Modern Control Engineering

Example Problems and Solutions 709

Assuming that the eigenvalues of an n*nmatrixAare distinct, substitute Aforlin the
polynomialpk(l). Then we get

Notice that pk(A)is a polynomial in Aof degree m-1. Notice also that

Now define

(9–102)

Equation (9–102) is known as Sylvester’s interpolation formula. Equation (9–102) is equivalent
to the following equation:

(9–103)

Equations (9–102) and (9–103) are frequently used for evaluating functions f(A)of matrix A—
for example,(lI-A)–1,eAt, and so forth. Note that Equation (9–103) can also be written as

(9–104)

Show that Equations (9–102) and (9–103) are equivalent. To simplify the arguments, assume
thatm=4.

7

1 1    1 I

l 1

l 2







lm

A

l^21

l^22







l^2 m

A^2

p

p

p

p

p

lm 1 -^1

lm 2 -^1







lmm-^1

Am-^1

fAl 1 B

fAl 2 B







fAlmB

f(A)

7 = 0

8

1

l 1

l^21







lm 1 -^1

fAl 1 B

1

l 2

l^22







lm 2 -^1

fAl 2 B

p

p

p

p

p

1

lm

l^2 m







lmm-^1

fAlmB

I

A

A^2







Am-^1

f(A)

8 = 0

= a

m

k= 1

fAlkB

AA-l 1 IBpAA-lk- 1 IBAA-lk+ 1 IBpAA-lm IB

Alk-l 1 BpAlk-lk- 1 BAlk-lk+ 1 BpAlk-lmB

f(A)= a

m

k= 1

fAlkBpk(A)

pkAli IB=b

I,

0 ,

ifi=k

ifiZk

pk(A)=

AA-l 1 IB p AA-lk- 1 IBAA-lk+ 1 IB p AA-lm IB

Alk-l 1 BpAlk-lk- 1 BAlk-lk+ 1 BpAlk-lmB