基于Bures-Wasserstein距离的LMI约束下的可控性格拉姆矩阵整形
A Controllability Gramain Shaping with LMI Constraints under Bures--Wasserstein Distance
AI总结：
本文提出基于Bures-Wasserstein距离与LMI约束的可控性格拉姆矩阵整形控制器设计方法，可实现各向异性可控性设计，且能趋近$H_2$控制，适用于制导机器人系统。
AI中文摘要：
本文提出一种控制器设计方法，用于将可控性格拉姆矩阵整形为期望形式，以设计从外部输入到系统状态的效应。利用Bures-Wasserstein距离，我们将整形问题建模为最小化系统格拉姆矩阵与期望格拉姆矩阵之间的距离，该目标函数在对称正定矩阵集合上被证明是严格凸的。此外，通过线性矩阵不等式（LMI）推导半定规划形式，提高了计算效率，且可纳入额外的LMI约束。当外部输入建模为高斯白噪声时，所提框架与$H_2$控制密切相关，可被解释为最优传输的特殊情况。数值示例验证了制导机器人的各向异性可控性设计，并证实了通过LMI施加额外方向约束的能力，还确认当期望格拉姆矩阵趋于零时，所提方法趋近于$H_2$控制。
英文摘要：
This paper proposes a controller design method for shaping the controllability Gramian into a desired form to design the effect from exogenous inputs to the system state. Using the Bures--Wasserstein distance, we formulate the shaping problem as the minimization of the distance between the system Gramian and a desired Gramian, and the objective function is shown to be strictly convex on the set of symmetric positive definite matrices. In addition, by deriving a semidefinite programming formulation via a linear matrix inequality (LMI), computational efficiency is improved and additional LMI constraints can be incorporated. When the exogenous input is modeled as Gaussian white noise, the proposed framework is closely related to $H_2$ control, which can be interpreted as a special case of optimal transport. Numerical examples demonstrate anisotropic controllability design for a guidance robot and verify the ability to impose additional directional constraints through LMIs. The numerical examples also confirm that the proposed method approaches $H_2$ control as the desired Gramian tends to zero.
