3.1.1. Weighting WS

*W*S is the weighting function of the sensitivity function *S*. In the RTHS used for earthquake engineering, the frequency of the commanded signals and disturbances is generally very low. Hence, to track the reference signal with high accuracy and to suppress the external disturbances, the sensitivity function *S* should be small enough over the lowfrequency range, while in the high-frequency band beyond the command signal, there are no additional restraints for the sensitivity function *S*. Therefore, the weighting function *W*S should have a high gain over the low-frequency range. Hence, *W*S can be selected as

$$\mathcal{W}\_{\rm S} = \frac{as + b\omega\_{\rm d}}{s + c} \tag{6}$$

where *a*, *b*, and *c* are adjustable parameters, and *ω*d is the desired bandwidth of the system. The singular value of *W*S −1 is shown in Figure 3. It is seen from the figure that, if *c* is small enough, that the sensitivity function *S* will tend to zero when the frequency is not very high, which means that the controlled system can realize the command perfectly. Typically, –60 dB is small enough for the sensitivity function *S* when the frequency is low; hence, it is recommended that *c* is less than 0.01.

**Figure 3.** Singular value of *W*S−1.
