Modern Control Engineering

Example Problems and Solutions 55

which can be rewritten as

By taking the inverse Laplace transforms of the preceding four equations, we obtain

Thus, a state-space model of the system in the standard form is given by

It is important to note that this is not the only state-space representation of the system. Infinite-

ly many other state-space representations are possible. However, the number of state variables is

the same in any state-space representation of the same system. In the present system, the num-

ber of state variables is three, regardless of what variables are chosen as state variables.

A–2–9. Obtain a state-space model for the system shown in Figure 2–27(a).

Solution.First, notice that (as+b)/s^2 involves a derivative term. Such a derivative term may be

avoided if we modify (as+b)/s^2 as

Using this modification, the block diagram of Figure 2–27(a) can be modified to that shown in

Figure 2–27(b).

Define the outputs of the integrators as state variables, as shown in Figure 2–27(b). Then from

Figure 2–27(b) we obtain

which may be modified to

Y(s)=X 1 (s)

sX 2 (s)=-bX 1 (s)+bU(s)

sX 1 (s)=X 2 (s)+aCU(s)-X 1 (s)D

Y(s)=X 1 (s)

X 2 (s)

U(s)-X 1 (s)

=

b

s

X 1 (s)

X 2 (s)+aCU(s)-X 1 (s)D

=

1

s

as+b

s^2

= aa+

b

s

b

1

s

C

x 1

x 2

x 3

y=[1 0 0] S

C

x# 1

x# 2

x# 3

S = C

- 5

0

1

10

0

0

0

- 1

- 1

SC

x 1

x 2

x 3

S +C

0

1

0

Su

y =x 1

x# 3 =x 1 - x 3

x

2 =-x 3 +u

x# 1 =-5x 1 +10x 2

Y(s)=X 1 (s)

sX 3 (s)=X 1 (s)-X 3 (s)

sX 2 (s)=-X 3 (s)+U(s)

sX 1 (s)=-5X 1 (s)+10X 2 (s)