56 Chapter 2 / Mathematical Modeling of Control Systems
Taking the inverse Laplace transforms of the preceding three equations, we obtain
Rewriting the state and output equations in the standard vector-matrix form, we obtain
A–2–10. Obtain a state-space representation of the system shown in Figure 2–28(a).
Solution.In this problem, first expand (s+z)/(s+p)into partial fractions.
Next, convert K/Cs(s+a)Dinto the product of K/sand1/(s+a).Then redraw the block diagram,
as shown in Figure 2–28(b). Defining a set of state variables, as shown in Figure 2–28(b), we ob-
tain the following equations:
y =x 1
x# 3 =-(z-p)x 1 - px 3 +(z-p)u
x# 2 =-Kx 1 +Kx 3 +Ku
x# 1 =-ax 1 +x 2
s+z
s+p
= 1 +
z-p
s+p
y =[1 0]B
x 1
x 2
R
B
x# 1
x# 2
R =B
- a
- b
1
0
RB
x 1
x 2
R +B
a
b
Ru
y =x 1
x
2 =-bx 1 +bu
x# 1 =-ax 1 +x 2 +au
U(s) Y(s)
as+b^1
s^2
(a)
(b)
a
U(s) Y(s)
b
s
1
s
X 2 (s) X 1 (s)
+–
+–
++
Figure 2–27
(a) Control system;
(b) modified block
diagram.
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