Page 913 - The Mechatronics Handbook

0066_Frame_C30 Page 24 Thursday, January 10, 2002 4:44 PM

offers an infinite upward gain margin and a downward gain margin of 1/6 (−15.56 dB). The resulting
unity gain crossover frequency is w g = 35 = 5.92 rad/s and the associated phase margin is about 99.59°.
Not bad.
The following example extends the LQG/LTR ideas presented in Example 30.3 to the general MIMO
2
setting—enabling the design of feedback loops (with nominal robustness margins) via H optimization.

Example 30.4 (MIMO LQG and LQG/LTR Control Design Via HH 2 Optimization)
We consider a MIMO plant P defined by the state space representation

x ˙ = Ax + Bu (30.139)

y = Cx (30.140)

It is assumed that the plant P = [A, B, C] is stabilizable and detectable.
2
The goal is to demonstrate how the H optimal output feedback solution that has been presented may
be used to solve MIMO LQG control problems. We specifically would like to present a method which
lends itself to the concept of LTR—whereby we use a model-based LQG controller to recover a target
loop transfer function matrix with desirable closed loop properties. Our motivation is not optimal
stochastic LQG control problems; it is the design of control laws with desirable closed loop properties.
Construction of Generalized Plant G. With our final objective being a model-based compensator defined
by a control gain matrix G c and a filter gain matrix H f , we consider the following generalized plant:

x ˙ = Ax + Lξ + Bu (30.141)

Mx
z = --------------------- (30.142)
rI n × n
u u
y = Cx + µq (30.143)

where u is the control, x is the (generalized) plant state, w 1 = ξ represents process noise in the state
n×n
n×n
equation, w 2 = q represents sensor noise in the measurement equation, A ∈R n×n , L ∈ R u , B ∈ R u ,
M ∈ R n ×n , C ∈ R n ×n , n y = n u , r > 0, µ > 0.
y
y
Design Parameter Assumptions. It is assumed that either:
(A, L) has no imaginary uncontrollable modes and (A, M) is detectable, or
(A, L) is stabilizable and (A, M) has no imaginary unobservable modes.
Here, L, M, m, and r should be viewed as “design parameters” that are selected in order to obtain control
and filter gain matrices G, and H f such that the resulting model-based compensator exhibits desirable
closed loop properties.
Two-Port State Space Representation for Generalized Plant G. The above model may be rewritten in
two-port state space form as follows

A L 0 n × B
n
y
x
x ˙ 0 n × n
= M 0 n × n u 0 n × n y y u x (30.144)
y
y
z
0 n × n 0 n × n 0 n × n rIn × n q
y u u u u y u u
u
C 0 n × mIn × n
y n u y y
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