EEL5173 Fall 2009 Lecture _25

EEL5173 Fall 2009 Lecture _25 - ii. Observer-Form...

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ii. Observer-Form Realization (continue) (1) Row-reduced Non-Singular Matrix of Polynomials given a co-prime LMFD, G ( s )= Q L - 1 P L . Q L can be made row-reduced through elementary row operations . Let l i be the degree of the i th row of P L and p i be the degree of the i th row of Q L , i = 1, 2, . .., p . G ( s ) is proper if and only if . , , 2 , 1 , p i p l i i L = G ( s ) is strictly proper if and only if . , , 2 , 1 , p i p l i i L = < (2) Properness of Matrices of Rational Functions
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(3) Coefficient Matrices of a LMFD be strictly proper, co-prime, and with Q L being row-reduced. Let Q L be represented by its rows as shown: L L P Q s G 1 ) ( = = Lp L L L Q Q Q Q r M r r 2 1 Assume that { } . , , 2 , 1 , deg p i Q p Let Li i L r = =
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We have {} p p p p p p p p hr Lhr hr Lhr p p p p rpp p rp p rp p p r p r p r p p r p r p r r r r L s s s diag s D where Q s D Q s s s diag s h s h s h s h s h s h s h s h s h s H where s L s H s Q L L L M O M M L L 2 1 2 1 2 2 2 1 1 1 ) ( ) ( ) ( ), ( ) ( ) ( 2 1 2 22 21 1 12 11 = = = = + =
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Then [ ] [] ij Llr lri ij p ij p ij ij p p ij p ij p p ij p ij rij rij r L r Q s D s s s s s s L L s H s Q s L i i i i i i i i r r M L L ) ( 1 ) ( ) ( ) ( 1 2 1 2 1 1 2 2 1 1 = = + + + + = = = ρ ρρ
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{} Llr lr L lrp lr lr lr pp Llr p Llr p Llr p Llr Llr Llr p Llr Llr Llr Llr Llr lr r P s D s P e Furthermor D D D diag block s D Q Q Q Q Q Q Q Q Q Q Q s D s L = = = = ) ( ) ( , ) ( ) ( ) ( 2 1 2 1 2 22 21 1 12 11 r L r r r L r r M O M M r L r r r L r r
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Ex. (continue) p 1 =2 p 2 =3 = + = + + + + = 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 2 2 1 5 0 4 4 0 4 0 1 0 0 0 0 0 1 0 1 1 0 0 0 2 2 5 4 4 4 0 0 0 2 2 3 2 2 3 2 s s s P s s s s s s s s s s s Q L L l 1 =1 l 2 =2 + + + + = 2 1 2 2 1 0 ) 2 ( ) 2 ( ) 1 ( ) 2 ( 0 s s s s s s s P Q L L H r L r D hr Q Lhr Q Llr P Llr D lr
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(4) Realization () [] L Lhr L def Llr Llr lr hr Lhr Llr lr Llr lr
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This note was uploaded on 07/15/2011 for the course EEL 5173 taught by Professor Tung during the Fall '09 term at FSU.

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EEL5173 Fall 2009 Lecture _25 - ii. Observer-Form...

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