Two-sided model order reduction for two-directional difference system

This paper defines a two-directional difference system and constructs the projection matrix. Then the original system is projected into the smaller system, and we discuss its moment-matching properties. Next we define the dual system, and discuss the dual relation between the dual system and the original system. Then we can construct the projection matrix with the above mentioned dual relation, and project the dual system into the respectively smaller system, hence derive the moment-matching properties. Finally synthesizing the above two moment-matching properties we obtain the main results that the number of moments matched is twice as much as the number of the generating terms of the constructed projection subspace. We apply this result to the two-sided model order reduction for parameter time delay system, and obtain the result that the reduced system can preserve twice moments as the number of the generating terms of the constructed projection subspace. Finally we derive an algorithm to compute the basis of the subspace involved in the reduction process.