Modern Control Engineering

88 Chapter 3 / Mathematical Modeling of Mechanical Systems and Electrical Systems

A–3–3. Obtain a state-space representation of the system shown in Figure 3–22.

Solution.The system equations are

The output variables for this system are y 1 andy 2 .Define state variables as

Then we obtain the following equations:

Hence, the state equation is

and the output equation is

A–3–4. Obtain the transfer function Xo(s)/Xi(s)of the mechanical system shown in Figure 3–23(a). Also

obtain the transfer function Eo(s)/Ei(s)of the electrical system shown in Figure 3–23(b). Show that

these transfer functions of the two systems are of identical form and thus they are analogous systems.

B

y 1

y 2

R= B

1

0

0

0

0

1

0

0

RD

x 1

x 2

x 3

x 4

T

D

x# 1

x# 2

x# 3

x# 4

T = F

0

-

k

m 1

0

k

m 2

1

-

b

m 1

0

0

0

k

m 1

0

• 
  

k

m 2

0

0

1

0

VD

x 1

x 2

x 3

x 4

T + E

0

0

0

1

m 2

Uu

x# 4 =

1

m 2

C-kAy 2 - y 1 B+uD=

k

m 2

x 1 -

k

m 2

x 3 +

1

m 2

u

x# 3 =x 4

x# 2 =

1

m 1

C-by# 1 - kAy 1 - y 2 BD=-

k

m 1

x 1 -

b

m 1

x 2 +

k

m 1

x 3

x# 1 =x 2

x 4 =y# 2

x 3 =y 2

x 2 =y# 1

x 1 =y 1

m 2 y$ 2 +kAy 2 - y 1 B=u

m 1 y$ 1 +by# 1 +kAy 1 - y 2 B= 0

m 1 m 2

k

y 1

b

u

y 2

Figure 3–22

Mechanical system.

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