$H_2$ Model Reduction of Linear Network Systems by Moment Matching and Optimization

In this article, we compute the reduced-order stable approximation of a linear network system, preserving the topology and optimal w.r.t. the H
₂
-norm of the approximation error. Our approach is based on time-domain moment matching, where we optimize over families of parameterized reduced-order models, matching moments at arbitrary interpolation points. The low-order models are parametrized in the free parameters (i.e., the elements of the input matrix) and the interpolation matrix. We formulate an optimization-based problem with the H
₂
-norm of the error as the objective function and with structural and physical properties as the constraints. The problem is nonconvex and we write it in terms of the Gramians of the error system. We propose two solutions. The first solution assumes that the error system admits a block diagonal observability Gramian, allowing for a simple convex reformulation as semidefinite programming, but at the cost of some performance loss. We also derive the sufficient conditions to guarantee the block diagonalization of the Gramian. The second solution employs a gradient projection method for a smooth reformulation, yielding the (locally) optimal interpolation points and free parameters. The potential of the methods is illustrated on a positive network and a power network.