EEL5173 Fall 2009 Lecture _23

EEL5173 Fall 2009 Lecture _23 - ─ given a co-prime RMFD,...

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Unformatted text preview: ─ given a co-prime RMFD, G ( s )= P R Q R- 1 . i. Controller-Form Realization (continue) (1) Column-reduced Non-Singular Matrix of Polynomials (2) Properness of Matrices of Rational Functions- how to check whether a given RMFD represents a proper or strictly proper matrix of rational functions. ─ Q R can be made column-reduced through elementary column operations . Theorem. Given an RMFD with Q R being column-reduced, let k i be the degree of the i th column of P R and q i be the degree of the i th column of Q R , i = 1, 2, ..., q , , ) ( 1 − = R R Q P s G G ( s ) is proper if and only if . , , 2 , 1 , q i q k i i L = ≤ G ( s ) is strictly proper if and only if . , , 2 , 1 , q i q k i i L = < Ex. For the following RMFD, determine whether G is proper, strictly proper, or not proper at all. q 1 =3 q 2 =1 , 1 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = Rhc Q singular. Therefore, Q R is not column-reduced. [ ] , 1 1 1 1 1 2 ) ( 1 2 3 2 1 − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + − ⋅ + = = s s s s s Q P s G R R k 1 =2 k 2 =0 The theorem to test the properness of a transfer function matrix of rational functions cannot be applied. ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ × − + + × − − ⇒ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + − = × 1 1 ~ ~ 1 1 1 1 1 1 1 1 1 1 2 2 2 3 2 3 Rhc R R R Q Q s s s s s s s s s s s s s Q q 1 =1 q 2 =1 non-singular, is column-reduced. R Q ~ [ ] [ ] [ ] R R R R P s s s s P R s ~ 1 1 1 1 1 1 2 1 1 2 1 1 1 1 2 2 2 2 2 = + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⋅ + ⇒ + = = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⇒ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ × × Since is column-reduced and k 1 > q 1 , G ( s ) in not proper....
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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 _23 - ─ given a co-prime RMFD,...

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