Smith-Type Methods for Balanced Truncation of Large Sparse Systems

The matrices P ∈ R and Q ∈ R are called the reachability and observability gramians, respectively. Under the assumptions that A is asymptotically stable, i.e. λi(A) ∈ C− (the open left half-plane), and that Σ is minimal (that is the pairs (A,B) and (C,A) are, respectively, reachable and observable), the gramians P, Q are unique and positive definite. In many applications, such as circuit simulation or time dependent PDE control problems, the dimension, n, of Σ is quite large, in the order of tens of thousands or higher, while the number of inputs m and outputs p usually satisfy m, p ? n. In these largescale settings, it is often desirable to approximate the given system with a much lower dimensional system

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