Modern Control Engineering

Example Problems and Solutions 57

Rewriting gives

Notice that the output of the integrator and the outputs of the first-order delayed integrators

C1/(s+a)and(z-p)/(s+p)Dare chosen as state variables. It is important to remember that

the output of the block (s+z)/(s+p)in Figure 2–28(a) cannot be a state variable, because this

block involves a derivative term,s+z.

A–2–11. Obtain the transfer function of the system defined by

Solution.Referring to Equation (2–29), the transfer function G(s)is given by

In this problem, matrices A,B,C, and Dare

A= C

- 1

0

0

1

- 1

0

0

1

- 2

S, B= C

0

0

1

S, C=[1 0 0], D= 0

G(s)=C(sI-A)-^1 B+D

y=[ 1 0 0 ]C

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S= C

- 1

0

0

1

- 1

0

0

1

- 2

SC

x 1

x 2

x 3

S + C

0

0

1

Su

y=[ 1 0 0 ]C

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S = C

• a


• K


• (z-p)

1

0

0

0

K

• p

SC

x 1

x 2

x 3

S +C

0

K

z-p

Su

uy

uy

(a)

(b)

s + z

s + p

K

s(s + a)

z – p

s + p

K

s

1

s + a

x 3 x 2 x 1

+–

+– +
Figure 2–28 +
(a) Control system;
(b) block diagram
defining state
variables for the
system.