State-Variable Representation - Control of Color Imaging Systems: Analysis and Design - page 180

R 1 =1 Ω L =2 H

N ode 1

+ +

i L ( t )

+

+

v C 1 ( t )

C 1 =1 F

u ( t )

- v C 2 ( t ) = y ( t )

C 2 =1 F

-

R 2 =3 Ω

- -

Second loop

FIGURE 4.3 Electric circuit with three states.

Kirchhoff is voltage law (KVL) around the second loop is

'

x 1 (t) ¼ v C 2 (t) ¼ v C 1 (t) þ v R 2 (t) ¼ x 2 (t) þ R 2 C 1 x 2 (t) ¼ x 2 (t) þ 3 x 2 (t) (4 : 17)

Therefore, we have

1
3 x 1 (t)

1
3 x 2 (t)

x 2 (t) ¼ (4 : 18)

Kirchhoff s current law (KCL) at Node 1 yields

'

¼ C 1 x 2 (t) þ C 2 x 1 (t) ¼ x 2 (t) þ x 1 (t)

x 3 (t) (4 : 19)

Therefore,

1
3 x 1 (t)

1
3 x 2 (t)

x 1 (t) ¼ x 3 (t) x 2 (t) ¼ þ þ x 3 (t) (4 : 20)

Finally, writing the KVL around the first loop results in

R 1 x 3 (t) þ Lx 3 (t) þ x 2 (t) þ R 2 x 2 (t) ¼ u(t) (4 : 21)

Hence, we have

x 3 (t)

1
2 x 3 (t)

1
2 x 2 (t)

3
2

x 2 (t)

1
2 u(t)

1
2 x 1 (t)

1
2 x 3 (t)

1
2 u(t)

¼ þ ¼ þ (4 : 22)

Equations 4.18, 4.20, and 4.22 can be written in matrix form as

2
4

3
5 ¼

2
4

3
5

2
4

3
5 þ

2
4

3
5 u(t)

1
3

1
3

x 1 (t)
x 2 (t)
x 3 (t)

1 x 1 (t)

x 2 (t)
x 3 (t)

0
0
1
2

1
3

1
3

0 (4 : 23)

1
2

1
2

0