3.1.2. Weighting WT

*W*T is the weighting function of the complementary sensitivity function *T*, which represents the characteristic of the multiplicative uncertainties [24]. Hence, the weighting *W*T can be selected as

$$
\sigma \left( \frac{G(j\omega)}{G\_0(j\omega)} - 1 \right) < \sigma (W\_T(j\omega))\_\prime \,\forall \omega \tag{7}
$$

where *G* is the actual plant, *G*0 is the nominal or analytical plant, and *σ* is the singular value. Once *W*T is determined, it can remain unchanged, because it is related to the model of the control plant.

In RTHS, it is expected that *T* should be close to 1 to achieve the reference command, especially over the concerned frequency band. Afterward, *T* should be small enough to suppress the modeling errors over the high-frequency range, which will also diminish the effect of measurement noise. Furthermore, if the gain of *W*T decreases quickly over the high-frequency range, the measurement noise will be suppressed effectively. Hence, for practical purposes, the recommended form of function *W*T is given by

$$\mathcal{W}\_T = hs^2 + ms + n \tag{8}$$

where *h*, *m*, and *n* are adjustable parameters. It should be noted that *W*T*P* should be a rational function. If not, the form in Equation (8) should be modified. Examples can be found in the subsequent sections.
