A Solution Methodology for Modified Lyapunov Equations

This paper develops a solution methodology for modified Lyapunov equations in which the modification term T(Q) is a linear function of the solution Q. Equations of this form arise in robustness analysis and in homotopy algorithms developed for solving the nonstandard Riccati and Lyapunov equations arising in robust reduced-order design. The methodology relies on decomposing T(Q) as T(Q) = g(¿(Q)) where ¿(Q) is an m-dimensional vector. If m is small, then it is shown that the new solution procedure will be much more efficient than solutions based on a straightforward transformation of the modified Lyapunov equation to a linear vector equation in n(n+1)/2 unknowns.