Residual-based iterations for the generalized Lyapunov equation

TL;DR

This paper extends iterative solution methods for the generalized Lyapunov equation, establishing theoretical foundations, connections to control system optimality, and proposing a residual-based Krylov subspace method.

Contribution

It generalizes the ALS method to the stable generalized Lyapunov equation, links it to H2-optimality, and introduces a residual-based Krylov subspace approach.

Findings

Theoretical justification for ALS in the generalized setting.

Connection between ALS energy minimization and H2-optimality.

Proposal of a residual-based Krylov subspace method.

Abstract

This paper treats iterative solution methods to the generalized Lyapunov equation. Specifically it expands the existing theoretical justification for the alternating linear scheme (ALS) from the stable Lyapunov equation to the stable generalized Lyapunov equation. Moreover, connections between the energy-norm minimization in ALS and the theory to H2-optimality of an associated bilinear control system are established. It is also shown that a certain ALS-based iteration can be seen as iteratively constructing rank-1 model reduction subspaces for bilinear control systems associated with the residual. Similar to the ALS-based iteration, the fixed-point iteration can also be seen as a residual-based method minimizing an upper bound of the associated energy norm. Lastly a residual-based generalized rational-Krylov-type subspace is proposed for the generalized Lyapunov equation.