On maximal positive invariant set computation for rank-deficient linear systems

TL;DR

This paper introduces a robust algorithm for computing the maximal positively invariant set in rank-deficient linear systems, addressing challenges caused by zero eigenvalues and projections onto lower-dimensional subspaces.

Contribution

It explicitly handles singular cases in MPI computation using Schur decomposition, improving accuracy for systems with rank deficiency.

Findings

Algorithm effectively computes MPI sets in rank-deficient systems.

Addresses limitations of static feedback synthesis in invariant set analysis.

Applicable to polyhedral and constrained-zonotope representations.

Abstract

The maximal positively invariant (MPI) set is obtained through a backward reachability procedure involving the iterative computation and intersection of predecessor sets under state and input constraints. However, standard static feedback synthesis may place some of the closed-loop eigenvalues at zero, leading to rank-deficient dynamics. This affects the MPI computation by inducing projections onto lower-dimensional subspaces during intermediate steps. By exploiting the Schur decomposition, we explicitly address this singular case and propose a robust algorithm that computes the MPI set in both polyhedral and constrained-zonotope representations.