Data-driven constrained optimal model reduction

Model reduction by moment matching can be interpreted as the problem of finding a reduced-order model which possesses the same steady-state output response of a given full-order system for a prescribed class of input signals. Little information regarding the transient behavior of the system is systematically preserved, limiting the use of reduced-order models in control applications. In this paper we formulate and solve the problem of constrained optimal model reduction. Using a data-driven approach we determine an estimate of the moments and of the transient response of a possibly unknown system. Consequently we determine a reduced-order model which matches the estimated moments at the prescribed interpolation signals and, simultaneously, possesses the estimated transient. We show that the resulting system is a solution of the constrained optimal model reduction problem. Detailed results are obtained when the optimality criterion is formulated with the time-domain ℓ₁, ℓ₂, ℓ_∞ norms and with the frequency-domain H₂ norm. The paper is illustrated by two examples: the reduction of the model of the vibrations of a building and the reduction of the Eady model (an atmospheric storm track model).