Modern Control Engineering

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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