Modern Control Engineering

674 Chapter 9 / Control Systems Analysis in State Space

Substituting 0 for l 1 and–2forl 2 in this last equation, we obtain

Expanding the determinant, we obtain

or

An alternative approach is to use Equation (9–48). We first determine a 0 (t)anda 1 (t)from

Sincel 1 =0andl 2 =–2,the last two equations become

Solving for a 0 (t)anda 1 (t)gives

Then can be written as

Linear Independence of Vectors. The vectors x 1 ,x 2 ,p,xnare said to be linearly

independent if

wherec 1 ,c 2 ,p,cnare constants, implies that

Conversely, the vectors x 1 ,x 2 ,p,xnare said to be linearly dependent if and only if xican

be expressed as a linear combination of xj(j=1, 2,p,n; jZi),or

xi= a

n

j= 1

jZi

cj xj

c 1 =c 2 =p=cn= 0

c 1 x 1 +c 2 x 2 +p+cn xn= 0

eAt=a 0 (t) I+a 1 (t) A=I+

1

2

A 1 - e-2tB A=B

1

0

1

2 A^1 - e

• 2tB
  e-2t

R

eAt

a 0 (t)=1, a 1 (t)=

1

2

A 1 - e-2tB

a 0 (t)- 2 a 1 (t)=e-2t

a 0 (t)= 1

a 0 (t)+a 1 (t)l 2 =el^2 t

a 0 (t)+a 1 (t)l 1 =el^1 t

= B

1

0

1

2 A^1 - e

• 2tB
  e-2t

R

=

1

2

bB

0

0

1

- 2

R+ B

2

0

0

2

R - B

0

0

1

- 2

Re-2tr

eAt=^12 AA+ 2 I-Ae-2tB

• 2eAt+A+ 2 I-Ae-2t= 0

3

1

1

I

0

- 2

A

1

e-2t

eAt

3 = 0

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