Eigenvalues of Dual Hermitian Matrices with Application in Formation Control
Liqun Qi, Chunfeng Cui
TL;DR
The paper develops a unified framework for dual Hermitian matrices over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{Q}$ and introduces the supplement matrix method (SMM) to compute the $n$ dual-number eigenvalues of a dual Hermitian matrix $A=A_s+A_d\epsilon$, leveraging the eigenstructure of $A_s$ and a supplement matrix $W^*A_dW$. It shows that the standard parts of the eigenvalues correspond to eigenvalues of $A_s$, while the dual parts are given by the eigenvalues of the supplement matrix, enabling practical computation of dual eigenvalues even when $A_s$ has multiplicities. The authors apply this framework to the relative configuration problem in multi-agent formation control, linking dual Hermitian eigenproblems to dual quaternion unit gain graphs and using spectral properties to assess the reasonableness of desired configurations. Numerical experiments on dual complex and dual quaternion unit gain graphs demonstrate the accuracy and efficiency of SMM and illustrate balance conditions in the gain graphs. The work provides a cross-disciplinary toolkit for both eigenvalue computation of dual matrices and feasibility analysis of formation configurations with potential impact in robotics and networked control.
Abstract
We propose a supplement matrix method for computing eigenvalues of a dual Hermitian matrix, and discuss its application in multi-agent formation control. Suppose we have a ring, which can be the real field, the complex field, or the quaternion ring. We study dual number symmetric matrices, dual complex Hermitian matrices and dual quaternion Hermitian matrices in a unified frame of dual Hermitian matrices. An $n \times n$ dual Hermitian matrix has $n$ dual number eigenvalues. We define determinant, characteristic polynomial and supplement matrices for a dual Hermitian matrix. Supplement matrices are Hermitian matrices in the original ring. The standard parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of the standard part Hermitian matrix in the original ring, while the dual parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of those supplement matrices. Hence, by applying any practical method for computing eigenvalues of Hermitian matrices in the original ring, we have a practical method for computing eigenvalues of a dual Hermitian matrix. We call this method the supplement matrix method. In multi-agent formation control, a desired relative configuration scheme may be given. People need to know if this scheme is reasonable such that a feasible solution of configurations of these multi-agents exists. By exploring the eigenvalue problem of dual Hermitian matrices, and its link with the unit gain graph theory, we open a cross-disciplinary approach to solve the relative configuration problem. Numerical experiments are reported.