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Eigenstructure perturbations for a class of Hamiltonian matrices and solutions of related Riccati inequalities

Volker Mehrmann, Hongguo Xu

TL;DR

This paper analyzes the set of Hermitian solutions to algebraic Riccati inequalities and equations arising in passivity analysis of linear time-invariant systems by studying the eigenstructure of the associated Hamiltonian matrix. It develops a perturbation framework for the Hamiltonian, leveraging unitary block decompositions and Lagrangian invariant subspaces to characterize extremal solutions and the feasible perturbation region. The key contributions include (i) conditions under which all ARE solutions are positive definite and (ii) a constructive procedure to obtain extremal solutions, plus (iii) a perturbation path that leads to a vertex where the Hamiltonian has all eigenvalues on the imaginary axis and a unique solution. These results provide a rigorous link between Riccati solution sets and eigenvalue perturbations, with implications for robust port-Hamiltonian representations and passivity-based control design.

Abstract

The characterization of the solution set for a class of algebraic Riccati inequalities is studied. This class arises in the passivity analysis of linear time invariant control systems. Eigenvalue perturbation theory for the Hamiltonian matrix associated with the Riccati inequality is used to analyze the extremal points of the solution set.

Eigenstructure perturbations for a class of Hamiltonian matrices and solutions of related Riccati inequalities

TL;DR

This paper analyzes the set of Hermitian solutions to algebraic Riccati inequalities and equations arising in passivity analysis of linear time-invariant systems by studying the eigenstructure of the associated Hamiltonian matrix. It develops a perturbation framework for the Hamiltonian, leveraging unitary block decompositions and Lagrangian invariant subspaces to characterize extremal solutions and the feasible perturbation region. The key contributions include (i) conditions under which all ARE solutions are positive definite and (ii) a constructive procedure to obtain extremal solutions, plus (iii) a perturbation path that leads to a vertex where the Hamiltonian has all eigenvalues on the imaginary axis and a unique solution. These results provide a rigorous link between Riccati solution sets and eigenvalue perturbations, with implications for robust port-Hamiltonian representations and passivity-based control design.

Abstract

The characterization of the solution set for a class of algebraic Riccati inequalities is studied. This class arises in the passivity analysis of linear time invariant control systems. Eigenvalue perturbation theory for the Hamiltonian matrix associated with the Riccati inequality is used to analyze the extremal points of the solution set.
Paper Structure (11 sections, 10 theorems, 136 equations, 1 figure)

This paper contains 11 sections, 10 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical solution theory
  3. Solvability of algebraic Riccati equations
  4. Positive definite solutions of the ARE (\ref{['are']})
  5. Extreme solutions
  6. Eigenvalue perturbation theory for Hamiltonian matrices
  7. General perturbations
  8. Perturbations of a purely imaginary eigenvalue
  9. Perturbation theory for pre-existing purely imaginary eigenvalues
  10. Perturbation of eigenvalues that are not purely imaginary
  11. Conclusion

Key Result

theorem 1

Assume that the system (GenSys) is minimal. The matrix inequality has a solution $X= X^H> 0$ if and only if (GenSys) is a passive system. In this case ${\mathcal{H}}(x) = \frac{1}{2} {x^H} X x$ defines a storage function for (GenSys) associated with the supply rate ${\mathop{\rm Re}\nolimits({y^H}u)}$, satisfying (DissipIneq). Furthermore, for passive systems ther

Figures (1)

  • Figure 1: Feasible region ${\cal R}$

Theorems & Definitions (17)

  • theorem 1
  • proposition 1
  • proof
  • lemma 1: BeaMX15_pptLanR95
  • lemma 2
  • proof
  • lemma 3: BoyEFB94LanR95
  • Example 1
  • lemma 4
  • lemma 5
  • ...and 7 more