This paper focuses on the development of minimum time control profiles for point-to-point motion of a gantry crane system in the presence of uncertainties in modal parameters. Assuming that the velocity of the trolley of the crane can be commanded and is subject to limits, an optimal control problem is posed to determine the bang-off-bang control profile to transition the system from a point of rest to the terminal states with no residual vibrations. Both undamped and underdamped systems are considered and the variation of the structure of the optimal control profiles as a function of the final displacement is studied. As the magnitude of the rigid body displacement is increased, the collapse and birthing of switches in the optimal control profile are observed and explained. Robustness to uncertainties in modal parameters is accounted for by forcing the state sensitivities at the terminal time to zero. The observation that the time-optimal control profile merges with the robust time-optimal control is noted for specific terminal displacements and the migration of zeros of the time-delay filter parameterizing the optimal control profile are used to explain this counter intuitive result. A two degree of freedom gantry crane system is used to experimentally validate the observations of the numerical studies and the tradeoff of increase in maneuver time to the reduction of residual vibrations is experimentally illustrated.

中文翻译：

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具有速率约束的门式起重机系统的最小时间控制

本文的重点是在模态参数存在不确定性的情况下，为龙门起重机系统的点对点运动开发最小时间控制曲线。假设起重机小车的速度可以被控制并且受到限制，一个最优控制问题被提出来确定 bang-off-bang 控制曲线以将系统从静止点转换到终端状态，没有残余振动。考虑了无阻尼和欠阻尼系统，并研究了最佳控制曲线的结构随最终位移的变化。随着刚体位移幅度的增加，观察并解释了最佳控制配置文件中开关的崩溃和诞生。通过强制终端时间的状态灵敏度为零来考虑模态参数不确定性的鲁棒性。时间最优控制曲线与鲁棒时间最优控制合并的观察结果针对特定的终端位移和参数化最优控制曲线的延时滤波器的零点迁移用于解释这种反直觉的结果。二自由度龙门起重机系统用于通过实验验证数值研究的观察结果，并通过实验说明增加机动时间与减少残余振动之间的权衡。时间最优控制曲线与鲁棒时间最优控制合并的观察结果针对特定的终端位移和参数化最优控制曲线的延时滤波器的零点迁移用于解释这种反直觉的结果。二自由度龙门起重机系统用于通过实验验证数值研究的观察结果，并通过实验说明增加机动时间与减少残余振动之间的权衡。时间最优控制曲线与鲁棒时间最优控制合并的观察结果针对特定的终端位移和参数化最优控制曲线的延时滤波器的零点迁移用于解释这种反直觉的结果。二自由度龙门起重机系统用于通过实验验证数值研究的观察结果，并通过实验说明增加机动时间与减少残余振动之间的权衡。