Modern Control Engineering

Example Problems and Solutions 53

we find

Referring to Equation (2–35), we have

Referring to Equation (2–34), we define

Then referring to Equation (2–36),

Hence, the state-space representation of the system is

This is one possible state-space representation of the system. There are many (infinitely many)
others. If we use MATLAB, it produces the following state-space representation:

See MATLAB Program 2-4. (Note that all state-space representations for the same system are
equivalent.)

C

x 1

x 2

x 3

y=[- 7 - 9 - 2] S +2u

C

x# 1

x# 2

x# 3

S =C

- 4

1

0

- 5

0

1

- 2

0

0

SC

x 1

x 2

x 3

S +C

1

0

0

Su

y=[ 1 0 0 ]C

x 1

x 2

x 3

S + 2 u

C

x# 1

x

2

x# 3

S= C

0

0

- 2

1

0

- 5

0

1

- 4

SC

x 1

x 2

x 3

S+ C

- 7

19

- 43

Su

=-2x 1 - 5x 2 - 4x 3 - 43u

x# 3 =-a 3 x 1 - a 2 x 2 - a 1 x 3 +b 3 u

x# 2 =x 3 +19u

x# 1 =x 2 - 7u

x 3 =x# 2 - b 2 u=x# 2 - 19u

x 2 =x# 1 - b 1 u=x# 1 +7u

x 1 =y-b 0 u=y-2u

= 2 - 4 19 - 5 (-7)- 2 * 2 =- 43

b 3 =b 3 - a 1 b 2 - a 2 b 1 - a 3 b 0

b 2 =b 2 - a 1 b 1 - a 2 b 0 = 1 - 4 *(-7)- 5 * 2 = 19

b 1 =b 1 - a 1 b 0 = 1 - 4 * 2 =- 7

b 0 =b 0 = 2

b 0 =2, b 1 =1, b 2 =1, b 3 = 2

a 1 =4, a 2 =5, a 3 = 2