961ls07Realization2 - Fall 2007 Linear Systems Chapter 07...

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Fall 2007 線性系統 Linear Systems Chapter 07 Minimal Realization & Coprime Fractions Feng-Li Lian NTU-EE Sep07 – Jan08 Materials used in these lecture notes are adopted from “Linear System Theory & Design,” 3rd. Ed., by C.-T. Chen (1999) NTUEE-LS7-Realization-2 Feng-Li Lian © 2007 ± Introduction (7.1) ± Implications of Coprimeness (7.2) ± Computing Comprime Fractions (7.3) ± Balanced Realization (7.4) ± Degree of Transfer Matrices (7.6) ± Minimal Realizations – Matrix Case (7.7, 7.8, 7.9) Outline
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NTUEE-LS7-Realization-3 Feng-Li Lian © 2007 Introduction (7.1) In Example 4.6: N ( s ) N 1 N 2 N 3 NTUEE-LS7-Realization-4 Feng-Li Lian © 2007 N 1 N 2 N 3 −α 1 I 2 −α 2 I 2 −α 3 I 2 Introduction – 2 The first realization is a six-dimensional realization
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NTUEE-LS7-Realization-5 Feng-Li Lian © 2007 Introduction – 3 In Example 4.7 NTUEE-LS7-Realization-6 Feng-Li Lian © 2007 Overall Realization: Introduction – 4 The second realization is a four-dimensional realization
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NTUEE-LS7-Realization-7 Feng-Li Lian © 2007 Introduction – 5 The third, fourth, fifth, . .. realizations are n 3 , n 4 , n 5 , … -dimensional realizations The question is what is the minimal (-dimensional) realization if exists. NTUEE-LS7-Realization-8 Feng-Li Lian © 2007 Coprimeness of Proper Transfer Functions (SISO) (7.2) strictly proper part Direct transmission part, “D-matrix” in realization Without loss of generality, only discuss
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NTUEE-LS7-Realization-9 Feng-Li Lian © 2007 Coprimeness of Proper Transfer Functions (SISO) – 2 NTUEE-LS7-Realization-10 Feng-Li Lian © 2007 Coprimeness of Proper Transfer Functions (SISO) – 3
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NTUEE-LS7-Realization-11 Feng-Li Lian © 2007 Coprimeness of Proper Transfer Functions (SISO) – 4 Which is called a controllable canonical form because with det( C ) = 1 Thus, a realization is NTUEE-LS7-Realization-12 Feng-Li Lian © 2007 Theorem 7.1 above Proof :
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NTUEE-LS7-Realization-13 Feng-Li Lian © 2007 Theorem 7.1 – 2 NTUEE-LS7-Realization-14 Feng-Li Lian © 2007 Coprimeness of Proper Transfer Functions (SISO Systems) – 5 Observable canonical form More canonical realizations:
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NTUEE-LS7-Realization-15 Feng-Li Lian © 2007 Coprimeness of Proper Transfer Functions (SISO Systems) – 6 With the equivalence transformation Which may become another observable canonical form reverse-labeling x i ’s And, another controllable canonical form NTUEE-LS7-Realization-16 Feng-Li Lian © 2007 Minimal Realizations (7.2.1) - Polynomial fraction : N ( s )/ D ( s ) - Coprime fraction : coprime N ( s )/ D ( s ) - Characteristic polynomial of N ( s )/ D ( s ) : the part of D ( s ) after the g.c.d. of N ( s ) and D ( s ) is factored out - Degree of N ( s )/ D ( s
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This note was uploaded on 07/21/2009 for the course EE 901-43400 taught by Professor Feng-lilian during the Fall '08 term at National Taiwan University.

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961ls07Realization2 - Fall 2007 Linear Systems Chapter 07...

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