GriD-LMIA: A Gridding-Based Assembler for Solving Differentiable Parameter-Dependent Linear Matrix Inequalities

Abstract

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.