Modern Control Engineering

696 Chapter 9 / Control Systems Analysis in State Space

Solution.The partial-fraction expansion of Equation (9–91) becomes

which may be written as

(9–92)

Define

Notice that the following relationships exist among X 1 (s),X 2 (s), and X 3 (s):

Then, from the preceding definition of the state variables and the preceding relationships, we

obtain

sXn(s)=-pn Xn(s)+U(s)







sX 4 (s)=-p 4 X 4 (s)+U(s)

sX 3 (s)=-p 1 X 3 (s)+U(s)

sX 2 (s)=-p 1 X 2 (s)+X 3 (s)

sX 1 (s)=-p 1 X 1 (s)+X 2 (s)

X 2 (s)

X 3 (s)

=

1

s+p 1

X 1 (s)

X 2 (s)

=

1

s+p 1

Xn(s)=

1

s+pn

U(s)







X 4 (s)=

1

s+p 4

U(s)

X 3 (s)=

1

s+p 1

U(s)

X 2 (s)=

1

As+p 1 B^2

U(s)

X 1 (s)=

1

As+p 1 B^3

U(s)

+

c 3

s+p 1

U(s)+

c 4

s+p 4

U(s)+p+

cn

s+pn

U(s)

Y(s)=b 0 U(s)+

c 1

As+p 1 B^3

U(s)+

c 2

As+p 1 B^2

U(s)

Y(s)

U(s)

=b 0 +

c 1

As+p 1 B^3

+

c 2

As+p 1 B^2

+

c 3

s+p 1

+

c 4

s+p 4

+p+

cn

s+pn

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