Model Reduction for Discrete-Time Systems via Optimization over Grassmann Manifold

In this paper, we investigate $h_2$-optimal model reduction methods for discrete-time linear time-invariant systems. Similar to the continuous-time case, we will formulate this problem as an optimization problem over a Grassmann manifold. We consider constructing reduced systems by both one-sided and two-sided projections. For one-sided projection, by utilizing the principle of the Grassmann manifold, we propose a gradient flow method and a sequentially quadratic approximation approach to solve the optimization problem. For two-sided projection, we apply the strategies of alternating direction iteration and sequentially quadratic approximation to the minimization problem and develop a numerically efficient method. One main advantage of these methods, based on the formulation of optimization over a Grassmann manifold, is that stability can be preserved in the reduced system. Several numerical examples are provided to illustrate the effectiveness of the methods proposed in this paper.