Characterizing the Solution Set of Algebraic Riccati Inequalities Using Eigenstructure Perturbations of Hamiltonian Matrices

Core Concepts

The paper studies the characterization of the solution set for a class of algebraic Riccati inequalities that arise in the passivity analysis of linear time-invariant control systems. Eigenvalue perturbation theory for the associated Hamiltonian matrix is used to analyze the extremal points of the solution set.

Abstract

The paper focuses on the characterization of the solution set for a class of algebraic Riccati inequalities (ARIs) and the related algebraic Riccati equations (AREs). The main goals are:

To study the perturbation theory for the eigenstructure of the Hamiltonian matrix associated with the Riccati inequality, in order to analyze the extremal points of the solution set.

To investigate the difference between the solutions of the ARE and the ARI, particularly when the Hamiltonian matrix has purely imaginary eigenvalues.

The key insights are:

The paper shows that the solvability of the ARE may be more restrictive than the solvability of the ARI, as the ARE requires additional conditions on the controllability and observability of the system.

When the Hamiltonian matrix has only purely imaginary eigenvalues, the solution of the ARE is unique and corresponds to the extremal solutions of the ARI.

The paper develops a perturbation theory for the Hamiltonian matrix that allows characterizing the region of perturbations for which the ARI has a solution.
The analysis provides a comprehensive understanding of the solution set of the ARI and the relationship between the solutions of the ARE and ARI, which is important for applications in passivity analysis and port-Hamiltonian system modeling.

Stats

The Hamiltonian matrix H associated with the Riccati inequality is defined as:
H = [F G; -K -F^H]
The system (1.4) is called passive if there exists a continuously differentiable storage function H(x) such that the dissipation inequality (1.5) holds.

Quotes

"The characterization of the solution set for a class of algebraic Riccati inequalities is studied."
"The main goal of this paper is to characterize the set of positive definite solutions of (1.1) and (1.2) and we do this by studying the perturbation theory for the eigenstructure of the associated Hamiltonian matrix."

Key Insights Distilled From

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

by Volker Mehrm... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2311.14202.pdf

Deeper Inquiries

How can the insights from this paper be extended to more general classes of Riccati inequalities and equations beyond the specific form considered here

The insights from this paper can be extended to more general classes of Riccati inequalities and equations by considering different structural properties of the matrices involved. For instance, the controllability and observability conditions can be relaxed to accommodate systems with different structural constraints. Additionally, the perturbation theory developed in the paper can be adapted to handle more complex perturbations in the system matrices. By generalizing the concepts of controllability, observability, and stability, the analysis can be applied to a broader range of systems beyond the specific form considered in the paper.

What are the implications of the unique solution property for the ARE when the Hamiltonian matrix has only purely imaginary eigenvalues, in terms of the robustness and optimality of the associated port-Hamiltonian system representations

The unique solution property for the Algebraic Riccati Equation (ARE) when the Hamiltonian matrix has only purely imaginary eigenvalues has significant implications for the robustness and optimality of the associated port-Hamiltonian system representations. In this case, the system exhibits a special structure that allows for a unique solution to the ARE, leading to a well-defined and stable representation of the system dynamics. This uniqueness ensures that the system has a clear and robust behavior, making it easier to analyze and optimize the system's performance.

Can the perturbation analysis techniques developed in this paper be applied to study the sensitivity of the solution set of the ARI with respect to other types of parameter variations in the system matrices

The perturbation analysis techniques developed in this paper can be applied to study the sensitivity of the solution set of the Algebraic Riccati Inequality (ARI) with respect to other types of parameter variations in the system matrices. By introducing different types of perturbations, such as variations in the system parameters or external disturbances, the analysis can provide insights into how the solution set of the ARI responds to these changes. This sensitivity analysis can help in understanding the robustness of the system and in designing control strategies that can adapt to different operating conditions.