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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 = =
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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