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


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)

0 0

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