Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case

This paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic Riccati and Lyapunov equations that have been reported during the last two decades. Also presented are bounds for the unified Riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete Riccati equations can be unified by them.

Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results

Design of decentralized observation schemes for large-scale interconnected systems: some new results

Qualitative analysis and decentralized controller synthesis for a class of large-scale systems with symmetrically interconnected subsystems

In this note, the problem of solution to the matrix equation AX+XTC=B is considered by the Moore–Penrose generalized inverse matrix. A general solution to this equation is obtained. At the same time, some useful conclusions are made, which play important roles in the linear system theories and applications.

The solution to matrix equation AX+XTC=B

An explicit solution to the algebraic Lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices. This enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the controllability matrix. Based on this explicit formula, upper and lower bounds for each eigenvalue of the solution are derived, which always give nontrivial estimates.

M. Kuwahara

Papers

Explicit solution and eigenvalue bounds in the Lyapunov matrix equation