Page 921 - The Mechatronics Handbook

0066_Frame_C30 Page 32 Thursday, January 10, 2002 4:44 PM

−1
u ˆ
u ˆ
–
that hold at (e.g., open loop transfer function matrix at is L i = G c (sI A) B ) in Fig. 30.7
get transferred to the plant input u (compensator output) within the feedback loop.
Comment 30.13 (Stability Margins and Peak Sensitivity)
The peak on the sensitivity plot is very important in the design of a feedback system. A large peak, for
example, may be due to a closed loop pole near the imaginary axis. This certainly is undesirable. We thus
want the peak to be “small.” It can be shown that the peak necessarily establishes gain and phase margin
bounds.
Suppose that the peak sensitivity is bounded above by a ≥ 1; i.e., s max S(jw) < a for all w. It can be
shown that the feedback loop then enjoys the following nominal multivariable stability robustness (gain
and phase) margin bounds:

a
↑GM > ------------ (30.191)
–
a 1
a
↓GM < ------------ (30.192)
a + 1
−1 1
PM > 2sin   -------  (30.193)
2a

These bounds may be easily motivated using SISO Nyquist [2,8] ideas as follows.
If

Sjw) < a (30.194)
(
for all ω, then it follows that

1
--- < 1 + Ljω( ) (30.195)
α
for all ω. This, however, implies that the Nyquist plot associated with L cannot penetrate a circle centered
at −1 with radius 1/α, left most end point at −[(α+1)/a], and right most end point at −[(α −1)/a]. The
upward gain margin bound follows from the right most point of the circle. The downward gain margin
bound follows from the left most point of the circle. The phase margin bound can be obtained with a
little geometry.
2
The following example considers the application of H theory to a robotic manipulator.
2
Example 30.5 (HH -LQG/LTR Design for PUMA 560 Robotic Manipulator)
2
In this example, we show how H optimization may be used to design an LQG/LTR controller for a
PUMA 560 robotic manipulator. The manipulator is shown in Fig. 30.8.
A two degree-of-freedom (dof) linear model P = [A p , B p , C p ] was used to initiate the design process.
Linearizing the PUMA’s nonlinear model [9] about the equilibrium point θ 1 = 90° θ 2 = 0° (both links
vertical), results in the following linear model:

x ˙ p = A p x p + B p m p (30.196)
y p = C p x p (30.197)

u p = [ t 1 t 2 ] T (30.198)
x p = [ q 1 q 2 q 1 q 2 ] T (30.199)
˙
˙
y p = [ q 1 q 2 ] T (30.200)

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