Page 917 - The Mechatronics Handbook

0066_Frame_C30 Page 28 Thursday, January 10, 2002 4:44 PM

for all w. This, in turn, implies that the associated sensitivity singular values satisfy

1
(
[
s max S KF jw)] = -------------------------------------- ≤ 1 ( 0dB) (30.156)
s min S KF jw([ )] −1
for all w, where

(
(
S KF jw) = [ I + G KF jw)] −1 (30.157)
• From the above sensitivity singular value relationship, we obtain the follow celebrated KBF loop
margins:
infinite upward gain margin,
1
at least (6 dB) downward gain margin,
--
2
at least ±60° phase margin.
The above gain margins apply to simultaneous and independent gain perturbations when the
loop is broken at the output. The same holds for the above phase margins. The above margins
are NOT guaranteed for simultaneous gain and phase perturbations. It should be noted that
these margins can be easily motivated using elementary SISO Nyquist stability arguments [2,8].
• From the above sensitivity singular value relations, we obtain the following complementary
sensitivity singular value relationship:

(
(
(
[
[
[
–
s max T KF jw)] = s max IS KF jw)] ≤ 1 + s max S KF jw)] ≤ 2 ( 6dB) (30.158)
for all w, where
T KF = IS KF = G KF 1 + G KF ] −1 (30.159)
[
–
2. Recovery of Target Loop L o Using Model-Based Compensator. The second step is to use a model-
based compensator K opt = [ A − BG c − H f C, H f , G c ] where G c is found by solving the CARE
with A, B, M = C, R = rI n × n with ρ a small positive scalar. Since r is small, we call this a
u u
cheap control problem.
• If the plant P = [A, B, C] is minimum phase, then it can be shown that
lim X = 0 (30.160)
r → 0 +

lim rG c = WC (30.161)
r → 0 +
T
for some orthonormal W (i.e., W W = WW = I )
T
lim PK opt = L o (30.162)
r → 0 +

In such a case, PK opt ≈ L o for small ρ and hence PK opt will possess stability margins that are
close to those of L o (at the plant output)—whatever method was used to design L o . It must
be noted that the minimum phase condition on the plant P is a suficient condition. It is
not necessary. Moreover, G c need not be computed using a CARE. In fact, any G c which (1)
satisfies a limiting condition lim rG c = WC for some invertible matrix W and which (2)
r → 0 +
ensures that A − BG c is stable (for small r), will result in LTR at the plant output. This result

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