Modern Control Engineering

710 Chapter 9 / Control Systems Analysis in State Space

Solution.Equation (9–103), where m=4, can be expanded as follows:

Since

and

we obtain

= 0

+fAl 1 BCAA-l 4 IBAA-l 3 IBAA-l 2 IBAl 4 - l 3 BAl 4 - l 2 BAl 3 - l 2 BD

• fAl 2 BCAA-l 4 IBAA-l 3 IBAA-l 1 IBAl 4 - l 3 BAl 4 - l 1 BAl 3 - l 1 BD

+fAl 3 BCAA-l 4 IBAA-l 2 IBAA-l 1 IBAl 4 - l 2 BAl 4 - l 1 BAl 2 - l 1 BD

• fAl 4 BCAA-l 3 IBAA-l 2 IBAA-l 1 IBAl 3 - l 2 BAl 3 - l 1 BAl 2 - l 1 BD

¢=f(A)CAl 4 - l 3 BAl 4 - l 2 BAl 4 - l 1 BAl 3 - l 2 BAl 3 - l 1 BAl 2 - l 1 BD

4

1

li

l^2 i

l^3 i

1

lj

l^2 j

l^3 j

1

lk

l^2 k

l^3 k

I

A

A^2

A^3

4 =AA-lk IBAA-lj IBAA-li IBAlk-ljBAlk-liBAlj-liB

4

1

l 1

l^21

l^31

1

l 2

l^22

l^32

1

l 3

l^23

l^33

1

l 4

l^24

l^34

4 =Al 4 - l 3 BAl 4 - l 2 BAl 4 - l 1 BAl 3 - l 2 BAl 3 - l 1 BAl 2 - l 1 B

+fAl 1 B 4

1

l 2

l^22

l^32

1

l 3

l^23

l^33

1

l 4

l^24

l^34

I

A

A^2

A^3

4

+fAl 3 B 4

1

l 1

l^21

l^31

1

l 2

l^22

l^32

1

l 4

l^24

l^34

I

A

A^2

A^3

4 - fAl 2 B 4

1

l 1

l^21

l^31

1

l 3

l^23

l^33

1

l 4

l^24

l^34

I

A

A^2

A^3

4

=f(A) 4

1

l 1

l^21

l^31

1

l 2

l^22

l^32

1

l 3

l^23

l^33

1

l 4

l^24

l^34

4 - fAl 4 B 4

1

l 1

l^21

l^31

1

l 2

l^22

l^32

1

l 3

l^23

l^33

I

A

A^2

A^3

4

¢= 5

1

l 1

l^21

l^31

fAl 1 B

1

l 2

l^22

l^32

fAl 2 B

1

l 3

l^23

l^33

fAl 3 B

1

l 4

l^24

l^34

fAl 4 B

I

A

A^2

A^3

f(A)

5

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