ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Matrix Equations

Tue 16:00–16:30, Room AV 02.17
Lyapunov equations in model order reduction of stochastic systems
Tobias Damm (University of Kaiserslautern)

We consider various approaches to balanced truncation of stochastic linear systems of the form

\begin{align*} dx = Ax\,dt+ Nx\,dw + Bu\,dt, \quad y = Cx. \end{align*}

To this end, we introduce different generalizations of the reachability Gramian. In particular we analyse the following two pairs of matrix inequalities

\begin{align*} A^TQ+QA+N^TQN&\le-C^TC,\\ AP+PA^T+NPN^T&\le-BB^T \end{align*}

and

\begin{align*} A^TQ+QA+N^TQN&\le-C^TC,\\ A^TP^{-1}+P^{-1}A+N^TP^{-1}N&\le-P^{-1}BB^TP^{-1}. \end{align*}

Performing truncation based on balancing of $Q$ and $P$ in the two different cases, we observe that both approaches preserve asymptotic stability, but only the second leads to a stochastic $H^\infty$-type bound for the approximation error of the truncated system. Further properties and numerical issues shall be discussed in the talk.