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With feedback only, on the other hand, the variance of y is simply
in the frequency domain and
|Y(f)| 2 = |Tywt(f)| 2 Swtwt (f) (220)
Var(y) =
∞
−∞
|Tywt(f)| 2 Swtwt (f)df (221)
in the time domain. Section 10.2.3 provides an explanation for treating reduction in variance
as a control objective.
By comparing Eq. (218) with Eq. (220), the frequencies where feedforward control with
ˆwt = wm is beneficial can be found. Ideal feedforward control with no prefilter reduces the
variance of y at the frequencies where Eq. (222) is satisfied:
Swtwt (f)+Swmwm (f)−2ℜ{Swtwm (f)} < Swtwt (f). (222)
Equation (222) can be rearranged as
1
2 < ℜ{Swtwm (f)}
. (223)
Swmwm (f)
10.2.2 Feedforward Control with Imperfect Measurements
By introducing a prefilter Hpre, shown in Fig. 134, to form an estimate of wt based on the
measured wm, the variance of the output variable y can be reduced below the value without
prefiltering so that feedforward control is always an improvement over feedback only (Simley
and Pao, 2013b). With prefiltering, the output variable y is given as
In the frequency domain, the variance of y is given by
y = Tywt(wt −Hprewm). (224)
|Y(f)| 2 = |Tywt(f)| 2 |Wt(f)−Hpre(f)Wm(f)| 2 . (225)
The optimal prefilter should minimize the variance of y in Eq. (225) at all frequencies. The
classical solution to this minimization problem is the transfer function
Hpre(f) = Swtwm (f)
. (226)
Swmwm (f)
When the optimal minimum-variance prefilter is employed, the variance of y in the frequency
domain reduces to
|Y(f)| 2 = |Tywt(f)| 2 Swtwt (f) 1−γ 2 wtwm (f)
(227)
where γ2 (f) is the coherence function between the true wind disturbance and the lidar
wtwm
measurement. The coherence function describes the correlation between two signals as a
function of frequency, with an output of 0 indicating no correlation and 1 describing perfect
correlation. The coherence between signals a(t) and b(t) is defined as
γ 2 ab(f) =
2
|Sab(f)|
. (228)
Saa(f)Sbb(f)
Eq. (227) reveals that when measurement coherence is 1 (perfect correlation), the variance
of y can be reduced to zero. When measurement coherence is 0 (the measurement contains
no information about the true disturbance), on the other hand, feedforward control is not
used and the variance of y reduces to the feedback only case (Eq. (220)).
Inreality,theoptimalprefilterdefinedbyEq.(226)cannotberealizedbecauseofconstraints
on the preview time available to the filter (Simley and Pao, 2013b). Instead, the optimal filter
can be calculated using time domain methods where the available preview time appears as a
hard constraint. Further details on implementation of the optimal measurement filter can be
found in Simley and Pao (2013b).
DTU Wind Energy-E-Report-0029(EN) 197