Modern Control Engineering

Section 9–5 / Some Useful Results in Vector-Matrix Analysis 673

= 0 (9–49)

Equation (9–49) can be solved for by expanding it about the last column.

It is noted that, just as in case 1, solving Equation (9–49) for is the same as writing

(9–50)

and determining the ak(t)’s (k=0, 1, 2,p, m-1)from

The extension to other cases where, for example, there are two or more sets of multiple

roots will be apparent. Note that if the minimal polynomial of Ais not found, it is possible

to substitute the characteristic polynomial for the minimal polynomial. The number of

computations may, of course, be increased.

EXAMPLE 9–8 Consider the matrix

Compute using Sylvester’s interpolation formula.

From Equation (9–47), we get

3

1

1

I

l 1

l 2

A

el^1 t

el^2 t

eAt

3 = 0

eAt

A= B

0

0

1

- 2

R

a 0 (t)+a 1 (t)lm+a 2 (t)lm^2 +p+am- 1 (t)lmm-^1 =elm^ t







a 0 (t)+a 1 (t)l 4 +a 2 (t)l 42 +p+am- 1 (t)l 4 m-^1 =el^4 t

a 0 (t)+a 1 (t)l 1 +a 2 (t)l 12 +p+am- 1 (t)l 1 m-^1 =el^1 t

a 1 (t)+ 2 a 2 (t)l 1 + 3 a 3 (t)l 12 +p+(m-1)am- 1 (t)l 1 m-^2 =tel^1 t

a 2 (t)+ 3 a 3 (t)l 1 +p+

(m-1)(m-2)

2

am- 1 (t)l 1 m-^3 =

t^2

2

el^1 t

eAt=a 0 (t) I+a 1 (t) A+a 2 (t) A^2 +p+am- 1 (t) Am-^1

eAt

eAt

(m- 1 )(m- 2 )

2

l 1 m-^3

(m- 1 )l 1 m-^2

l 1 m-^1

l 4 m-^1







lmm-^1

Am-^1

t^2

2

el^1 t

tel^1 t

el^1 t

el^4 t







elmt

eAt

0 0 1 1    1 I

0

1

l 1

l 4







lm

A

1

2 l 1

l 12

l 42







lm^2

A^2

3 l 1

3 l 12

l 13

l 43







lm^3

A^3

p p p p p p p p p