Modern Control Engineering

702 Chapter 9 / Control Systems Analysis in State Space

where the greatest common divisor of the n^2 elements (which are functions of l) of B(l)is unity.

Since

(lI-A)adj(lI-A)=|lI-A|I

we obtain

(9–98)

from which we find that is divisible by d(l). Let us put

(9–99)

Because the coefficient of the highest-degree term in lofd(l)has been chosen to be 1, the

coefficient of the highest-degree term in lofc(l)is also 1. From Equations (9–98) and (9–99),

we have

Hence,

Note that c(l)can be written as

wherea(l)is of lower degree than f(l). Since c(A)= 0 andf(A)= 0 , we must have a(A)= 0.

Also, since f(l)is the minimal polynomial,a(l)must be identically zero, or

Note that because f(A)= 0 , we can write

Hence,

Noting that , we obtain

Since the greatest common divisor of the n^2 elements of B(l)is unity, we have

Therefore,

Then, from this last equation and Equation (9–99), we obtain

f(l)=

∑l I-A∑

d(l)

c(l)=f(l)

g(l)= 1

B(l)=g(l) C(l)

(l I-A) B(l)=c(l) I

c(l) I=g(l)f(l) I=g(l)(l I-A) C(l)

f(l) I=(l I-A) C(l)

c(l)=g(l)f(l)

c(l)=g(l)f(l)+a(l)

c(A)= 0

(l I-A) B(l)=c(l) I

∑l I-A∑=d(l)c(l)

∑l I-A∑

d(l)(l I-A) B(l)=∑l I-A∑ I

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