AI 中文总结
针对依赖参数的可微线性矩阵不等式求解需求，提出GriD-LMIA工具包，通过网格划分与伯恩斯坦多项式将连续域约束转换为有限LMI，可导出多种验证证书并分析相关参数平衡关系。
AI 中文摘要
依赖参数的线性矩阵不等式（PD-LMIs）需要在连续域上成立。当调度参数随时间变化时，依赖参数的决策变量的导数也会进入约束条件。由于半定规划求解器需要有限个约束，我们提出了GriD-LMIA，即基于网格划分的可微PD-LMI组装器。它将需要在连续域上成立的约束条件转换为MATLAB中的有限个充分LMI。该工具包首先用超矩形网格划分域，并在每个单元格上用张量伯恩斯坦多项式表示已知数据和决策变量，通过YALMIP导出直接伯恩斯坦、波利亚提升及基于平方和的验证证书。通过实例研究了网格密度、决策阶数与证书选择之间的平衡关系。
英文摘要
Parameter-dependent linear matrix inequalities (PD-LMIs) require holding over a continuous domain. When the scheduling parameters vary with time, derivatives of parameter-dependent decisions may also enter the conditions. Since semidefinite programming solvers require finitely many constraints, we introduce GriD-LMIA, the Gridding-based Differentiable PD-LMI Assembler. It converts the conditions that need to hold on a continuous domain into finitely many sufficient LMIs in MATLAB. It first partitions the domain with a hyper-rectangular grid and represents known data and decisions on each cell with tensor Bernstein polynomials. The package exports direct Bernstein, Pólya-elevated, and sum-of-squares-based certificates through YALMIP. Examples are given to examine the balance among grid density, decision degree, and certificate choice.
