National Instruments Xmath - page 98

Chapter 6 Tutorial
© National Instruments Corporation 6-3 Xmath Model Reduction Module
With a state weighting matrix,
Q = 1e-3*diag([2,2,80,80,8,8,3,3]);
R = 1;
(and unity control weighting), a state-feedback control-gain is determined
through a linear-quadratic performance index minimization as:
[Kr,ev] = regulator(sys,Q,R);
A–B × Kr is stable. Next, with an input noise variance matrix Q=WtBBWt
where,
and measurement noise covariance matrix =1, an estimation gain Ke
(so that A–KeC is stable) is determined:
Qhat = Wt*b*b'*Wt;
Rhat = 1;
[Ke,ev] = estimator(sys,Qhat,Rhat,{skipChks});
The keyword skipChks circumvents syntax checking in most functions.
Itis used here because we know that Qhat does not fulfill positive
semidefiniteness due to numerics).
sysc=lqgcomp(sys,Kr,Ke);
poles(sysc)
ans (a column vector) =
-0.296674 + 0.292246 j
-0.296674 - 0.292246 j
-0.15095 + 0.765357 j
-0.15095 - 0.765357 j
-0.239151 + 1.415 j
-0.239151 - 1.415 j
-0.129808 + 1.84093 j
-0.129808 - 1.84093 j
The compensator itself is open-loop stable. A brief explanation of how Q
and Wt are chosen is as follows. First, Q is chosen to ensure that the loop
gain (which would be relevant were the state measurable)
meets the constraints as far as possible. However, it is not possible to obtain
a 40 dB per decade roll-off at high frequencies, as LQ design virtually
always yields a 20 dB per decade roll-off. Second, a loop transfer recovery
approach to the choice of as for some large ρ is modified through
the introduction of the diagonal matrix Wt. The larger entries of Wt, because
of the modal coordinate system, in effect promote better loop transfer
WtDIAG 0.346 0.346 0.024 0.0240.042 0.042,0.042 0.042,,,,[]()=
R
ˆ
KrjωIA–()
1–B
Q
ˆρBB′