Reduction Through Truncation

In this chapter we consider a structural model in modal coordinates, namely, modal models 1, 2, and 3, as in (2.52), (2.53), and (2.54)). The states of the model are ordered as follows:

modal norm indicator modal norm indicator

where xt is the state corresponding to the ith mode. It consists of two states; see (2.55), (2.56), and (2.57):

Let ||Gi| denote either H2, Hm, or Hankel norms of the ith mode, and order the states in the state vector (6.1) in the descending norm order. Now, the norm of the first mode is the largest one, and the norm of the last mode is the smallest, which is marked in (6.1) with the norm value indicator located to the right of the equation. In the indicator the largest norm is marked in black and the smallest norm in white.

We obtain a reduced-order model by evaluating the modal states and truncating the least important. Since the modes with the smallest norm are the last ones in the state vector, a reduced-order model is obtained here by truncating the last states in the modal vector. How many of them? This will be determined later in this section by evaluating the reduction errors. Let (A, B, C) be the modal representation (the subscript m is dropped for simplicity of notation) corresponding to the modal state vector x as in (6.1). Let x be partitioned as follows:

where xr is the vector of the retained states and xt is a vector of truncated states. If there are k < n retained modes, xr is a vector of 2k states, and xt is a vector of 2(n - k) states. Let the state triple (A, B, C) be partitioned accordingly,

We obtain the reduced model by deleting the last 2(n - k) rows of A, B, and the last 2(n - k) columns of A, C. Formally, this operation can be written as follows:

Modal reduction by truncation of stable models always produces a stable reduced model, since the poles of the reduced model are a subset of the poles of the full-order model.

The problem is to order the states so that the retained states xr will be the best reproduction of the full system response. The choice depends on the definition of the reduction index.

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