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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 #4

Problem 1 (1) We have � � �� � � � � �� �

2 2 22 ( )( , ) t t tT t e e

Therefore � �� �� � � � �� � � �2 2(( ) 0 ) ( )( ,0) ( , )t tT t e e T t

(2) This weighting pattern IS NOT realizable by a finite dimensional linear system (time varying or time invariant). The reason is that it is impossible to find two functions N(t) and G(�) (1 x n and n x 1 respectively) such that T(t,����� N(t)G(�).

Problem 2

(1) We have �

� ��

� �� �0

1( )( ) ( )1

t

t

bx t u d and � �� 1( ) ( ) 1 ( )y t c t x t

� ��

� � � � ��

� � � ��

� �0 0

11 1 1

( )( ) ( ) 1 ( ) ( ( ) ( ) 1) ( )

1

t t

t t

by t c t u d c t b u d

For the given 1( )c t and 1( ),b � � �( , ) 1T t , for all values of t and � . (2) The above state space model IS NOT a minimal realization of the weighting pattern ( , )T t � in (1). The realization given has dimension 2. On the other hand we can select N(t)=1 and D(�����everywhere to represent T(t, �������t)D(��. The middle dimension of this factorization is 1<2 . (3) Due to the factorization given in (2) the order of the weighting pattern of (1) is 1.

Problem 3

(1)

2

(2)

3

Problem 4

4

Problem 5

5

Problem 6

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