Unstable modes in projection-based reduced-order models: How many can there be, and what do they tell you?
For researchers using projection-based model reduction, this work provides theoretical guarantees on stability loss and a new interpretation of unstable modes as useful signals.
This paper investigates the number of unstable modes that can appear in reduced-order models constructed via orthogonal projection, providing rigorous upper bounds for both continuous- and discrete-time systems. It also demonstrates that these unstable modes can serve as indicators of transient growth in the original system.
Projection methods provide an appealing way to construct reduced-order models of large-scale linear dynamical systems: they are intuitively motivated and fairly easy to compute. Unfortunately, the resulting reduced models need not inherit the stability of the original system. How many unstable modes can these reduced models have? This note investigates this question, using theory originally motivated by iterative methods for linear algebraic systems and eigenvalue problems, and illustrating the theory with a number of small examples. From these results follow rigorous upper bounds on the number of unstable modes in reduced models generated via orthogonal projection, for both continuous- and discrete-time systems. Can anything be learned from the unstable modes in reduced-order models? Several examples illustrate how such instability can helpfully signal transient growth in the original system.