## Placement for Large Structures

In the case of the placement of a very large number of sensors, the maximization of the performance index alone may be either a sufficient or satisfactory criterion. Suppose that a specific sensor location gives a high-performance index. Inevitably, locations close to it will have a high-performance index as well. But the locations in the neighborhood of the original sensor are not necessarily the best choice, since the sensors at these locations can be replaced by the appropriate gain adjustment of the original sensor. We want to find sensor locations that cannot be compensated for by original sensor gain adjustment. These locations we determine using an additional criterion, which is based on the correlation of each sensor modal norm. We define a vector of the ith sensor norms, which is composed of the squares of the modal norms

0 20 40 60 80 100 node number i 0.5

0 20 40 60 80 100 node number

0 20 40 60 80 100 node number

0 20 40 60 80 100 node number

Figure 7.7. Sensor placement H2 indices as a function of sensor locations: (a) For the first mode; (b) for the first two modes; (c) for the first three modes; and (d) for the first four modes.

INI2 Gil2

where Gik denotes the transfer function of the kth mode at the ith sensor. The norm || | denotes the H2, or Hankel norms. Next, we define the correlation coefficient rik as follows:

Denote a small positive number, s, say s- 0.01 - 0.20. Denote the membership index I(k), k= 1,...,R, where R is the number of sensors. We define this index as follows:

f 0 if rik > 1 -s for <rk < cri and for k > i, I (k) = ^ ik k ' (7.24)

[ 1 elsewhere, for k > i. If I(k) = 1, the kth sensor is accepted, and if I(k) = 0, the kth sensor is rejected (in this case the two locations i and k are either highly correlated, or the ith location has a higher performance <ri).

Based on the above analysis we establish the placement strategy. For technical and economic reasons the number of sensors significantly exceeds the number of actuators. Therefore, the actuator selection comes first, as a less flexible procedure.

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