Page 896 - The Mechatronics Handbook

0066_Frame_C30 Page 7 Thursday, January 10, 2002 4:43 PM

Feedback System Performance Issues. Generally, in designing a feedback controller K as shown in
Fig. 30.2, a designer must consider each of the following closed loop performance issues:

• Closed Loop Stability. The closed loop system should be stable. This involves all closed loop transfer
function matrices since we generally want all of them to be stable. A stricly proper closed loop
2
transfer function matrix whose H norm is infinite, for example, implies that the transfer function
matrix is unstable (or marginally stable). Stable strictly proper transfer function matrices neces-
3
sarily have a finite H norm.
• Command Following. The closed loop system should exhibit good low frequency reference com-
mand following; i.e., the output y (not to be confused with generalized plant measurements)
should track low frequency reference commands r that are issued to the feedback system. This
typically requires that the sensitivity transfer function matrix

def
S = I +[ PK] – 1 (30.25)

be small at low frequencies.
• Disturbance Attenuation. The closed loop system should exhibit good low frequency disturbance
attenuation. For disturbances d o modeled at the plant output, this requires that the sensitivity
transfer function matrix be small at low frequencies. For disturbances d i modeled at the plant
input, this requires that

def
T d y = SP (30.26)
i
be small at low frequencies.
• Sensor Noise Attenuation. The closed loop system should exhibit good high frequency noise n
attenuation. This typically requires that the complementary sensitivity transfer function matrix

def
T = IS (30.27)
–
be small at high frequencies.
• Stability Robustness. The closed loop system should exhibit robustness with respect to high fre-
quency unmodeled dynamics (e.g., flexible modes, parasitic dynamics, time delays, etc.); This
typically requires that the “peak” of some closed loop transfer function matrix be small at high
frequencies.
• Multiplicative Modeling Error. For a plant modeled as

P act = [ I + ∆]P (30.28)

where P act represents the actual plant, P represents a nominal model, and ∆ represents a stable
multiplicative perturbation at the plant output, the relevant closed loop transfer function matrix
(that seen by ∆) is T.
• Additive Modeling Error. For a plant modeled as

P act = P + ∆ (30.29)

©2002 CRC Press LLC