Page 894 - The Mechatronics Handbook

0066_Frame_C30 Page 5 Thursday, January 10, 2002 4:43 PM

Generalized Plant
The generalized plant G is assumed to possess the following two-port state space structure:

A B 1 B 2
G 11 G 12 AB
G = = C 1 0 n ×n D 12 = (30.10)
z w
G 21 G 22 CD
0 n ×n
C 2 D 21
y u
–
∈
(
∈
∈
1
z
y
z
–
where G ij s() = C i sI A) B j , A R n×n , B 1 R n×n w , B 2 ∈ R n×n u , C 1 ∈ R n ×n , C 2 R n ×n , D 12 ∈R n ×n u ,
D 21 ∈ R n ×n z .
u
Comment 30.6 (Weighting Functions: Satisfying Closed Loop Design Specifications)
As stated earlier, the generalized plant G may contain frequency dependent weighting functions as well
as a model for the physical system P (plant) being controlled. Typically P = G 22 = [A, B 2 , C 2 ]. Weighting
functions within G may be viewed as design parameters (mathematical knobs) that may be manipulated
3
by a designer to influence the H problem in a manner which results in a controller that is not just
optimal—a notion that is often irrelevant in practical applications—but which satisfies desired closed
loop design specifications. Weighting functions may be used to weight (penalize) tracking errors, actuator
and other signal levels, state estimation errors, etc. By making the weight on a signal large in a specific
frequency range, we are indirectly telling the optimization problem to find a controller that makes the signal
small in that frequency range. By making the weight on a signal small in a specific frequency range, we
are indirectly conveying our willingness to tolerate a signal which is large in that frequency range. This
idea can be illustrated via example.
Comment 30.7 (D 11 == == 0 Necessary, D 22 == == 0 Not Necessary)
D 11 = 0 Necessary. Note that we have assumed that D 11 = 0; i.e., there is no direct path from the exogenous
2
signals w to the regulated signals z. This assumption is essential for the H norm of the closed loop
2
will be infinite and the H problem will be
transfer function T wz to be finite. If D 11 ≠ 0, then T wz 2
H
ill-posed; i.e., make no sense. If we have a nonzero D 11 , adding strictly proper filters on either w or z (e.g.,
[1000/(s + 1000)]I) will result in D 11 = 0. In this sense, the assumption is not restrictive.
D 22 = 0 Not Necessary. It has also been assumed that D 22 = 0; i.e., the transfer function matrix D 22 from
controls u to measurements y is strictly proper. This assumption is very realistic since G 22 (our plant P)
is typically strictly proper in practice. If not, high frequency dynamics (e.g., actuator dynamics, flexible
modes, parasitics, etc.) may be included to make it strictly proper. One might even include a simple high
bandwidth low pass filter (e.g., 1000/(s + 1000)) to make G 22 strictly proper. If this is not desirable because
of the increased dimension, there is an alternative that does not increase the dimension of G.
ˆ ˆ
• One can always remove D 22 from G 22 to obtain a new generalized plant G with D 22 = 0. The term
ˆ
D 22 is then absorbed into an augmented controller K by noting that u is related to y as follows:
−1
u = K[y + D u] = [I − KD ] Ky (30.11)
22
22
Noting this, it follows that the augmented controller, denoted K ˆ , is given by
ˆ – 1
K = [ IKD 22 ] K (30.12)
–
ˆ
2
The H problem can then be carried out for G and K ˆ (without regard to D 22 ). When the optimal
ˆ ˆ
G
K
controller for is obtained, one can compute the optimal controller K for G using the relationship
[
ˆ
ˆ
K = K I + D 22 K] – 1 (30.13)
With this stated, the assumption D 22 = 0 is made without any loss of generality.
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