## Controllability and Observability of a Second Order Modal Model

In this section we present the controllability and observability properties of a structure given by the second-order model.

### 4.6.1 Grammians

The grammians and the balanced models are defined exclusively in the state-space representation, and they do not exist in the second-order form. This is a certain disadvantage since the second-order structural equations are popular forms of structural modeling. We will show, however, that for flexible structures one can find a second-order model which is almost balanced, and for which Hankel singular values can be approximately determined without using a state-space representation. First, we determine the grammians for the second-order modal model.

Property 4.6. Controllability and Observability Grammians of the Second-Order Modal Model. The controllability (wc) and observability (wo) grammians of the second-order modal model are given as wc = 0.25Z-1Q-1diag(B,X ), w0 = 0.25Z-1Q-1diag(CmCm ), mCm )

where diag(BtbT) and diag(C^Cm) denote the diagonal part of BtbT and

CTmCm, respectively, Bm is given by (2.23) Cm = \_Cmq^ Cmv ] , and Cmq ' Cmv are defined as in (2.24) and (2.25). Therefore, the ith diagonal entries of wc and wo are

b 112

mi 112

II IP

Cmi 112

and bmi is the ith row of Bm, and cmi is the ith column of Cm.

Proof. In order to show this we introduce a state-space representation by defining the following state vector:

The following state-space representation is associated with the above vector

By inspection, for this representation, the grammians are diagonally dominant, in the form

Wo 0

0 Wo where wc and wo are the diagonally dominant matrices, wc = diag(wci) and wo = diag(woi). Introducing the last two equations to the Lyapunov equations (4.5) we obtain (4.58). 0

Having the grammians for the second-order models, the Hankel singular values are determined approximately from (4.58) as wciwoi =—-¡7--, i = 1,..., n. (4.59)

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