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

Bogdan Gheorghe, Daniel Ioan, Cristian Flutur, Ionela Prodan, Florin Stoican

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.

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

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.
Paper Structure (11 sections, 2 theorems, 38 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 2 theorems, 38 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem formulation and justification
  4. Main idea
  5. The polyhedral case
  6. Constrained zonotopic case
  7. Analysis
  8. Illustrative example
  9. Numerical example in $\mathbb R^6$
  10. The CSE example Leibfritz2006COMPleibCM
  11. Conclusions

Key Result

Proposition 1

Consider the singular closed-loop matrix whose Schur decomposition has the $(2,2)$ block $S_{22}\in \mathbb{R}^{d_2\times d_2}$ nilpotent of degree $p+1<n$, i.e., The scalars $d_1>0$, $d_2>0$ satisfy $d_1+d_2=n$. Introducing the shorthand notation the MPI set associated with the state matrix eq:schur and constraints eq:hrep is given by with The pair $(F_z, \theta_z)$ defines the MPI set, com

Figures (3)

  • Figure 1: Example for computing the MPI set.
  • Figure 2: MPI set $\Phi_z$ in $\mathbb R^3$.
  • Figure 3: Plot number of iterations and commutation time.

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3