Modern Control Engineering

708 Chapter 9 / Control Systems Analysis in State Space

Similarly, if

then

A–9–12. Consider the following polynomial in lof degree m-1, where we assume l 1 ,l 2 ,p,lmto be

distinct:

wherek=1,2,p,m. Notice that

Then the polynomial f(l)of degree m-1,

takes on the values fAlkBat the points lk. This last equation is commonly called Lagrange’s

interpolation formula. The polynomial f(l)of degree m-1is determined from mindependent

data fAl 1 B, fAl 2 B, p, fAlmB. That is, the polynomial f(l) passes through mpoints

fAl 1 B,fAl 2 B,p,fAlmB. Since f(l)is a polynomial of degree m-1, it is uniquely determined.

Any other representations of the polynomial of degree m-1can be reduced to the Lagrange

polynomialf(l).

= a

m

k= 1

fAlkB

Al-l 1 BpAl-lk- 1 BAl-lk+ 1 BpAl-lmB

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

f(l)= a

m

k= 1

fAlkBpk(l)

pkAliB= b

1,

0,

ifi=k

ifiZk

pk(l)=

Al-l 1 BpAl-lk- 1 BAl-lk+ 1 BpAl-lmB

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

eJt= G

el^1 t

0

0

0

tel^1 t

el^1 t

0

1

2 t

(^2) el 1 t
tel^1 t
el^1 t
el^4 t
0
tel^4 t
el^4 t
el^6 t
0

0

0

el^7 t

W

J=G

l 1

0

0

0

1

l 1

0

0

1

l 1

l 4

0

1

l 4

l 6

0

l 7

W

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