Page 928 - The Mechatronics Handbook

0066_Frame_C30 Page 39 Thursday, January 10, 2002 4:44 PM

sensitivity and complementary sensitivity singular values are desirable in that they suggest that the target
loop will possess:
• good low frequency command following properties,
• good low frequency disturbance attenuation properties,
• good high frequency sensor noise attenuation properties, and
• good MIMO stability margins (nearly infinite upward gain margin, at least 6 dB downward gain
margin, and at least ±60° phase margin) at the output.
The complementary sensitivity singular values suggest that a reference command prefilter W would reduce
overshoot due to step reference commands. The design of such a filter will be considered below.
Step 3: Solve Cheap Control Problem to Recover Target Loop at Plant Output
Next we solve an appropriately formulated “cheap LQR control problem” that would produce a control
2
gain matrix G c such that the H optimal model-based compensator K d = [A − BG c − H f C, H f , G c ] with
P d = [A, B, C] approximates (“recovers”) the target loop transfer function matrix L o = G KF ; i.e.,
P d K d ≈ L o = G KF (30.222)
−13
This was done by solving the following CARE (using the “lqr” command) with R = ρI 2×2 (ρ = 10 ):
XA + A X + C CXBR B X = 0 (30.223)
–
1
T
T
T
–
for X ≥ 0 and forming the control gain matrix
G c = R B X (30.224)
1
T
–
= 987.9832 – 543.0034 3162945.2928 56.9921 13941.9005 2069.8324 (30.225)
– 543.0034 3657.5891 11.7867 3162634.3919 2069.7987 3765.7978
Doing so yields the following closed loop regulator poles (l i (A − BG c )):
s = −440.8808, −220.4404 ± j381.7871, −1881.9053, −940.9527 ± j1629.7168 (30.226)

All have damping factors greater than or equal to ζ = 0.5. As a practical note to facilitate real-time imple-
mentation of the resulting controller, one might use model reduction techniques [10] to remove some
of the very high frequency poles in the compensator. Doing so would permit using a larger integration
step size in any real-time embedded system or microprocessor implementation.

Step 4: Construct Final Controller K
Next we form the final controller as follows:

K = K d (30.227)
-----
s
[ ABG c – H f C, H f , G c ]
–
= ----------------------------------------------------------- (30.228)
s
= [ A k , B k , C k ] (30.229)

A state space representation for this controller is given by

A K = 0 2 × 2 G c , B K = 0 2 × 2 (30.230)
–
0 6 × 2 ABG c – H f C H f
C K = I 2 × 2 0 2 × 6 (30.231)

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