Modern Control Engineering

830 Chapter 10 / Control Systems Design in State Space

The left-hand side of Equation (10–154) is

The right-hand side of Equation (10–154) is

Thus, we see that Equation (10–154) holds true. Hence, we have proved Equation (10–153).

Next we shall show that

or

Notice that

Hence, we have shown that

Next define

Then Equation (10–151) becomes

(10–155)

and Equation (10–152) becomes

(10–156)

Referring to Equation (10–153),Equation (10–155) becomes

C

xˆ

1

xˆ

2

xˆ

3

S = C

0

1

0

0

0

1

• a 3


• a 2


• a 1

SC

xˆ 1

xˆ 2

xˆ 3

S +C

g 3

g 2

g 1

Su

y=CQxˆ+Du

xˆ

=Q-^1 AQxˆ +Q-^1 Bu

x=Qxˆ

[ 0 0 1 ]=C(WN* )-^1 =CQ

=[1 0 0]C

C

CA

CA^2

S =C

0 0 1 =[ 0 0 1 ]C

a 2

a 1

1

a 1

1

0

1

0

0

SC

C

CA

CA^2

S

C(WN*)-^1 =[ 0 0 1 ]

CQ=[0 0 1]

= C

• a 3
  0
  0

0

a 1

1

0

1

0

S

C

0

1

0

0

0

1

• a 3


• a 2


• a 1

S W= C

0

1

0

0

0

1

• a 3


• a 2


• a 1

SC

a 2

a 1

1

a 1

1

0

1

0

0

S

= C

• a 3
  0
  0

0

a 1

1

0

1

0

S

W C

0

0

• a 3

1

0

• a 2

0

1

• a 1

S = C

a 2

a 1

1

a 1

1

0

1

0

0

SC

0

0

• a 3

1

0

• a 2

0

1

• a 1

S

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