high accuracy and of high compu- tation efficiency to solve the RHC problem and the stability of closed-loop systems is guaranteed.
文摘Aiming at increasing the calculation efficiency of the pseudospectral methods, a multiple- interval Radau pseudospectral method （RPM） is presented to generate a reusable launch vehicle （RLV） ＇s optimal re-entry trajectory. After dividing the optimal control problem into many intervals, the state and control variables are approximated using many fixed- and low-degree Lagrange polyno- mials in each interval. Convergence of the numerical discretization is then achieved by increasing the number of intervals. With the application of the proposed method, the normal nonlinear program- ming （NLP） problem transcribed from the optimal control problem can avoid being dense because of the low-degree approximation polynomials in each interval. Thus, the NLP solver can easily compute a solution. Finally, simulation results show that the optimized re-entry trajectories satisfy the path constraints and the boundary constraints successfully. Compared with the single interval RPM, the multiple-interval RPM is significantly faster and has higher calculation efficiency. The results indicate that the multiple-interval RPM can be applied for real-time trajectory generation due to its high effi- ciency and high precision.
基金supported by the Natural Science Foundation of Guangdong Province under Grant 2020A1515010773the Key-Area Research and Development Program of Guangdong Province under Grant 2019B090912001.
文摘The study of series–parallel plug-in hybrid electric vehicles(PHEVs)has become a research hotspot in new energy vehicles.The global optimal Pareto solutions of energy management strategy(EMS)play a crucial role in the development of PHEVs.This paper presents a multi-objective global optimization algorithm for the EMS of PHEVs.The algorithm combines the Radau Pseudospectral Knotting Method(RPKM)and the Nondominated Sorting Genetic Algorithm(NSGA)-II to optimize both energy conservation and battery lifespan under the suburban driving conditions of the New European Driving Cycle.The driving conditions are divided into stages at evident mode switching points and the optimal objectives are computed using RPKM.The RPKM results serve as the fitness values in iteration through the NSGA-II method.The results of the algorithm applied to a PHEV simulation show a 26.74%–53.87%improvement in both objectives after 20 iterations compared to the solutions obtained using only RPKM.The proposed algorithm is evaluated against the weighting dynamic programming and is found to be close to the global optimality,with the added benefits of faster and more uniform solutions.
文摘In this paper, we derive optimal order a posteriori error estimates for the local dis- continuous Galerkin （LDC） method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver- gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 （2015）, pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to （p ＋ 1）-degree right and left Radau interpolating polynomials under mesh re- finement. The order of convergence is proved to be p ＋ 2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to （p ＋ 1）-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at （.9（hp＋2） rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at （9（h） rate. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be p＋3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using PP polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.
文摘In this study,the problem of time-optimal reconnaissance trajectory design for the aeroassisted vehicle is considered.Different from most works reported previously,we explore the feasibility of applying a high-order aeroassisted vehicle dynamic model to plan the optimal flight trajectory such that the gap between the simulated model and the real system can be narrowed.A highly-constrained optimal control model containing six-degree-of-freedom vehicle dynamics is established.To solve the formulated high-order trajectory planning model,a pipelined optimization strategy is illustrated.This approach is based on the variable order Radau pseudospectral method,indicating that the mesh grid used for discretizing the continuous system experiences several adaption iterations.Utilization of such a strategy can potentially smooth the flight trajectory and improve the algorithm convergence ability.Numerical simulations are reported to demonstrate some key features of the optimized flight trajectory.A number of comparative studies are also provided to verify the effectiveness of the applied method as well as the high-order trajectory planning model.
基金supported by the National Natural Science Foundation of China(No.61773345)the Zhejang Provincial Natural Science Foundation(No.LR17F030004).
文摘Optimal trajectory planning of high-speed trains(HSTs)aims to obtain such speed curves that guarantee safety,punctuality,comfort and energy-saving of the train.In this paper,a new shrinking horizon model predictive control(MPC)algorithm is proposed to plan the optimal trajectories of HSTs using real-time traffic information.The nonlinear longitudinal dynamics of HSTs are used to predict the future behaviors of the train and describe variable slopes and variable speed limitations based on real-time traffic information.Then optimal trajectory planning of HSTs is formulated as the shrinking horizon optimal control problem with the consideration of safety,punctuality,comfort and energy consumption.According to the real-time position and running time of the train,the shrinking horizon is updated to ensure the recursive feasibility of the optimization problem.The optimal speed curve of the train is computed by online solving the optimization problem with the Radau Pseudo-spectral method(RPM).Simulation results demonstrate that the proposed method can satisfy the requirements of energy efficiency and punctuality of the train.
基金supported by National Natural Science Foundation of China(Grant Nos.11971132,U1637208,71773024,51605114 and 11501149)the National Key Research and Development Program of China(Grant No.2017YFB1401801)。
文摘This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations.By the technique of constructing some special correction functions,we prove the(2k+1)-th-order superconvergence for the cell averages,and the numerical traces in the discrete L^(2) norm.In addition,superconvergence of orders k+2 and k+1 is obtained for the error and its derivative at generalized Radau points.All the theoretical findings are confirmed by numerical experiments.
