582hw2 - EEE 582 Problem 2.1 HW # 2 SOLUTIONS y (n) (t) +...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
EEE 582 HW # 2 SOLUTIONS Problem 2.1 y ( n ) ( t )+ a n ¡ 1 ( t ) y ( n ¡ 1) ( t ¢¢¢ + a 0 ( t ) y ( t )= b 0 ( t ) u ( t b 1 ( t ) u (1) ( t ) Let x n ( t y ( n ¡ 1) ( t ) ¡ b 1 ( t ) u ( t )and x 1 ( t y ( t )then _ x 1 ( t )= _ y ( t x 2 ( t ) _ x 2 ( t )=Ä y ( t x 3 ( t ) . . . _ x n ¡ 2 ( t x n ¡ 1 ( t ) _ x n ¡ 1 ( t x n + b 1 ( t ) u ( t ) _ x n ( t y ( n ) ( t ) ¡ b (1) 1 u ( t ) ¡ b 1 ( t ) u (1) ( t ) = ¡ n ¡ 1 X i =0 a i ( t ) y ( i ) ( t b 0 ( t ) u ( t ) ¡ b (1) 1 ( t ) u ( t ) = ¡ n ¡ 2 X i =0 a i ( t ) x i ( t a n ¡ 1 ( t )( x n ( t b 1 ( t ) u ( t )) + ³ b 0 ( t ) ¡ b (1) 1 ( t ) ´ u ( t ) then we can write A ( t 2 6 6 6 6 6 4 01 0 00 0 . . . . . . . . . 1 ¡ a 0 ( t ) ¡ a 1 ( t ) ¢¢¢ ¡ a n ¡ 1 ( t ) 3 7 7 7 7 7 5 ;B ( t 2 6 6 6 6 6 4 0 0 . . . b 1 ( t ) b 0 ( t ) ¡ b (1) 1 ( t a n ¡ 1 ( t ) b 1 ( t ) 3 7 7 7 7 7 5 C ( t £ 10 0 ¤ ;D ( t )=0 Problem 2.8 Identity dc-gain means that for a given ~ u , 9 ~ x ,suchthat A ~ x + B ~ u =0 , C ~ x =~ u , this implies that the matrix · AB C 0 ¸ is invertible. 1. If K 2 IR m £ n is such that ( A + BK ) is invertible, then C ( A + ) ¡ 1 B is invertible. Since · C 0 ¸ is invertible , for any K , · A + BK B C 0 ¸ is invertible, this from · A + C 0 ¸ = · C 0 ¸· I 0 KI ¸ Then · A + C 0 R 1 R 2 R 3 R 4 ¸ = · I 0 0 I ¸ so ( A + ) R 1 + BR 3 = I ( A + ) R 2 + 4 ) R 2 = ¡ ( A + ) ¡ 1 4 CR 1 2 = I C ( A + ) ¡ 1 4 = I hence C ( A + ) ¡ 1 B is invertible. 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2. We need to show that there exits N such that 0=( A + BK )~ x + BN ~ u ~ u = C ~ x The ¯rst equation gives ~ x = ¡ ( A + ) ¡ 1 ~ u .T h u sw en e e d t o c h o o s e N such that ¡ C ( A + ) ¡ 1 =~ u .F rompa r t1 .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 04/28/2011.

Page1 / 3

582hw2 - EEE 582 Problem 2.1 HW # 2 SOLUTIONS y (n) (t) +...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online