基于改进Gauss伪谱法的高超声速飞行器轨迹优化与制导
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摘要
近二十年来，临近空间被各大国所特别关注，由于大气稀薄，空气阻力小，是实现全球范围内快速飞行的理想空间，高超声速飞行器既具有入轨的能力，又可以再入大气层快速攻击地面目标，具有大升阻比、机动性强、全球快速到达、威力大等特点，可以进行全球实时侦察、快速部署和远程精确打击能力，对未来的作战方式将产生重要影响。因此，高超声速飞行器相关技术和理论得到了广泛研究，其中，高超声速飞行器快速轨迹优化与高精度制导是保证其完成任务的重要前提，由于运动方程的非线性、环境复杂性和多种约束条件的存在，使得难以求解，本文针对该问题进行相关研究。
第一，研究了高超声速飞行器运动和气动建模问题。对于实际项目中的高超声速飞行器，给出了上升段轨迹优化问题的数学描述。在半速度坐标系下，建立了高超声速飞行器再入运动模型。通过分析气动系数数据，综合考虑攻角和速度两个因素，建立了双变量气动系数模型，通过对比，表明该气动模型具有更高的精度。在所建立的气动模型的基础上，分析了高超声速飞行器的升阻比特点，对不同升阻比下的轨迹进行了分析。提出了高超声速飞行器再入轨迹优化与制导问题的数学描述。
第二，研究了含有路径约束非线性最优控制问题的计算方法。传统的Gauss伪谱法中终端状态积分约束是非线性的，不利于计算，并且协态变量的计算方法与其他伪谱法不一致。为了提高收敛速度，利用状态微分方程矩阵和Gauss权重系数将非线性的终端状态积分约束等价转化为等价的线性约束形式，减少了非线性约束的个数，基于线性终端状态积分约束，提出了改进的Gauss伪谱法(IGPM)，证明了新的协态映射定理，进一步完善了Gauss伪谱法。新的协态变量计算方法与其他伪谱法具有统一形式。针对初始时刻协态变量不易计算的问题，提出了简单的计算初始时刻协态变量的方法；由于控制变量没有在边界时刻取值，根据极大值原理，考虑动态微分方程约束和路径约束，提出了端点时刻控制变量的计算方法。
第三，研究了标准的Bolza型多段非线性最优控制问题的多段轨迹优化方法。对于含有复杂路径约束的非线性最优控制问题，只增加离散点个数，导致微分矩阵维数增大，计算效率下降。因此，引入分段的思想，基于IGPM，提出了多段改进Gauss伪谱法(MIGPM)，证明了多段协态映射定理，可以方便地计算协态变量，并提出了分割点处协态和控制变量的计算方法。为了提高解的精度，提出了基于MIGPM的多段轨迹优化算法，并求解高超声速飞行器再入轨迹纵向航程最大问题，解的精度和收敛速度均优于全局轨迹优化方法。
第四，研究了能够高精度处理路径约束和确定间断点能力的动态轨迹优化方法。对于有高精度路径约束要求，并含有不连续控制的问题，全局轨迹优化方法对路径约束的精度不高，并且多段轨迹优化算法难以确定间断点。以综合相对误差向量作为参数更新的判断条件，提出离散点和阶段数量更新方法。重点考虑解和路径约束的满足精度，基于MIGPM提出了动态轨迹优化算法。通过实例验证了该算法的快速收敛性、寻找间断点和处理路径约束的能力。
第五，研究了竖直平面和三维空间的高超声速飞行器上升段最省燃料的轨迹优化问题，并研究了再入轨迹纵向航程和横向航程最大的轨迹优化问题。采用基于IGPM的迭代算法一和动态轨迹优化算法分别求解竖直平面上升段最优轨迹和上升段同时转向的最优轨迹，结果表明最省燃料的轨迹均属于先加速后上升的类型。为了提高计算速度和精度，提出了两种全局轨迹优化迭代算法，仿真结果表明，两种迭代算法都能很快地收敛到最优解。应用动态轨迹优化算法和最新的hp自适应伪谱法求解再入纵向航程和横向航程最大轨迹优化问题，结果表明，动态轨迹优化算法的计算效率、收敛半径和处理路径约束的能力都优于hp自适应伪谱法。分析了再入的最优升阻比特性，再入中段最优升阻比基本上等于最大升阻比，为再入轨迹优化问题的初值选取提供了借鉴。
第六，研究了高超声速飞行器再入过程在线最优反馈制导问题。为了处理模型中的不确定性和干扰，采用Carathe′odory π轨迹的概念，推导采样时间的取值方法，从而保证在存在计算延迟的条件下，系统的跟踪精度仍然能够满足误差要求。将最优控制与在线最优反馈相结合，利用IGPM快速收敛的特点，提出了在线最优反馈制导算法。将提出的轨迹优化方法与在线最优反馈制导算法相结合，提出了一种满足在线计算要求的高超声速飞行器再入闭环制导系统结构，为提高高超声速飞行器再入过程的自主性、可靠性和最优性提供了解决方案。将在线最优反馈制导算法应用于横向航程最大制导问题、定点定向攻击制导问题和在线自主避障再入制导问题，仿真结果表明该制导方法能够很好地满足指标要求。
综上所述，本文提出了高超声速飞行器轨迹优化方法和在线最优反馈制导方法，为高超声速飞行器轨迹在线优化和高精度制导奠定了一定基础。
In the past twenty years, near space is focused by some great countries. It is the idealspace to realize the hypersonic flying because of the thin air and the small resistance. Thehypersonic vehicle not only has the ability to entry the space but also can reentry the at-mosphere for attacking the ground targets fast. It has the big lift-to-drag ratio (LDR), wellmaneuverability, fast reaching the global world and great power, etc. It can be used in thefields of the global real-time scout, rapid deployment and remote precision attacking andimpacts the future war greatly. Therefore, much researches on the related technology andtheory of the hypersonic vehicle have won wide studies. It is the essential preconditionfor the mission to realize the trajectory optimization and high-precision guidance. Thisproblem is difcult to solve because of the nonlinear property of the motion equations,complexity of the environment and various nonlinear constraints. The thesis takes someresearches on this problem.
1. The motion models and the aerodynamic model of the hypersonic vehicle arediscussed. The mathematical descriptions of the ascent trajectory optimization problem ofthe hypersonic vehicle are given for the engineering problems. The reentry motion modelis presented under the half velocity coordinate system. Considering the attacking angleand velocity variables, the bivariate aerodynamic coefcients model is built by analyzingthe characteristics of the data, The comparison shows that the proposed aerodynamiccoefcients model can well describe the regularity of the aerodynamic coefcients, whichprovides the basic model for the next trajectory optimization. The comparison resultsshow that the bivariate aerodynamic coefcients model has the high accuracy. On thebasis of the aerodynamic model, the characteristic of the LDR and the reentry trajectoryof the hypersonic vehicle with the diferent LDR are analyzed. The reentry trajectoryoptimization and the guidance problems of the hypersonic vehicle are described.
2. The algorithm for solving the nonlinear optimal control problem with the pathconstraints is studied. The final state state integral constraints are nonlinear in the tradi-tional Gauss pseudospectral method (GPM). It isn’t in favour of the fast computation. Thecomputation of the costate doesn’t agree with those of the other pseudospectral methods.In order to increase the convergence rate, the nonlinear final state integral constraint istransformed into the linear one equivalently using the state diferential matrix and Gaussweights. Then, the number of the nonlinear constraints decreases. The improved Gauss pseudospectral method (IGPM) is proposed based on the linear final state integral con-straint. The new costate mapping theorem is proved to complete the GPM. The newcostate computation is the same with the other pseudospectral methods The simple com-putation of the initial costate is given to alleviate the complexity of the original method.Since the control doesn’t has the values at the boundaries time, the computations of theboundaries time are proposed based on the maximum principle considering the dynamicdiferential equations and path constraints.
3. The multiphase trajectory trajectory optimization method of the standard Bolzamultiphase nonlinear optimal control problem is investigated. The dimension of the d-iferential matrix becomes too big when the number of the discrete points increases forthe nonlinear optimal control problem with the complex path constraints. And the com-putation efciency decreases. Therefore, The multiphase improved Gauss pseudospectralmethod (MIGPM) is proposed based on the IGPM by ideal of the multiple phases. Themultiphase costate mapping theorem is proved to compute the costate easily. The meth-ods of the costates and the controls at the break points are given. In order to increasethe accuracy of the solution, the multiphase trajectory optimization algorithm is proposedbased on MIGPM. It is used to solve the reentry trajectory optimization of the maximaldownrange problem of the hypersonic vehicle. The accuracy and the convergence of thesolution are better than those of the global trajectory optimization method.
4. The dynamic trajectory optimization method with the ability of making the pathconstraints high accuracy and finding the discontinuous points is researched. The accura-cy of the path constraints isn’t good using the global trajectory optimization method forthe problem with the requirement of high precision and discontinuous control. The multi-phase trajectory optimization method hardly find the discontinuous points. The updatingmethods of the discrete points and the number of the phases are given by the compositiverelative error vector which is used as the judging conditions. Considering the satisfy-ing accuracy of the solution and the path constraints especially, the dynamic trajectoryoptimization method is proposed based on MIGPM. The examples illustrate that this al-gorithm has the ability of converging fast, finding the discontinuous points and dealingwith the path constraints.
5. The minimal fuel ascent trajectory optimization problems of the hypersonic vehi-cle are studied in the vertical plane and the three-dimension space and reentry trajectorymaximal downrange and crossrange problems of the hypersonic vehicle are solved. Theascent optimal trajectory in the vertical plane and the ascent trajectory with the turning operation are solved by the iterative algorithm one based on the IGPM and the dynamictrajectory optimization algorithm. The results show that the minimal fuel trajectories areboth the velocity first and ascent type. In order to increase the convergence and preci-sion, two global trajectory optimization iterative algorithms are put forward. The simula-tion results show that two iterative algorithms can converge to the optimal solution well.The reentry trajectory optimization problems with maximal downrange and crossrangeare solved by the dynamic trajectory optimization algorithm and latest hp adaptive pseu-dospectral method. The dynamic trajectory optimization algorithm behaves better thanthose of the hp adaptive pseudospectral method in the field of the computation efciency,the converging radius and the ability of dealing with the path constraints. The propertyof the reentry optimal LDR is analyzed and the optimal LDR is almost the same with themaximal LDR during the reentry middle phase. It provides the reference for choosing theinitial value of the reentry trajectory optimization problem.
6. The reentry on-line optimal feedback guidance problem of the hypersonic vehicleis researched. In order to suppress the model uncertainty and the disturbance, the methodof choosing the sample period is derived using the concept of the Caratheodory trajec-tory. The sampling period can guarantee that the tracking precision satisfies the errorrequirement under the condition of the computation delay. The on-line optimal feedbackguidance algorithm is put forward by the benefit of fast convergence of the pseudospec-tral method and the combination of the optimal control theory and the on-line optimalfeedback. The proposed trajectory optimization method and the on-line optimal feedbackguidance algorithm are combined to construct the framework of the reentry closed loopguidance system of the hypersonic vehicle, which meets the requirement of the on-linecomputation. It provides the solution for the reentry of the hypersonic vehicle with theautonomy, reliability and optimality. The on-line optimal feedback guidance method isapplied to the guidance problems，which are the maximal crossrange problem, the fixedpoints fixed direction attacking problem and the on-line autonomous obstacle-avoidancereentry guidance problem. The results show the proposed guidance framework can satisfythe requirement of the index.
In conclusion, this thesis puts forward the trajectory optimization methods and theon-line optimal feedback guidance method for hypersonic vehicle, which lay the founda-tion for the on-line trajectory optimization and high-accuracy guidance.
第一，研究了高超声速飞行器运动和气动建模问题。对于实际项目中的高超声速飞行器，给出了上升段轨迹优化问题的数学描述。在半速度坐标系下，建立了高超声速飞行器再入运动模型。通过分析气动系数数据，综合考虑攻角和速度两个因素，建立了双变量气动系数模型，通过对比，表明该气动模型具有更高的精度。在所建立的气动模型的基础上，分析了高超声速飞行器的升阻比特点，对不同升阻比下的轨迹进行了分析。提出了高超声速飞行器再入轨迹优化与制导问题的数学描述。
第二，研究了含有路径约束非线性最优控制问题的计算方法。传统的Gauss伪谱法中终端状态积分约束是非线性的，不利于计算，并且协态变量的计算方法与其他伪谱法不一致。为了提高收敛速度，利用状态微分方程矩阵和Gauss权重系数将非线性的终端状态积分约束等价转化为等价的线性约束形式，减少了非线性约束的个数，基于线性终端状态积分约束，提出了改进的Gauss伪谱法(IGPM)，证明了新的协态映射定理，进一步完善了Gauss伪谱法。新的协态变量计算方法与其他伪谱法具有统一形式。针对初始时刻协态变量不易计算的问题，提出了简单的计算初始时刻协态变量的方法；由于控制变量没有在边界时刻取值，根据极大值原理，考虑动态微分方程约束和路径约束，提出了端点时刻控制变量的计算方法。
第三，研究了标准的Bolza型多段非线性最优控制问题的多段轨迹优化方法。对于含有复杂路径约束的非线性最优控制问题，只增加离散点个数，导致微分矩阵维数增大，计算效率下降。因此，引入分段的思想，基于IGPM，提出了多段改进Gauss伪谱法(MIGPM)，证明了多段协态映射定理，可以方便地计算协态变量，并提出了分割点处协态和控制变量的计算方法。为了提高解的精度，提出了基于MIGPM的多段轨迹优化算法，并求解高超声速飞行器再入轨迹纵向航程最大问题，解的精度和收敛速度均优于全局轨迹优化方法。
第四，研究了能够高精度处理路径约束和确定间断点能力的动态轨迹优化方法。对于有高精度路径约束要求，并含有不连续控制的问题，全局轨迹优化方法对路径约束的精度不高，并且多段轨迹优化算法难以确定间断点。以综合相对误差向量作为参数更新的判断条件，提出离散点和阶段数量更新方法。重点考虑解和路径约束的满足精度，基于MIGPM提出了动态轨迹优化算法。通过实例验证了该算法的快速收敛性、寻找间断点和处理路径约束的能力。
第五，研究了竖直平面和三维空间的高超声速飞行器上升段最省燃料的轨迹优化问题，并研究了再入轨迹纵向航程和横向航程最大的轨迹优化问题。采用基于IGPM的迭代算法一和动态轨迹优化算法分别求解竖直平面上升段最优轨迹和上升段同时转向的最优轨迹，结果表明最省燃料的轨迹均属于先加速后上升的类型。为了提高计算速度和精度，提出了两种全局轨迹优化迭代算法，仿真结果表明，两种迭代算法都能很快地收敛到最优解。应用动态轨迹优化算法和最新的hp自适应伪谱法求解再入纵向航程和横向航程最大轨迹优化问题，结果表明，动态轨迹优化算法的计算效率、收敛半径和处理路径约束的能力都优于hp自适应伪谱法。分析了再入的最优升阻比特性，再入中段最优升阻比基本上等于最大升阻比，为再入轨迹优化问题的初值选取提供了借鉴。
第六，研究了高超声速飞行器再入过程在线最优反馈制导问题。为了处理模型中的不确定性和干扰，采用Carathe′odory π轨迹的概念，推导采样时间的取值方法，从而保证在存在计算延迟的条件下，系统的跟踪精度仍然能够满足误差要求。将最优控制与在线最优反馈相结合，利用IGPM快速收敛的特点，提出了在线最优反馈制导算法。将提出的轨迹优化方法与在线最优反馈制导算法相结合，提出了一种满足在线计算要求的高超声速飞行器再入闭环制导系统结构，为提高高超声速飞行器再入过程的自主性、可靠性和最优性提供了解决方案。将在线最优反馈制导算法应用于横向航程最大制导问题、定点定向攻击制导问题和在线自主避障再入制导问题，仿真结果表明该制导方法能够很好地满足指标要求。
综上所述，本文提出了高超声速飞行器轨迹优化方法和在线最优反馈制导方法，为高超声速飞行器轨迹在线优化和高精度制导奠定了一定基础。
In the past twenty years, near space is focused by some great countries. It is the idealspace to realize the hypersonic flying because of the thin air and the small resistance. Thehypersonic vehicle not only has the ability to entry the space but also can reentry the at-mosphere for attacking the ground targets fast. It has the big lift-to-drag ratio (LDR), wellmaneuverability, fast reaching the global world and great power, etc. It can be used in thefields of the global real-time scout, rapid deployment and remote precision attacking andimpacts the future war greatly. Therefore, much researches on the related technology andtheory of the hypersonic vehicle have won wide studies. It is the essential preconditionfor the mission to realize the trajectory optimization and high-precision guidance. Thisproblem is difcult to solve because of the nonlinear property of the motion equations,complexity of the environment and various nonlinear constraints. The thesis takes someresearches on this problem.
1. The motion models and the aerodynamic model of the hypersonic vehicle arediscussed. The mathematical descriptions of the ascent trajectory optimization problem ofthe hypersonic vehicle are given for the engineering problems. The reentry motion modelis presented under the half velocity coordinate system. Considering the attacking angleand velocity variables, the bivariate aerodynamic coefcients model is built by analyzingthe characteristics of the data, The comparison shows that the proposed aerodynamiccoefcients model can well describe the regularity of the aerodynamic coefcients, whichprovides the basic model for the next trajectory optimization. The comparison resultsshow that the bivariate aerodynamic coefcients model has the high accuracy. On thebasis of the aerodynamic model, the characteristic of the LDR and the reentry trajectoryof the hypersonic vehicle with the diferent LDR are analyzed. The reentry trajectoryoptimization and the guidance problems of the hypersonic vehicle are described.
2. The algorithm for solving the nonlinear optimal control problem with the pathconstraints is studied. The final state state integral constraints are nonlinear in the tradi-tional Gauss pseudospectral method (GPM). It isn’t in favour of the fast computation. Thecomputation of the costate doesn’t agree with those of the other pseudospectral methods.In order to increase the convergence rate, the nonlinear final state integral constraint istransformed into the linear one equivalently using the state diferential matrix and Gaussweights. Then, the number of the nonlinear constraints decreases. The improved Gauss pseudospectral method (IGPM) is proposed based on the linear final state integral con-straint. The new costate mapping theorem is proved to complete the GPM. The newcostate computation is the same with the other pseudospectral methods The simple com-putation of the initial costate is given to alleviate the complexity of the original method.Since the control doesn’t has the values at the boundaries time, the computations of theboundaries time are proposed based on the maximum principle considering the dynamicdiferential equations and path constraints.
3. The multiphase trajectory trajectory optimization method of the standard Bolzamultiphase nonlinear optimal control problem is investigated. The dimension of the d-iferential matrix becomes too big when the number of the discrete points increases forthe nonlinear optimal control problem with the complex path constraints. And the com-putation efciency decreases. Therefore, The multiphase improved Gauss pseudospectralmethod (MIGPM) is proposed based on the IGPM by ideal of the multiple phases. Themultiphase costate mapping theorem is proved to compute the costate easily. The meth-ods of the costates and the controls at the break points are given. In order to increasethe accuracy of the solution, the multiphase trajectory optimization algorithm is proposedbased on MIGPM. It is used to solve the reentry trajectory optimization of the maximaldownrange problem of the hypersonic vehicle. The accuracy and the convergence of thesolution are better than those of the global trajectory optimization method.
4. The dynamic trajectory optimization method with the ability of making the pathconstraints high accuracy and finding the discontinuous points is researched. The accura-cy of the path constraints isn’t good using the global trajectory optimization method forthe problem with the requirement of high precision and discontinuous control. The multi-phase trajectory optimization method hardly find the discontinuous points. The updatingmethods of the discrete points and the number of the phases are given by the compositiverelative error vector which is used as the judging conditions. Considering the satisfy-ing accuracy of the solution and the path constraints especially, the dynamic trajectoryoptimization method is proposed based on MIGPM. The examples illustrate that this al-gorithm has the ability of converging fast, finding the discontinuous points and dealingwith the path constraints.
5. The minimal fuel ascent trajectory optimization problems of the hypersonic vehi-cle are studied in the vertical plane and the three-dimension space and reentry trajectorymaximal downrange and crossrange problems of the hypersonic vehicle are solved. Theascent optimal trajectory in the vertical plane and the ascent trajectory with the turning operation are solved by the iterative algorithm one based on the IGPM and the dynamictrajectory optimization algorithm. The results show that the minimal fuel trajectories areboth the velocity first and ascent type. In order to increase the convergence and preci-sion, two global trajectory optimization iterative algorithms are put forward. The simula-tion results show that two iterative algorithms can converge to the optimal solution well.The reentry trajectory optimization problems with maximal downrange and crossrangeare solved by the dynamic trajectory optimization algorithm and latest hp adaptive pseu-dospectral method. The dynamic trajectory optimization algorithm behaves better thanthose of the hp adaptive pseudospectral method in the field of the computation efciency,the converging radius and the ability of dealing with the path constraints. The propertyof the reentry optimal LDR is analyzed and the optimal LDR is almost the same with themaximal LDR during the reentry middle phase. It provides the reference for choosing theinitial value of the reentry trajectory optimization problem.
6. The reentry on-line optimal feedback guidance problem of the hypersonic vehicleis researched. In order to suppress the model uncertainty and the disturbance, the methodof choosing the sample period is derived using the concept of the Caratheodory trajec-tory. The sampling period can guarantee that the tracking precision satisfies the errorrequirement under the condition of the computation delay. The on-line optimal feedbackguidance algorithm is put forward by the benefit of fast convergence of the pseudospec-tral method and the combination of the optimal control theory and the on-line optimalfeedback. The proposed trajectory optimization method and the on-line optimal feedbackguidance algorithm are combined to construct the framework of the reentry closed loopguidance system of the hypersonic vehicle, which meets the requirement of the on-linecomputation. It provides the solution for the reentry of the hypersonic vehicle with theautonomy, reliability and optimality. The on-line optimal feedback guidance method isapplied to the guidance problems，which are the maximal crossrange problem, the fixedpoints fixed direction attacking problem and the on-line autonomous obstacle-avoidancereentry guidance problem. The results show the proposed guidance framework can satisfythe requirement of the index.
In conclusion, this thesis puts forward the trajectory optimization methods and theon-line optimal feedback guidance method for hypersonic vehicle, which lay the founda-tion for the on-line trajectory optimization and high-accuracy guidance.
引文
[1] Moses P L, Rausch V L, Nguyen L T, et al. NASA Hypersonic Flight Demon-strators—overview, Status, and Future Plans[J]. Acta Astronautica,2004,55(3-9):619–630.
[2] McClinton C R, Rausch V L, Sitz J, et al. Hyper-X Program Status[C]//10thAIAA/NAL/NASD-ISAS International Space Planes and Hypersonic Systems andTechnologies Conference. Kyoto: AIAA Press, AIAA Paper No.2001-1910,2001:1–11.
[3] Hueter U, McClinton C R. NASA’s Advanced Space Transportation HypersonicProgram[C]//11th AIAA/AAAF International Conference. Orleans, France: AIAAPress, AIAA Paper No.2002-5175,2002:1–16.
[4] Curran E T. Scramjet Engines: the First Forty Years[J]. Journal of Propulsion andPower,2001,17(6):1138–1148.
[5] Randall T V, Lawrence D H, Charles R M. X-43A Hypersonic Vehicle TechnologyDevelopment[J]. Acta Astronautica,2006,59(1-5):181–191.
[6]黄伟,罗世彬,柳军.基于乘波技术的高超声速巡航飞行器发展趋势与管理模式[J].导弹与航天运载技术,2009,302(4):23–29.
[7]李菁菁,任章,黎科峰.高超声速飞行器再入段的动力学建模与仿真[J].系统仿真学报,2009,21(2):534–537.
[8]李佳峰,陈万春.带约束的高超声速飞行器最优弹道的数值算法[J].航空学报,2009,30(9):1605–1611.
[9]黄显林,葛东明.输入受限高超声速飞行器鲁棒变增益控制[J].系统工程与电子技术,2011,33(8):1829–1836.
[10]马广富,佘智勇.基于改进型伴随方法的高超声速飞行器上升段轨迹优化[J].固体火箭技术,2011,34(3):273–276.
[11]马辉,袁建平,方群.高超声速飞行器最优周期巡航轨迹优化[J].飞行力学,2008,26(4):31–34.
[12]康炳南,唐硕.基于非线性规划的高超声速滑翔轨迹优化[J].飞行力学,2008,26(3):49–52.
[13]孟中杰,黄攀峰,闫杰.基于改进稀疏A*算法的高超声速飞行器航迹规划技术[J].西北工业大学学报,2010,28(2):182–186.
[14]刘鹏,谷良贤.高超声速飞行器动态输出反馈最优跟踪控制[J].哈尔滨工业大学学报,2011,43(7):131–134.
[15]雍恩米,唐国金,陈磊.基于Gauss伪谱方法的高超声速飞行器再入轨迹快速优化[J].宇航学报,2008,29(6):1766–1772.
[16]雍恩米.高超声速滑翔式再入飞行器轨迹优化与制导方法研究[D].长沙:国防科学技术大学,2008:1–170.
[17] Tian B L, Zong Q. Optimal Guidance for Reentry Vehicles Based on IndirectLegendre Pseudospectral Method[J]. Acta Astronautica,2011,68:1176–1184.
[18]宗群,田苓,窦立谦.基于Gauss伪谱法的临近空间飞行器上升段轨迹优化[J].宇航学报,2010,31(7):1775–1781.
[19]郑总准,谢富强,王永骥.高超声速飞行器多约束参考轨迹快速规划算法[J].计算技术与自动化,2009,28(1):88–91.
[20]谭湘敏,易建强,赵冬斌, et al.高超声速飞行器轨迹跟踪控制仿真研究[J].系统仿真学报,2011,23(4):745–749.
[21] Todd H G. Advancement and Analysis of a Gauss Pseudospectral Transcription forOptimal Control Problems[D]. Cambridge,MA: Massachusetts Institute of Tech-nology,2007:23–32.
[22] Betts J T. Survey of Numerical Methods for Trajectory Optimization[J]. Journal ofGuidance, Control, and Dynamics,1998,21(2):193–207.
[23] Ross I M, Fahroo F. A Perspective on Methods for Trajectory Optimiza-tion[C]//AIAA/AAS Astrodynamics Specialist Conference and Exhibit. Monterey:AIAA Press, AIAA Paper No.2002-4727,2002:1–7.
[24]李佳峰,周浩,陈万春.高超声速飞行器滑行段最优弹道的间接算法[J].飞行力学,2009,27(3):41–44.
[25]李惠峰,李昭莹.高超声速飞行器上升段最优制导间接法研究[J].宇航学报,2011,32(2):297–301.
[26] Betts J T, Hufman W P. Application of Sparse Nonlinear Programming to Trajec-tory Optimization[J]. J. Guidance,1992,15(1):198–206.
[27] Betts J T, Frank P D. A Sparse Nonlinear Optimization Algorithm[J]. J. Optim.Theory and Applics.,1994,82(5):519–541.
[28]陈刚,万自明.基于遗传算法的RLV再入轨迹优化设计[J].系统工程与电子技术,2006,28(8):1240–1243.
[29] Gill P E, Murray W, Saunders M A. SNOPT: An SQP Algorithm for LargeScaleConstrained Optimization[J]. SIAM Journal on Optimization,2002,12(4):979–1006.
[30] A. S. Theory and Implementation of Numerical Methods Based on Runge-KuttaIntegration for Solving Optimal Control Problems[D]. Berkeley: University ofCalifornia,1996:1–100.
[31] Teo K L, Goh C J, Wong K H. A Unified Computational Approach to OptimalControl Problems[M].1nd. HongKong: Longman Scientific and Technical,1991:10–200.
[32] Hager W W. Runge-Kutta Methods in Optimal Control and the Transformed Ad-joint System[J]. Numer. Math.,2000,87:247–282.
[33] Seywald H. Trajectory Optimization Based on Diferential Inclusion[J]. Journal ofGuidance,Control and Dynamics,1994,17(3):780–785.
[34] Conway B A, Larson K M. Collocation Versus Diferential Inclusion in DirectOptimization[J].1998,21(5):780–785.
[35] Enright P J, Conway B A. Discrete Approximations to Optimal TrajectoriesUsing Direct Transcription and Nonlinear Programming[J]. Journal of Guid-ance,Control,and Dynamics,1992,15(4):994–1002.
[36] Reddien G W. Collocation at Gauss Points as a Discretization in Optimal Con-trol[J]. SIAM Journal of Control and Optimization,1979,17(2):1–6.
[37] Cuthrell J E, Biegler L T. Simultaneous Optimization and Solution Methods forBatch Reactor Control Profiles[J]. Computers and Chemical Engineering,1989,13(1):49–62.
[38] Elnagar G, Kazemi M A, Razzaghi M. The Pseudospectral Legendre Method forDiscretizing Optimal Control Problems[J]. IEEE Transactions On Automatic Con-trol,1995,40(10):1793–1796.
[39] Vlassenbroeck J. A Chebyshev Polynomial Method for Optimal Control with StateConstraints[J]. Automatica,1988,24(4):499–506.
[40] Fahroo F, Ross I M. Costate Estimation by a Legendre Pseudospectral Method[J].2001,24(2):270–277.
[41] Ross I M, Fahroo F. Legendre Pseudospectral Approximations of Optimal ControlProblems[J]. Lecture Notes in Control and Information Sciences,2003,295:327–342.
[42] Benson D A. A Gauss Pseudospectral Transcription for Optimal Control[D]. Cam-bridge, MIT: Massachusetts Institute of Technology,2005:49–55.
[43] Benson D A, Huntington G T, Thorvaldsen T P, et al. Direct Trajectory Optimiza-tion and Costate Estimation via an Orthogonal Collocation Method[J]. Journal ofGuidance,Control and Dynamics,2006,29(6):1435–1440.
[44] Huntington G T. Advancement and Analysis of a Gauss Pseudospectral Transcrip-tion for Optimal Control Problems[D]. Cambridge, MIT: Massachusetts Instituteof Technology,2007:50–85.
[45] Huntington G T, Rao A V. Optimal Spacecraft Formation Configuration Using aGauss Pseudospectral Method[C]//Proceedings of the2005AAS/AISS SpaceflightMechanics Meeting. Copper Mountain, Colorado: AIAA Press,2005:1–6.
[46] Zhang M R, Sun Y, Duan G R. Reentry Trajectory Optimization of HypersonicVehicle with Enhancing Parametrization Method[C]//Proceedings of the3rd In-ternational Symposium on Systems and Control in Aeronautics and Astronautics.Harbin: China,2010:40–45.
[47] Zheng Z Z, Yang Y, Wang Y J, et al. Entry Trajectory Genetic Algorithm Opti-mization Using Legendre Pseudospectral Method[C]//Proceedings of the2010In-ternational Conference on Modelling, Identification and Control. Okayama, Japan:Japan Press,2010:505–510.
[48]周文雅,杨涤,李顺利.利用高斯伪谱法求解具有最大横程的再入轨迹[J].系统工程与电子技术,2010,32(5):1038–1042.
[49] Ma G F, Zhuang Y F, Li C J, et al. Pseudospectral Method for Optimal Motion Plan-ning of a Rigid Underactuated Spacecraft[C]//20108th IEEE International Confer-ence on Control and Automation. Xiamen, China: IEEE Press,2010:684–688.
[50] Jorris T R, Cobb R G. Three-Dimensional Trajectory Optimization Satisfying Way-point and No-Fly Zone Constraints[J]. Journal of Guidance,Control and Dynamics,2009,32(2):551–572.
[51]水尊师,周军,葛致磊.基于高斯伪谱方法的再入飞行器预测校正制导方法研究[J].宇航学报,2006,32(6):1249–1255.
[52] Jorris T R. Constrained Trajectory Optimization Using Pseudospectral Method-s[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit. Honolulu,Hawaii: AIAA Press, AIAA Paper No.2008-6218,2008:1–33.
[53] Ross I M, D’Souza C, Fahroo F, et al. A Fast Approach to Multi-Stage Launch Ve-hicle Trajectory Optimization[C]//AIAA Guidance, Navigation, and Control Con-ference and Exhibit. Austin, Texas: AIAA Press, AIAA Paper No.2003-5639,2003:1–9.
[54] Bollino K P. High-Fidelity Real-Time Trajectory Optimization for ReusableLaunch Vehicles[D]. California:: Naval Postgraduate School,2006:70–105.
[55] Ross I M. User’s Manual for DIDO: A MATLAB Application Package for SolvingOptimal Control Problems[J]. Tomlab Optimization,2004:1–86.
[56] Kang W, Bedrossian N. Pseudospectral Optimal Control Theory Makes DebutFlight-Saves NASA$1M in Under3Hours[J]. SIAM News,2007,40(7):1–5.
[57] Bedrossian N, Bhatt S, Lammers M, et al. Zero Propellant Maneuver Flight Resultsfor180ISS Rotation[C]//2007International Symposium on Space Flight Dynam-ics. Xiamen, China: USA Press,2007:684–688.
[58] Garg D, Patterson M A, Darby C L, et al. Direct Trajectory Optimizationand Costate Estimation of General Optimal Control Problems Using a RadauPseudospectralMethod[C]//AIAA Guidance, Navigation, and Control Conference.Chicago, Illinois: AIAA Press, AIAA Paper No.2009-5989,2009:1–29.
[59] Fahroo F, Ross I M. Direct Trajectory Optimization by a Chebyshev PseudospectralMethod[J]. Journal of Guidance,Control,and Dynamics,2002,25(1):160–166.
[60] H. P. Ma T Q, Zhang W. An Efcient Chebyshev Algorithm for the Solution ofOptimal Control Problems[J]. IEEE Transactions on Automatic Control,2011,56(3):675–680.
[61] Gong Q, Ross I M, Fahroo F. Costate Computation by a Chebyshev PseudospectralMethod[J]. Journal of Guidance,Control,and Dynamics,2010,33(2):623–628.
[62] Fahroo F, Ross I M. Trajectory Optimization by Indirect Spectral Colloca-tion Methods[C]//Proceedings of the AIAA/AAS Astrodynamics Specialist Con-ference. Denver, CO: AIAA Press, AIAA Paper No.2000-4028,2000:1–6.
[63] Yan H, Fahroo F, Ross I M. Optimal feedback control law by Legendre pseu-dospectral approximations[C]//Proceedings of the American Control Conference.Arlington, VA: AIAA Press, AIAA Paper No.2000-4028,2001:2388–2393.
[64] Huntington G T, Benson D, Rao A V. A Comparison of Accuracy and Compu-tational Efciency of Three Pseudsopectral Methods[C]//AIAA Guidance, Navi-gation and Control Conference and Exhibit. South Carolina: AIAA Press, AIAAPaper No.2007-6405,2007:1–25.
[65] Garg D, Patterson M, Hager W W, et al. A Unified Framework for the NumericalSolution of Optimal Control Problems Using Pseudospectral Methods[J]. Auto-matica,2010,46(1):1843––1851.
[66] Ross I M, Fahroo F. Pseudospectral Knotting Methods for Solving Optimal ControlProblems[J]. Journal of Guidance,Control and Dynamics,2004,27(3):397–405.
[67] Ross I M, Fahroo F. Advances In Pseudospectral Methods[C]//AIAA Guidance,Navigation, and Control Conference. Honolulu, Hawaii: AIAA Press, AIAA PaperNo.2008-7309,2008:1–23.
[68] Ross I M, Fahroo F. Convergence of the Costates do not Imply Convergence of theControls[J]. Journal of Guidance,Control and Dynamics,2004,31(4):1492–1497.
[69] Gong Q, Kang W, Ross I M. A Pseudospectral Method for the Optimal Controlof Constrained Feedback Linearizable Systems[J]. IEEE Trans. Autom. Control,2006,51(7):1115––1129.
[70] Gong Q, Kang W, Fahroo F. Connections Between the Covector Mapping Theo-rem and Convergence of Pseudospectral Methods for Optimal Control[J]. ComputOptim Appl,2008,41(1):307––335.
[71] Darby C L, Hager W W, Rao A V. A Preliminary Analysis of a Variable-OrderApproach to Solving Optimal Control Problems Using Pseudospectral Method-s[C]//AIAA/AAS Astrodynamics Specialist Conference. Toronto, Ontario Canada:AIAA Press, AIAA Paper No.2010-8272,2010:1–14.
[72] Cheng P, Shen Z J, Lavalle S M. RRT-Based Trajectory Design for AutonomousAutomobiles and Spacecraft[J]. Archives of Control Sciences,2001,11(3-4):167–194.
[73] Frazzoli E, Dahleh M A, Feron E. Real-Time Motion Planning for AgileAutonomous Vehicles[J]. Journal of Guidance,Control and Dynamics,2002,25(1):116–129.
[74]陈刚,胡莹.基于NSGA-II算法的RLV多目标再入轨迹优化设计[J].西北工业大学学报,2006,24(2):133–137.
[75] Rajesh K A. Reentry Trajectory Optimization: Evolutionary Approach[C]//9thAIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. At-lanta: AIAA Press, AIAA Paper No.2002-5466,2002:1–5.
[76] Yokoyama N, Suzuki S. Trajectory Optimization Via Modified Genetic Algorith-m[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit. Austin,Texas: AIAA Press, AIAA Paper No.2003-5493,2003:1–9.
[77] Ross I M, Fahroo F. Issues in the Real-time Computation of Optimal Control[J].Mathematical and Computer Modeling,2006,43(9):1172–1188.
[78] Gong Q, Kang W, Ross I M. A Pseudospectral Method for the Optimal Control ofConstrained Feedback Linearizable Systems[J]. IEEE Transactions on AutomaticControl,2006,51(7):1115–1129.
[79] Gong Q, Fahroo F, Ross I M. Spectral Algorithm for Pseudospectral Methods inOptimal Control[J]. Journal of Guidance,Control and Dynamics,2008,31(3):460–471.
[80] Darby C L, Garg D, Rao A V. Costate Estimation Using Multiple-Interval Pseu-dospectral Methods[C]//AIAA Guidance, Navigation, and Control Conference.Portland, Oregon: AIAA Press, AIAA Paper No.2011-6571,2011:1–24.
[81] Darby C L, Garg D, Rao A V. Costate Estimation Using Multiple-Interval Pseu-dospectral Methods[J]. Journal of Spacecraft and Rockets,2011,48(5):856–866.
[82]洪蓓,辛万青.基于hp自适应伪谱法的滑翔弹道快速优化设计[C]//Proceedings of the30th Chinese Control Conference. Yantai: China,2011:1956–1961.
[83] Zhao Y, Tsiotras P. A Density-Function Based Mesh Refinement Algorithm forSolving Optimal Control Problems[C]//Proceedings of the Infotech and AerospaceConference. Seattle, WA: AIAA press,2009:1956–1961.
[84] Zhao Y, Tsiotras P. Density Functions for Mesh Refinement in Numerical OptimalControl[J]. Journal of Guidance,Control and Dynamics,2011,34(1):271–277.
[85] Darby C L, Hager W W, Rao A V. An Hp-Adaptive Pseudospectral Method forSolving Optimal Control Problems[J]. Optimal Control Applications and Methods,2011,32:476–502.
[86] Darby C L, Hager W W, Rao A V. Direct Trajectory Optimization Using a Vari-able Low-Order Adaptive Pseudospectral Method[J]. Journal of Spacecraft andRockets,2011,48(3):433–445.
[87] Lu P. Entry Guidance and Trajectory Control for Reusable Launch Vehicle[J].Journal of Guidance,Control and Dynamics,1997,20(1):143–149.
[88] Lu P, Hanson J M. Entry guidance for X-33Vehicle[J]. Journal of Spacecraft andRockets,1998,35(3):342–349.
[89] Grimm W, Meulen J G, Roenneke A J. Optimal Update Scheme for Drag ReferenceProfile in an Entry Guidance[J]. Journal of Guidance,Control and Dynamics,2003,26(5):695–701.
[90] Jouhaud F. Closed Loop Reentry Guidance Law of a Space Plane:Application toHermes[J]. Acta Astronautica,1992,26(8-10):577–585.
[91] Harpold J C, Graves C A. Shuttle Entry Guidance[J]. Journal of AstronauticalSciences,1979,27(3):239–268.
[92]赵汉元.航天器再入制导方法综述[J].航天控制,1994,1:26–33.
[93]闫晓东,唐硕.基于反馈线性化的H-V返回轨道跟踪方法[J].宇航学报,2008,29(5):1546–1550.
[94] Sheu D L, Chen Y M, Chen J S. Optimal Three-Dimensional Glide for MaximumReachable Domain[C]//AIAA Atmospheric Flight Mechanics Conference and Ex-hibit. Portland, OR: AIAA Press, AIAA Paper No.1998-4462,1999:9–11.
[95]赵汉元.飞行器再入动力学与制导[M].第一版.长沙:国防科技大学出版社,1997:1–482.
[96] Vasile I. Optimal Skip Entry Into Atmosphere[C]//AIAA Atmospheric Flight Me-chanics Conference and Exhibit. Portland, OR: AIAA Press, AIAA Paper No.1999-4170,1999:806–813.
[97] Rao A V, Mease K D. Dichotomic Basis Approach To Solving HypersensitiveOptimal Control Problems[J]. Automatica,1999,35:633–642.
[98]周浩,陈万春,殷兴良.高超声速飞行器时间最短弹道优化设计[J].弹道学报,2007,19(4):26–29.
[99]周浩,陈万春,殷兴良.高超声速飞行器滑行航迹优化[J].北京航空航天大学学报,2006,32(5):513–517.
[100]闫晓东,唐硕.亚轨道飞行器返回轨道设计方法研究[J].宇航学报,2008,29(2):467–471.
[101] Roenneke A J. Adaptive On-board Guidance for Entry Vehicle[C]//AIAA Guid-ance, Navigation, and Control Conference and Exhibit,. Montreal, Canada: AIAAPress, AIAA Paper No.2001-4048,2001:1–11.
[102] Mease K D, Chen D T, Teufel P, et al. Reduced-Order Entry Entry TrajectoryPlanning for Acceleration[J]. Journal of Guidance,Control and Dynamics,2002,25(2):257–266.
[103]王琦,丁运亮.可重复使用运载器再入轨迹次实时优化[J].北京航空航天大学学报,2007,33(4):405–408.
[104] Shen Z J, Lu P. On-Board Generation of Three-Dimensional Constrained EntryTrajectories[C]//AIAA Guidance, Navigation, and Control Conference and Exhib-it. Monterey, California: AIAA Press, AIAA Paper No.2002-4455,2002:1–16.
[105] Shen Z J, Lu P. On-Board Entry Trajectory Planning Expanded To Sub-OrbitalFlight[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit.Austin, Texas: AIAA Press, AIAA Paper No.2003-5736,2003:1–13.
[106] Dukeman G A. Profile-Following Entry Guidance Using Linear Quadratic Regu-lator Theory[C]//AIAA Guidance, Navigation, and Control Conference. Monterey,CA: AIAA Press, AIAA Paper No.2002-4457,2002:1–10.
[107] Zimmerman C, Dukeman G A, Hanson J M. Automated Method to Com-pute Orbital Reentry Trajectories with Heating Constraints[J]. Journal of Guid-ance,Control and Dynamics,2003,26(4):523–529.
[108]方群,李新三.临近空间高超声速无动力滑翔飞行器最优轨迹设计及制导研究[J].宇航学报,2008,29(5):1485–1481.
[109] Hussein Y, Rajiv S C, Howard L, et al. Predictor-Corrector Entry Guidance forReusable Launch Vehicles[C]//AIAA Guidance, Navigation, and Control Confer-ence and Exhibit. Montreal,Canada: AIAA Press, AIAA Paper No.2001-4043,2001:2–17.
[110] Xue S B, Lu P. Constrained Predictor-Corrector Entry Guidance[C]//AIAA Guid-ance, Navigation, and Control Conference. Chicago, Illinois: AIAA Press, AIAAPaper No.2009-5767,2001:2–17.
[111]潘乐飞,李新国.可重复使用运载器预测-校正再入制导研究[J].飞行力学,2007,25(1):55–58.
[112] Joshi A, Sivan K, Amma S S. Predictor-Corrector Reentry Guidance Algorith-m with Path Constraints for Atmospheric Entry Vehicles[J]. Journal of Guid-ance,Control and Dynamics,2007,30(5):1307–1318.
[113] Ross I M, Fahroo F. A Unified Computational Framework for Real-Time OptimalControl[C]//Proceedings of the42nd IEEE Conierence on Decision and Control.Maui, Hawaii USA: IEEE Press,2003:2210–2215.
[114] Ross I M, Sekhavat P, Fleming A, et al. Pseudospectral Feedback Control: Foun-dations, Examples and Experimental Results[C]//Proceedings of the2006AIAAGuidance, Navigation, and Control Conference and Exhibit. Keystone, Colorado:AIAA Press, AIAA Paper No.2006-6354,2006:1–23.
[115] Ross I M, Gong Q, Fahroo F, et al. Practical Stabilization Through Real-TimeOptimal Control[C]//Proceedings of the2006American Control Conference. Min-neapolis, Minnesota, USA: IEEE press,2006:304–309.
[116] Ross I M, Sekhavat P, Fleming A, et al. Optimal Feedback Control: Foundation-s, Examples,and Experimental Results for a New Approach[J]. Journal of Guid-ance,Control and Dynamics,2008,31(2):1307–321.
[117] Bollino K, Ross I M, Doman D. Optimal Nonlinear Feedback Guidance for Reen-try Vehicles[C]//AIAA Guidance, Navigation, and Control Conference. Keystone,CO: AIAA Press, AIAA Paper No.2006-6074,2006:1–20.
[118] Bollino K P, Ross I M. A Pseudospectral Feedback Method for Real-Time Opti-mal Guidance of Reentry Vehicles[C]//Proceedings of the2007American ControlConference. New York City, USA: IEEE Press,2007:3861–3867.
[119] Karpenko M, Sekhavat P, Park J, et al. A Pseudospectral Feedback Method forReal-Time Optimal Guidance of Reentry Vehicles[C]//AIAA Guidance, Naviga-tion, and Control Conference. Toronto, Ontario Canada: AIAA Press, AIAA PaperNo.2010-8051,2010:1–11.
[120] P. Lu andH. S. Sun B T. Closed-loop endoatmospheric ascent guidance[J]. Journalof Guidance,Control, and Dynamics,2003,26(2):283–294.
[121] Dukeman G A. Atmospheric ascent guidance for rocket-powered launch vehi-cles[C]//AIAA Guidance, Navigation, and Conrol Conference and Exhibit. Mon-terey, California: AIAA Press, AIAA Paper No. AIAA2002-4559,2002:1–11.
[122] Lu P, Pan B. Highly Constrained Optimal Launch Ascent Guidance[C]//AIAAGuidance, Navigation, and Control Conference and Exhibit. Monterey, California:AIAA Press, AIAA Paper No.2009-5961,2009:1–26.
[123]王厚庆,何国强,刘佩进. RBCC为动力的巡航飞行器轨迹与质量分析[J].西北工业大学学报,2006,24(6):774–777.
[124]闫海江,唐硕,泮斌峰.亚轨道飞行器上升段轨迹快速生成方法研究[J].科学技术与工程,2011,11(10):2266–2270.
[125] Jorris T R. Common Aero Vehicle Autonomous Reentry Trajectory OptimizationSatisfying Waypoint and No-Fly Zone Constraints[D]. Wright-Patterson: Air ForceInstitute of Technology,2007:88–94.
[126] Jarlebring E, Damm T. The Lambertwfunction and the Spectrum of Some Multi-dimensional Time-Delay Systems[J]. Automatica,2007,43:2124–2128.
[127] Clarke F. Lyapunov Functions and Discontinuous Stabilizing Feedback[J]. AnnualReviews in Control,2011,35(1):13–33.
[2] McClinton C R, Rausch V L, Sitz J, et al. Hyper-X Program Status[C]//10thAIAA/NAL/NASD-ISAS International Space Planes and Hypersonic Systems andTechnologies Conference. Kyoto: AIAA Press, AIAA Paper No.2001-1910,2001:1–11.
[3] Hueter U, McClinton C R. NASA’s Advanced Space Transportation HypersonicProgram[C]//11th AIAA/AAAF International Conference. Orleans, France: AIAAPress, AIAA Paper No.2002-5175,2002:1–16.
[4] Curran E T. Scramjet Engines: the First Forty Years[J]. Journal of Propulsion andPower,2001,17(6):1138–1148.
[5] Randall T V, Lawrence D H, Charles R M. X-43A Hypersonic Vehicle TechnologyDevelopment[J]. Acta Astronautica,2006,59(1-5):181–191.
[6]黄伟,罗世彬,柳军.基于乘波技术的高超声速巡航飞行器发展趋势与管理模式[J].导弹与航天运载技术,2009,302(4):23–29.
[7]李菁菁,任章,黎科峰.高超声速飞行器再入段的动力学建模与仿真[J].系统仿真学报,2009,21(2):534–537.
[8]李佳峰,陈万春.带约束的高超声速飞行器最优弹道的数值算法[J].航空学报,2009,30(9):1605–1611.
[9]黄显林,葛东明.输入受限高超声速飞行器鲁棒变增益控制[J].系统工程与电子技术,2011,33(8):1829–1836.
[10]马广富,佘智勇.基于改进型伴随方法的高超声速飞行器上升段轨迹优化[J].固体火箭技术,2011,34(3):273–276.
[11]马辉,袁建平,方群.高超声速飞行器最优周期巡航轨迹优化[J].飞行力学,2008,26(4):31–34.
[12]康炳南,唐硕.基于非线性规划的高超声速滑翔轨迹优化[J].飞行力学,2008,26(3):49–52.
[13]孟中杰,黄攀峰,闫杰.基于改进稀疏A*算法的高超声速飞行器航迹规划技术[J].西北工业大学学报,2010,28(2):182–186.
[14]刘鹏,谷良贤.高超声速飞行器动态输出反馈最优跟踪控制[J].哈尔滨工业大学学报,2011,43(7):131–134.
[15]雍恩米,唐国金,陈磊.基于Gauss伪谱方法的高超声速飞行器再入轨迹快速优化[J].宇航学报,2008,29(6):1766–1772.
[16]雍恩米.高超声速滑翔式再入飞行器轨迹优化与制导方法研究[D].长沙:国防科学技术大学,2008:1–170.
[17] Tian B L, Zong Q. Optimal Guidance for Reentry Vehicles Based on IndirectLegendre Pseudospectral Method[J]. Acta Astronautica,2011,68:1176–1184.
[18]宗群,田苓,窦立谦.基于Gauss伪谱法的临近空间飞行器上升段轨迹优化[J].宇航学报,2010,31(7):1775–1781.
[19]郑总准,谢富强,王永骥.高超声速飞行器多约束参考轨迹快速规划算法[J].计算技术与自动化,2009,28(1):88–91.
[20]谭湘敏,易建强,赵冬斌, et al.高超声速飞行器轨迹跟踪控制仿真研究[J].系统仿真学报,2011,23(4):745–749.
[21] Todd H G. Advancement and Analysis of a Gauss Pseudospectral Transcription forOptimal Control Problems[D]. Cambridge,MA: Massachusetts Institute of Tech-nology,2007:23–32.
[22] Betts J T. Survey of Numerical Methods for Trajectory Optimization[J]. Journal ofGuidance, Control, and Dynamics,1998,21(2):193–207.
[23] Ross I M, Fahroo F. A Perspective on Methods for Trajectory Optimiza-tion[C]//AIAA/AAS Astrodynamics Specialist Conference and Exhibit. Monterey:AIAA Press, AIAA Paper No.2002-4727,2002:1–7.
[24]李佳峰,周浩,陈万春.高超声速飞行器滑行段最优弹道的间接算法[J].飞行力学,2009,27(3):41–44.
[25]李惠峰,李昭莹.高超声速飞行器上升段最优制导间接法研究[J].宇航学报,2011,32(2):297–301.
[26] Betts J T, Hufman W P. Application of Sparse Nonlinear Programming to Trajec-tory Optimization[J]. J. Guidance,1992,15(1):198–206.
[27] Betts J T, Frank P D. A Sparse Nonlinear Optimization Algorithm[J]. J. Optim.Theory and Applics.,1994,82(5):519–541.
[28]陈刚,万自明.基于遗传算法的RLV再入轨迹优化设计[J].系统工程与电子技术,2006,28(8):1240–1243.
[29] Gill P E, Murray W, Saunders M A. SNOPT: An SQP Algorithm for LargeScaleConstrained Optimization[J]. SIAM Journal on Optimization,2002,12(4):979–1006.
[30] A. S. Theory and Implementation of Numerical Methods Based on Runge-KuttaIntegration for Solving Optimal Control Problems[D]. Berkeley: University ofCalifornia,1996:1–100.
[31] Teo K L, Goh C J, Wong K H. A Unified Computational Approach to OptimalControl Problems[M].1nd. HongKong: Longman Scientific and Technical,1991:10–200.
[32] Hager W W. Runge-Kutta Methods in Optimal Control and the Transformed Ad-joint System[J]. Numer. Math.,2000,87:247–282.
[33] Seywald H. Trajectory Optimization Based on Diferential Inclusion[J]. Journal ofGuidance,Control and Dynamics,1994,17(3):780–785.
[34] Conway B A, Larson K M. Collocation Versus Diferential Inclusion in DirectOptimization[J].1998,21(5):780–785.
[35] Enright P J, Conway B A. Discrete Approximations to Optimal TrajectoriesUsing Direct Transcription and Nonlinear Programming[J]. Journal of Guid-ance,Control,and Dynamics,1992,15(4):994–1002.
[36] Reddien G W. Collocation at Gauss Points as a Discretization in Optimal Con-trol[J]. SIAM Journal of Control and Optimization,1979,17(2):1–6.
[37] Cuthrell J E, Biegler L T. Simultaneous Optimization and Solution Methods forBatch Reactor Control Profiles[J]. Computers and Chemical Engineering,1989,13(1):49–62.
[38] Elnagar G, Kazemi M A, Razzaghi M. The Pseudospectral Legendre Method forDiscretizing Optimal Control Problems[J]. IEEE Transactions On Automatic Con-trol,1995,40(10):1793–1796.
[39] Vlassenbroeck J. A Chebyshev Polynomial Method for Optimal Control with StateConstraints[J]. Automatica,1988,24(4):499–506.
[40] Fahroo F, Ross I M. Costate Estimation by a Legendre Pseudospectral Method[J].2001,24(2):270–277.
[41] Ross I M, Fahroo F. Legendre Pseudospectral Approximations of Optimal ControlProblems[J]. Lecture Notes in Control and Information Sciences,2003,295:327–342.
[42] Benson D A. A Gauss Pseudospectral Transcription for Optimal Control[D]. Cam-bridge, MIT: Massachusetts Institute of Technology,2005:49–55.
[43] Benson D A, Huntington G T, Thorvaldsen T P, et al. Direct Trajectory Optimiza-tion and Costate Estimation via an Orthogonal Collocation Method[J]. Journal ofGuidance,Control and Dynamics,2006,29(6):1435–1440.
[44] Huntington G T. Advancement and Analysis of a Gauss Pseudospectral Transcrip-tion for Optimal Control Problems[D]. Cambridge, MIT: Massachusetts Instituteof Technology,2007:50–85.
[45] Huntington G T, Rao A V. Optimal Spacecraft Formation Configuration Using aGauss Pseudospectral Method[C]//Proceedings of the2005AAS/AISS SpaceflightMechanics Meeting. Copper Mountain, Colorado: AIAA Press,2005:1–6.
[46] Zhang M R, Sun Y, Duan G R. Reentry Trajectory Optimization of HypersonicVehicle with Enhancing Parametrization Method[C]//Proceedings of the3rd In-ternational Symposium on Systems and Control in Aeronautics and Astronautics.Harbin: China,2010:40–45.
[47] Zheng Z Z, Yang Y, Wang Y J, et al. Entry Trajectory Genetic Algorithm Opti-mization Using Legendre Pseudospectral Method[C]//Proceedings of the2010In-ternational Conference on Modelling, Identification and Control. Okayama, Japan:Japan Press,2010:505–510.
[48]周文雅,杨涤,李顺利.利用高斯伪谱法求解具有最大横程的再入轨迹[J].系统工程与电子技术,2010,32(5):1038–1042.
[49] Ma G F, Zhuang Y F, Li C J, et al. Pseudospectral Method for Optimal Motion Plan-ning of a Rigid Underactuated Spacecraft[C]//20108th IEEE International Confer-ence on Control and Automation. Xiamen, China: IEEE Press,2010:684–688.
[50] Jorris T R, Cobb R G. Three-Dimensional Trajectory Optimization Satisfying Way-point and No-Fly Zone Constraints[J]. Journal of Guidance,Control and Dynamics,2009,32(2):551–572.
[51]水尊师,周军,葛致磊.基于高斯伪谱方法的再入飞行器预测校正制导方法研究[J].宇航学报,2006,32(6):1249–1255.
[52] Jorris T R. Constrained Trajectory Optimization Using Pseudospectral Method-s[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit. Honolulu,Hawaii: AIAA Press, AIAA Paper No.2008-6218,2008:1–33.
[53] Ross I M, D’Souza C, Fahroo F, et al. A Fast Approach to Multi-Stage Launch Ve-hicle Trajectory Optimization[C]//AIAA Guidance, Navigation, and Control Con-ference and Exhibit. Austin, Texas: AIAA Press, AIAA Paper No.2003-5639,2003:1–9.
[54] Bollino K P. High-Fidelity Real-Time Trajectory Optimization for ReusableLaunch Vehicles[D]. California:: Naval Postgraduate School,2006:70–105.
[55] Ross I M. User’s Manual for DIDO: A MATLAB Application Package for SolvingOptimal Control Problems[J]. Tomlab Optimization,2004:1–86.
[56] Kang W, Bedrossian N. Pseudospectral Optimal Control Theory Makes DebutFlight-Saves NASA$1M in Under3Hours[J]. SIAM News,2007,40(7):1–5.
[57] Bedrossian N, Bhatt S, Lammers M, et al. Zero Propellant Maneuver Flight Resultsfor180ISS Rotation[C]//2007International Symposium on Space Flight Dynam-ics. Xiamen, China: USA Press,2007:684–688.
[58] Garg D, Patterson M A, Darby C L, et al. Direct Trajectory Optimizationand Costate Estimation of General Optimal Control Problems Using a RadauPseudospectralMethod[C]//AIAA Guidance, Navigation, and Control Conference.Chicago, Illinois: AIAA Press, AIAA Paper No.2009-5989,2009:1–29.
[59] Fahroo F, Ross I M. Direct Trajectory Optimization by a Chebyshev PseudospectralMethod[J]. Journal of Guidance,Control,and Dynamics,2002,25(1):160–166.
[60] H. P. Ma T Q, Zhang W. An Efcient Chebyshev Algorithm for the Solution ofOptimal Control Problems[J]. IEEE Transactions on Automatic Control,2011,56(3):675–680.
[61] Gong Q, Ross I M, Fahroo F. Costate Computation by a Chebyshev PseudospectralMethod[J]. Journal of Guidance,Control,and Dynamics,2010,33(2):623–628.
[62] Fahroo F, Ross I M. Trajectory Optimization by Indirect Spectral Colloca-tion Methods[C]//Proceedings of the AIAA/AAS Astrodynamics Specialist Con-ference. Denver, CO: AIAA Press, AIAA Paper No.2000-4028,2000:1–6.
[63] Yan H, Fahroo F, Ross I M. Optimal feedback control law by Legendre pseu-dospectral approximations[C]//Proceedings of the American Control Conference.Arlington, VA: AIAA Press, AIAA Paper No.2000-4028,2001:2388–2393.
[64] Huntington G T, Benson D, Rao A V. A Comparison of Accuracy and Compu-tational Efciency of Three Pseudsopectral Methods[C]//AIAA Guidance, Navi-gation and Control Conference and Exhibit. South Carolina: AIAA Press, AIAAPaper No.2007-6405,2007:1–25.
[65] Garg D, Patterson M, Hager W W, et al. A Unified Framework for the NumericalSolution of Optimal Control Problems Using Pseudospectral Methods[J]. Auto-matica,2010,46(1):1843––1851.
[66] Ross I M, Fahroo F. Pseudospectral Knotting Methods for Solving Optimal ControlProblems[J]. Journal of Guidance,Control and Dynamics,2004,27(3):397–405.
[67] Ross I M, Fahroo F. Advances In Pseudospectral Methods[C]//AIAA Guidance,Navigation, and Control Conference. Honolulu, Hawaii: AIAA Press, AIAA PaperNo.2008-7309,2008:1–23.
[68] Ross I M, Fahroo F. Convergence of the Costates do not Imply Convergence of theControls[J]. Journal of Guidance,Control and Dynamics,2004,31(4):1492–1497.
[69] Gong Q, Kang W, Ross I M. A Pseudospectral Method for the Optimal Controlof Constrained Feedback Linearizable Systems[J]. IEEE Trans. Autom. Control,2006,51(7):1115––1129.
[70] Gong Q, Kang W, Fahroo F. Connections Between the Covector Mapping Theo-rem and Convergence of Pseudospectral Methods for Optimal Control[J]. ComputOptim Appl,2008,41(1):307––335.
[71] Darby C L, Hager W W, Rao A V. A Preliminary Analysis of a Variable-OrderApproach to Solving Optimal Control Problems Using Pseudospectral Method-s[C]//AIAA/AAS Astrodynamics Specialist Conference. Toronto, Ontario Canada:AIAA Press, AIAA Paper No.2010-8272,2010:1–14.
[72] Cheng P, Shen Z J, Lavalle S M. RRT-Based Trajectory Design for AutonomousAutomobiles and Spacecraft[J]. Archives of Control Sciences,2001,11(3-4):167–194.
[73] Frazzoli E, Dahleh M A, Feron E. Real-Time Motion Planning for AgileAutonomous Vehicles[J]. Journal of Guidance,Control and Dynamics,2002,25(1):116–129.
[74]陈刚,胡莹.基于NSGA-II算法的RLV多目标再入轨迹优化设计[J].西北工业大学学报,2006,24(2):133–137.
[75] Rajesh K A. Reentry Trajectory Optimization: Evolutionary Approach[C]//9thAIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. At-lanta: AIAA Press, AIAA Paper No.2002-5466,2002:1–5.
[76] Yokoyama N, Suzuki S. Trajectory Optimization Via Modified Genetic Algorith-m[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit. Austin,Texas: AIAA Press, AIAA Paper No.2003-5493,2003:1–9.
[77] Ross I M, Fahroo F. Issues in the Real-time Computation of Optimal Control[J].Mathematical and Computer Modeling,2006,43(9):1172–1188.
[78] Gong Q, Kang W, Ross I M. A Pseudospectral Method for the Optimal Control ofConstrained Feedback Linearizable Systems[J]. IEEE Transactions on AutomaticControl,2006,51(7):1115–1129.
[79] Gong Q, Fahroo F, Ross I M. Spectral Algorithm for Pseudospectral Methods inOptimal Control[J]. Journal of Guidance,Control and Dynamics,2008,31(3):460–471.
[80] Darby C L, Garg D, Rao A V. Costate Estimation Using Multiple-Interval Pseu-dospectral Methods[C]//AIAA Guidance, Navigation, and Control Conference.Portland, Oregon: AIAA Press, AIAA Paper No.2011-6571,2011:1–24.
[81] Darby C L, Garg D, Rao A V. Costate Estimation Using Multiple-Interval Pseu-dospectral Methods[J]. Journal of Spacecraft and Rockets,2011,48(5):856–866.
[82]洪蓓,辛万青.基于hp自适应伪谱法的滑翔弹道快速优化设计[C]//Proceedings of the30th Chinese Control Conference. Yantai: China,2011:1956–1961.
[83] Zhao Y, Tsiotras P. A Density-Function Based Mesh Refinement Algorithm forSolving Optimal Control Problems[C]//Proceedings of the Infotech and AerospaceConference. Seattle, WA: AIAA press,2009:1956–1961.
[84] Zhao Y, Tsiotras P. Density Functions for Mesh Refinement in Numerical OptimalControl[J]. Journal of Guidance,Control and Dynamics,2011,34(1):271–277.
[85] Darby C L, Hager W W, Rao A V. An Hp-Adaptive Pseudospectral Method forSolving Optimal Control Problems[J]. Optimal Control Applications and Methods,2011,32:476–502.
[86] Darby C L, Hager W W, Rao A V. Direct Trajectory Optimization Using a Vari-able Low-Order Adaptive Pseudospectral Method[J]. Journal of Spacecraft andRockets,2011,48(3):433–445.
[87] Lu P. Entry Guidance and Trajectory Control for Reusable Launch Vehicle[J].Journal of Guidance,Control and Dynamics,1997,20(1):143–149.
[88] Lu P, Hanson J M. Entry guidance for X-33Vehicle[J]. Journal of Spacecraft andRockets,1998,35(3):342–349.
[89] Grimm W, Meulen J G, Roenneke A J. Optimal Update Scheme for Drag ReferenceProfile in an Entry Guidance[J]. Journal of Guidance,Control and Dynamics,2003,26(5):695–701.
[90] Jouhaud F. Closed Loop Reentry Guidance Law of a Space Plane:Application toHermes[J]. Acta Astronautica,1992,26(8-10):577–585.
[91] Harpold J C, Graves C A. Shuttle Entry Guidance[J]. Journal of AstronauticalSciences,1979,27(3):239–268.
[92]赵汉元.航天器再入制导方法综述[J].航天控制,1994,1:26–33.
[93]闫晓东,唐硕.基于反馈线性化的H-V返回轨道跟踪方法[J].宇航学报,2008,29(5):1546–1550.
[94] Sheu D L, Chen Y M, Chen J S. Optimal Three-Dimensional Glide for MaximumReachable Domain[C]//AIAA Atmospheric Flight Mechanics Conference and Ex-hibit. Portland, OR: AIAA Press, AIAA Paper No.1998-4462,1999:9–11.
[95]赵汉元.飞行器再入动力学与制导[M].第一版.长沙:国防科技大学出版社,1997:1–482.
[96] Vasile I. Optimal Skip Entry Into Atmosphere[C]//AIAA Atmospheric Flight Me-chanics Conference and Exhibit. Portland, OR: AIAA Press, AIAA Paper No.1999-4170,1999:806–813.
[97] Rao A V, Mease K D. Dichotomic Basis Approach To Solving HypersensitiveOptimal Control Problems[J]. Automatica,1999,35:633–642.
[98]周浩,陈万春,殷兴良.高超声速飞行器时间最短弹道优化设计[J].弹道学报,2007,19(4):26–29.
[99]周浩,陈万春,殷兴良.高超声速飞行器滑行航迹优化[J].北京航空航天大学学报,2006,32(5):513–517.
[100]闫晓东,唐硕.亚轨道飞行器返回轨道设计方法研究[J].宇航学报,2008,29(2):467–471.
[101] Roenneke A J. Adaptive On-board Guidance for Entry Vehicle[C]//AIAA Guid-ance, Navigation, and Control Conference and Exhibit,. Montreal, Canada: AIAAPress, AIAA Paper No.2001-4048,2001:1–11.
[102] Mease K D, Chen D T, Teufel P, et al. Reduced-Order Entry Entry TrajectoryPlanning for Acceleration[J]. Journal of Guidance,Control and Dynamics,2002,25(2):257–266.
[103]王琦,丁运亮.可重复使用运载器再入轨迹次实时优化[J].北京航空航天大学学报,2007,33(4):405–408.
[104] Shen Z J, Lu P. On-Board Generation of Three-Dimensional Constrained EntryTrajectories[C]//AIAA Guidance, Navigation, and Control Conference and Exhib-it. Monterey, California: AIAA Press, AIAA Paper No.2002-4455,2002:1–16.
[105] Shen Z J, Lu P. On-Board Entry Trajectory Planning Expanded To Sub-OrbitalFlight[C]//AIAA Guidance, Navigation, and Control Conference and Exhibit.Austin, Texas: AIAA Press, AIAA Paper No.2003-5736,2003:1–13.
[106] Dukeman G A. Profile-Following Entry Guidance Using Linear Quadratic Regu-lator Theory[C]//AIAA Guidance, Navigation, and Control Conference. Monterey,CA: AIAA Press, AIAA Paper No.2002-4457,2002:1–10.
[107] Zimmerman C, Dukeman G A, Hanson J M. Automated Method to Com-pute Orbital Reentry Trajectories with Heating Constraints[J]. Journal of Guid-ance,Control and Dynamics,2003,26(4):523–529.
[108]方群,李新三.临近空间高超声速无动力滑翔飞行器最优轨迹设计及制导研究[J].宇航学报,2008,29(5):1485–1481.
[109] Hussein Y, Rajiv S C, Howard L, et al. Predictor-Corrector Entry Guidance forReusable Launch Vehicles[C]//AIAA Guidance, Navigation, and Control Confer-ence and Exhibit. Montreal,Canada: AIAA Press, AIAA Paper No.2001-4043,2001:2–17.
[110] Xue S B, Lu P. Constrained Predictor-Corrector Entry Guidance[C]//AIAA Guid-ance, Navigation, and Control Conference. Chicago, Illinois: AIAA Press, AIAAPaper No.2009-5767,2001:2–17.
[111]潘乐飞,李新国.可重复使用运载器预测-校正再入制导研究[J].飞行力学,2007,25(1):55–58.
[112] Joshi A, Sivan K, Amma S S. Predictor-Corrector Reentry Guidance Algorith-m with Path Constraints for Atmospheric Entry Vehicles[J]. Journal of Guid-ance,Control and Dynamics,2007,30(5):1307–1318.
[113] Ross I M, Fahroo F. A Unified Computational Framework for Real-Time OptimalControl[C]//Proceedings of the42nd IEEE Conierence on Decision and Control.Maui, Hawaii USA: IEEE Press,2003:2210–2215.
[114] Ross I M, Sekhavat P, Fleming A, et al. Pseudospectral Feedback Control: Foun-dations, Examples and Experimental Results[C]//Proceedings of the2006AIAAGuidance, Navigation, and Control Conference and Exhibit. Keystone, Colorado:AIAA Press, AIAA Paper No.2006-6354,2006:1–23.
[115] Ross I M, Gong Q, Fahroo F, et al. Practical Stabilization Through Real-TimeOptimal Control[C]//Proceedings of the2006American Control Conference. Min-neapolis, Minnesota, USA: IEEE press,2006:304–309.
[116] Ross I M, Sekhavat P, Fleming A, et al. Optimal Feedback Control: Foundation-s, Examples,and Experimental Results for a New Approach[J]. Journal of Guid-ance,Control and Dynamics,2008,31(2):1307–321.
[117] Bollino K, Ross I M, Doman D. Optimal Nonlinear Feedback Guidance for Reen-try Vehicles[C]//AIAA Guidance, Navigation, and Control Conference. Keystone,CO: AIAA Press, AIAA Paper No.2006-6074,2006:1–20.
[118] Bollino K P, Ross I M. A Pseudospectral Feedback Method for Real-Time Opti-mal Guidance of Reentry Vehicles[C]//Proceedings of the2007American ControlConference. New York City, USA: IEEE Press,2007:3861–3867.
[119] Karpenko M, Sekhavat P, Park J, et al. A Pseudospectral Feedback Method forReal-Time Optimal Guidance of Reentry Vehicles[C]//AIAA Guidance, Naviga-tion, and Control Conference. Toronto, Ontario Canada: AIAA Press, AIAA PaperNo.2010-8051,2010:1–11.
[120] P. Lu andH. S. Sun B T. Closed-loop endoatmospheric ascent guidance[J]. Journalof Guidance,Control, and Dynamics,2003,26(2):283–294.
[121] Dukeman G A. Atmospheric ascent guidance for rocket-powered launch vehi-cles[C]//AIAA Guidance, Navigation, and Conrol Conference and Exhibit. Mon-terey, California: AIAA Press, AIAA Paper No. AIAA2002-4559,2002:1–11.
[122] Lu P, Pan B. Highly Constrained Optimal Launch Ascent Guidance[C]//AIAAGuidance, Navigation, and Control Conference and Exhibit. Monterey, California:AIAA Press, AIAA Paper No.2009-5961,2009:1–26.
[123]王厚庆,何国强,刘佩进. RBCC为动力的巡航飞行器轨迹与质量分析[J].西北工业大学学报,2006,24(6):774–777.
[124]闫海江,唐硕,泮斌峰.亚轨道飞行器上升段轨迹快速生成方法研究[J].科学技术与工程,2011,11(10):2266–2270.
[125] Jorris T R. Common Aero Vehicle Autonomous Reentry Trajectory OptimizationSatisfying Waypoint and No-Fly Zone Constraints[D]. Wright-Patterson: Air ForceInstitute of Technology,2007:88–94.
[126] Jarlebring E, Damm T. The Lambertwfunction and the Spectrum of Some Multi-dimensional Time-Delay Systems[J]. Automatica,2007,43:2124–2128.
[127] Clarke F. Lyapunov Functions and Discontinuous Stabilizing Feedback[J]. AnnualReviews in Control,2011,35(1):13–33.