Efficient computation of the solutions to modified Lyapunov equations

This paper develops a solution method for modified Lyapunov equations in which the modification term $\mathcal{F}( Q )$ 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 that arise in robust reduced-order design. The method relies on decomposing $\mathcal{F}( Q )$ as $\mathcal{F} ( Q ) = \mathcal{G} ( \phi ( Q ) )$, where $\phi ( Q )$ is an $m$-dimensional vector. It is shown that if $m$ is small, the new solution procedure is much more efficient than are solution procedures based on a straightforward transformation of the modified Lyapunov equation to a linear vector equation in $n( n + 1 )/2$ unknowns. The results are extended to develop an efficient procedure for computing the solutions to an arbitrary number of coupled Lyapunov equations in which the coupling terms are linear operators.