Eigenvalue problems of linear Hamiltonian systems arising from H∞ filtering

From the viewpoint of differential eigenvalue problem of Hamiltonian system, a linear finite-time H∞ filtering problem can be addressed by computing eigenvalue and solving Riccati differential equation of an associated linear Hamiltonian system. This paper shows how a problem of determining the minimum induced norm of the H∞ filter is formulated as a Hamiltonian differential eigenvalue problem. The H∞ filters concerned here include central filter, perturbed filter and decentralised filter. The methods presented in the paper are based on characteristics of eigensolution of the corresponding Hamiltonian system and Riccati differential equation. With eigensolution of the Hamiltonian system arising from the central H∞ filtering problem, variational methods are proposed to compute eigenvalues of perturbed Hamiltonian systems and large-scale Hamiltonian systems derived from perturbed H∞ filters and decentralised H∞ filters, respectively. Then, eigenvalues can be obtained by calculating stationary values of corresponding extended Rayleighʹs quotient with dual argument functions, which is essentially different from the well-known Rayleighʹs quotient of Lagrange systems with only one independent argument function. Numerical examples are also presented to illustrate the variational approaches presented in this paper.