Optimal time-domain moment matching with partial placement of poles and zeros

In this paper we consider a minimal, linear, time-invariant (LTI) system of order n, large. Our goal is to compute an approximation of order ν <; n that simultaneously matches ν moments, has ℓ poles and k zeros fixed, with ℓ + k <; ν, and achieves minimal H
₂
norm of the approximation error. For this, in the family of ν order parametrized models that match ν moments we impose ℓ+k linear constraints yielding a subfamily of models with ℓ poles and k zeros imposed. Then, in the subfamily of ν order models matching ν moments, with ℓ poles and k zeros imposed we propose an optimization problem that provides the model yielding the minimal H
₂
-norm of the approximation error. We analyze the first-order optimality conditions of this optimization problem and compute explicitly the gradient of the objective function in terms of the controllability and the observability Gramians of the error system. We then propose a gradient method that finds the (optimal) stable model, with fixed ℓ poles and k zeros.