Modern Control Engineering

654 Chapter 9 / Control Systems Analysis in State Space

EXAMPLE 9–2 Consider the following state-space representation of a system.

(9–13)

(9–14)

Equations (9–13) and (9–14) can be put in a standard form as

(9–15)

(9–16)

where

The eigenvalues of matrix Aare

l 1 =–1, l 2 =–2, l 3 =–3

Thus, three eigenvalues are distinct. If we define a set of new state variables z 1 ,z 2 ,andz 3 by the

transformation

or

x=Pz (9–17)

where

(9–18)

then, by substituting Equation (9–17) into Equation (9–15), we obtain

By premultiplying both sides of this last equation by P–1,we get

(9–19)

or

• C

3

- 3

1

2.5

- 4

1.5

0.5

- 1

0.5

SC

0

0

6

Su

C

z# 1

z# 2

z# 3

S =C

3

- 3

1

2.5

- 4

1.5

0.5

- 1

0.5

SC

0

0

- 6

1

0

- 11

0

1

- 6

SC

1

- 1

1

1

- 2

4

1

- 3

9

SC

z 1

z 2

z 3

S

z# =P-^1 APz+P-^1 Bu

Pz# =APz+Bu

P= C

1

l 1

l 12

1

l 2

l 22

1

l 3

l 32

S= C

1

- 1

1

1

- 2

4

1

- 3

9

S

C

x 1

x 2

x 3

S = C

1

- 1

1

1

- 2

4

1

- 3

9

SC

z 1

z 2

z 3

S

A= C

0

0

- 6

1

0

- 11

0

1

- 6

S, B= C

0

0

6

S, C=[1 0 0]

y=Cx

x# =Ax+Bu

y =[1 0 0]C

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S= C

0

0

- 6

1

0

- 11

0

1

- 6

SC

x 1

x 2

x 3

S +C

0

0

6

Su

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