$$\mathcal {H}_2$$ H 2 -gap Model Reduction for Stabilizable and Detectable Systems

We formulate here an approach to model reduction that is well-suited for linear time-invariant control systems that are stabilizable and detectable but may otherwise be unstable. We introduce a modified $$\mathcal {H}_2$$ H 2 -error metric, the $$\mathcal {H}_2$$ H 2 -gap, that provides an effective measure of model fidelity in this setting. While the direct evaluation of the $$\mathcal {H}_2$$ H 2 -gap requires the solutions of a pair of algebraic Riccati equations associated with related closed-loop systems, we are able to work entirely within an interpolatory framework, developing algorithms and supporting analysis that do not reference full-order closed-loop Gramians. This leads to a computationally effective strategy yielding reduced models designed so that the corresponding reduced closed-loop systems will interpolate the full-order closed-loop system at specially adapted interpolation points, without requiring evaluation of the full-order closed-loop system nor even computation of the feedback law that determines it. The analytical framework and computational algorithm presented here provides an effective new approach toward constructing reduced-order models for unstable systems. Numerical examples for an unstable convection diffusion equation and a linearized incompressible Navier-Stokes equation illustrate the effectiveness of this approach.