Digital Control Systems [MEE 4003] - Kckong.info

2.2. LAPLACE TRANSFORMS AND TRANSFER FUNCTIONS 37
where p i ’s are the poles of G(s). Note that the new transfer function can be represented
with the following differential equations.
x˙
1 = p 1 x 1 +u
x˙
2 = p 2 x 2 +u
x˙
3 = p 3 x 3 +u
y = k 1 x 1 +k 2 x 2 +k 3 x 3
In matrix form:
⎡
d
⎣
dt
⎤
x 1
x 2
⎦ =
x 3
⎡ ⎤⎡
p 1 0 0
⎣ 0 p 2 0 ⎦⎣
0 0 p 3
} {{ }
F d
⎡
y = [ ]
k 1 k 2 k 3
⎣
} {{ }
H d
⎤ ⎡
x 1
x 2
⎦+ ⎣
x 3
⎤
x 1
x 2
⎦
x 3
1
1
1
⎤
} {{ }
G d
⎦u
Note thatH d (sI −F d ) −1 G d = k 1
s−p 1
+ k 2
s−p 2
+ k 3
s−p 3
.
Method 3-2: Jordan canonical form (JCF)
If the transfer function in (2.12) has a repeated pole (p 2 = p 3 = p m ), it can be reduced to
G(s) = k 1 k 2
+
s−p 1 (s−p m ) + k 3
2 s−p m
The new transfer function can be represented with the following differential equations.
x˙
1 = p 1 x 1 +u
x˙
2 = p 2 x 2 +x 3
x˙
3 = p 3 x 3 +u
y = k 1 x 1 +k 2 x 2 +k 3 x 3
In matrix form:
⎡
d
⎣
dt
⎤
x 1
x 2
⎦ =
x 3
⎡ ⎤⎡
p 1 0 0
⎣ 0 p 2 1 ⎦⎣
0 0 p 3
} {{ }
F j
⎡
y = [ ]
k 1 k 2 k 3
⎣
} {{ }
H j
⎤ ⎡
x 1
x 2
⎦+ ⎣
x 3
⎤
x 1
x 2
⎦
x 3
1
0
1
⎤
} {{ }
G j
⎦u
Note thatH j (sI −F j ) −1 G j = k 1
s−p 1
+ k 2
(s−p m) 2 + k 3
s−p m
.
Digital Control Systems, Sogang University
Kyoungchul Kong