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Electrical and Computer Engineering Department

University of Maryland College Park

ENEE 660

System Theory

Fall 2008

Professor John S. Baras

Solutions to Homework Set #5

Problem 1

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Problem 2

Problem 3

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So the answer to (a) is that =3.

And this construction provides the answer to (c).

To answer (b) we have from above that the least common multiple of the denominators of the

elements of is d(s) = (s + 1)(s + 2). So following Theorem 5.5.38, =N(s) /d(s), where

2 2( 2)( )

1 1

s sN s

s

+ + = +

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Then the g.c.d.s of the subdeterminants ofN(s) are

D1(s) = 1

D2(s) = (s +2)(s +1) + 2 (s +2) = (s +2)(s +3)

And therefore the Smith form ofN

(s) is

Therefore the McMillan canonical form of is then

So the poles (three) of are -1, -1, -2, while there is only one zero at -3.

Problem 4

1 0( )

0 ( 2)( 3)N s

s s

+ +

10

( 1)( 2)( )( 3)

0

( 1)

s sT s

s

s

+ +

+

+

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Problem 5

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Problem 6

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