Page 909 - The Mechatronics Handbook

0066_Frame_C30 Page 20 Thursday, January 10, 2002 4:43 PM

moment of inertia about its c.g. is very small. This assumption leads to a simple first order model. The
x ˙
missile’s angular velocity is assumed to be proportional to its attitude x and the process noise w I = ξ.
T
Regulated signals z = [z l z 2 ] include the vehicle’s pitch attitude z l = x and a weighted control input z 2 =
ru . Here, r > 0 is a design parameter to be selected below. The vehicle’s pitch attitude is measured.
The pitch attitude measurement y includes additive sensor noise w 2 = q.
Missile Model. The (generalized) missile model is given as follows:

x ˙ = x ++ u (30.94)
x

x
z = (30.95)
ru
y = x + mq (30.96)

where µ > 0 is design parameter to be selected below.
2
Design Specifications. The goal is to design a real-rational proper model-based H optimal compensator
) which results in a stable closed loop system with a dominant closed loop pole
(i.e., minimizes T wz 2
H
at s = −5 (settling time t s ≈ 1 s).
Construction of Generalized Plant. The above model may be rewritten as follows:

x ˙ = x + 10 x + u (30.97)
q

z = 1 x + 0 2 × 2 x + 0 u (30.98)
0 q s

y = x + x + 0 1 × u (30.99)
0 m 1
q

From this, it follows that

A = 1, B 1 = 10 , B 2 = 1 (30.100)

C 1 = 1 , D 11 = 0 2 × , D 12 = 0 (30.101)
2
0 r
C 2 = 1, D 21 = 0 m , D 22 = 0 1 × 1 (30.102)

2
2
H Problem Assumptions. We now check each of the H output feedback problem assumptions, as
stated in Assumption 30.2. From the above data, it follows that D 11 = 0 2 × 2 , D 22 = 0 1 × 1 , and (A, B 2 , C 2 )
is stabilizable and detectable, and

T
R = D 12 D 12 = r > 0 (30.103)
Θ = D 21 D 21 = m > 0 (30.104)
T

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