Page 899 - The Mechatronics Handbook

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

or

W 1 – W 1 P
z 1
= (30.42)
z 2 0 W 2 w
0 W 3 P u
z 3
y I – P

State Space Representation for Generalized Plant G
Next we obtain a two-port state space representation for G. To do so, we assume the following state space
representations:

P = [A , B , C , D ] with state x p (30.43)
p
p
p
p
W = [A , B , C , D ] with state x 1 (30.44)
1
1
1
1
1
W = [A , B , C , D ] with state x 2 (30.45)
2
2
2
2
2
W = [A , B , C , D ] with state x 3 (30.46)
3
3
3
3
3
( 3 {} , y)
3
x ˙ i
To obtain the desired state space representation for G, we need to express the signals {} i=1, x ˙ p, z i i=1
in terms of the signals {}( x i i=1 , x p , w, u). This is just a matter of simple bookkeeping. Doing so yields
3
the following:
(
x ˙ 1 = A 1 x 1 + B 1 y = A 1 x 1 + B 1 wC p x p – D p u) = A 1 x 1 – B 1 C p x p – B 1 D p u (30.47)
–
x ˙ 2 = A 2 x 2 + B 2 u (30.48)
(
x ˙ 3 = A 3 x 3 + B 3 z ˆ 3 = A 3 x 3 + B 3 C p x p + D p u) = A 3 x 3 + B 3 C p x p + B 3 D p u (30.49)
x ˙ p = A p x p + B p u (30.50)
(
z 1 = C 1 x 1 + D 1 y = C 1 x 1 + D 1 wC p x p – D p u) = C 1 x 1 – D 1 C p x p + D 1 wD 1 D p u (30.51)
–
–
z 2 = C 2 x 2 + D 2 u (30.52)
(
z 3 = C 3 x 3 + D 3 z ˆ 3 = C 3 x 3 + D 3 C p x p + D p u) = C 3 x 1 + D 3 C p x p + D 3 D p u (30.53)
y = w C p x p – D p u = – C p x p + wD p u (30.54)
–
–
The above equations may be written in standard two-port form:

x ˙ AB 11 B 12 x
= (30.55)
z C 11 D 11 D 12 w
y C 21 D 21 D 22 u

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