Chapter 4 ControllerSynthesis
MATRIXx Xmath Robust Control Module 4-6 ni.com
Selecting these weights has much the same effect here. Specifically, let Hzv
be the closed-loop transfer matrix (with u=Kγ) from inputs:
to outputs:
Thus,
Suppose that the controller u=Ky approximates Equation4 -2. Thus,
In many cases, this means that the maximum singular value of the
frequency response matrix (WoutHzvWin)( jω) is constant over all
frequencies. That is,
An interpretation is that the weighting filters Win and Wout determine the
shape of the closed-loop frequency response Hzw(jω), and γopt determines
the peak value. This observation helps motivate the selection of the weights
so as to shape the closed-loop frequency response matrix Hzw(jω).
Observe, however, that the elements of the frequency response matrix,
(WoutHzvWin)( jω), need not be constant. Instead, the maximum singular
value of at least one of the four subblocks is within 3 dB of γopt. For all ω,
vd
n
=
zyreg
u
=
Hzv
HyregdHyregn
Hud Hun
=
WoutHzvWin ∞γopt
≈
σmax
WregHyregdWdist WregHyregnWnoise
WactHudWdist WactHjnWnoise
jw()
⎝⎠
⎜⎟
⎜⎟
⎛⎞
γopt
≈
Mω()σ
max WoutHzvWin
()jω()[]2Mω()≤≤