Modern Control Engineering

Section 3–2 / Mathematical Modeling of Mechanical Systems 67

Next we shall obtain a state-space model of this system. We shall first compare the differen-
tial equation for this system

with the standard form

and identify a 1 , a 2 , b 0 , b 1 ,andb 2 as follows:

Referring to Equation (3–35), we have

Then, referring to Equation (2–34), define

From Equation (2–36) we have

and the output equation becomes

or

(3–3)

and

(3–4)

Equations (3–3) and (3–4) give a state-space representation of the system. (Note that this is not
the only state-space representation. There are infinitely many state-space representations for the
system.)

y=[1 0]B

x 1

x 2

R

B

x# 1

x# 2

R = C

0

-

k

m

1

-

b

m

SB

x 1

x 2

R + D

b

m

k

m

• a

b

m

b

2 Tu

y=x 1

x# 2 =-a 2 x 1 - a 1 x 2 +b 2 u=-

k

m

x 1 -

b

m

x 2 + c

k

m

• a

b

m

b

2

du

x# 1 =x 2 +b 1 u=x 2 +

b

m

u

x 2 =x# 1 - b 1 u=x# 1 -

b

m

u

x 1 =y-b 0 u=y

b 2 =b 2 - a 1 b 1 - a 2 b 0 =

k

m

• a

b

m

b

2

b 1 =b 1 - a 1 b 0 =

b

m

b 0 =b 0 = 0

a 1 =

b

m

, a 2 =

k

m

, b 0 =0, b 1 =

b

m

, b 2 =

k

m

y$+a 1 y# +a 2 y=b 0 u$+b 1 u# +b 2 u

y$+

b

m

y# +

k

m

y=

b

m

u# +

k

m

u