961ls04Solution2 - Fall 2007 Linear Systems Chapter 04...

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Fall 2007 線性系統 Linear Systems Chapter 04 State-Space Solutions & Realizations 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-LS4-Solution-2 Feng-Li Lian © 2007 ± Introduction ± Solution of LTI State Equations (4.2) ± Equivalent State Equations (4.3) ± Realizations (4.4) Outline
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NTUEE-LS4-Solution-3 Feng-Li Lian © 2007 Solution of LTI State Equations (4.2) – 1 Derivative of Exponential Function: NTUEE-LS4-Solution-4 Feng-Li Lian © 2007 Solution of LTI State Equations – 2 LTI State Equation and its Solution :
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NTUEE-LS4-Solution-5 Feng-Li Lian © 2007 Solution of LTI State Equations – 3 LTI State Equations : NTUEE-LS4-Solution-6 Feng-Li Lian © 2007 Solution of LTI State Equations – 4 Useful formulae: Verification :
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NTUEE-LS4-Solution-7 Feng-Li Lian © 2007 Solution of LTI State Equations – 5 Output Equation: NTUEE-LS4-Solution-8 Feng-Li Lian © 2007 Solution of LTI State Equations (4.2): By Laplace Transform Laplace transform :
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NTUEE-LS4-Solution-9 Feng-Li Lian © 2007 Computing e A t & ( s I - A ) -1 NTUEE-LS4-Solution-10 Feng-Li Lian © 2007 Computing e A t –1
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NTUEE-LS4-Solution-11 Feng-Li Lian © 2007 Computing e At –2 NTUEE-LS4-Solution-12 Feng-Li Lian © 2007 Computing e At –3
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NTUEE-LS4-Solution-13 Feng-Li Lian © 2007 Computing e At –4 NTUEE-LS4-Solution-14 Feng-Li Lian © 2007 Computing ( s I - A ) -1 –1
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NTUEE-LS4-Solution-15 Feng-Li Lian © 2007 Computing ( s I - A ) -1 –2 Ä Ä NTUEE-LS4-Solution-16 Feng-Li Lian © 2007 Computing ( s I - A )^-1 – 3 the Jordan form for ˆ A
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NTUEE-LS4-Solution-17 Feng-Li Lian © 2007 Computing ( s I - A )^-1 – 4 () ss s s 112 3 2 IA I A A −−− −= + + + " (Problem 3.26) NTUEE-LS4-Solution-18 Feng-Li Lian © 2007 Example 4.2
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NTUEE-LS4-Solution-19 Feng-Li Lian © 2007 Solution Characteristics If Re( λ i ) < 0 for all i , then every zero-input response will approach zero as t →∞ If Re( λ i ) > 0 for some i , then part of zero-input response may grow unbounded as t NTUEE-LS4-Solution-20 Feng-Li Lian © 2007 Solution Characteristics If Re( λ i ) 0 for all i , and λ j with Re( λ j ) = 0 has only index 1 , then zero-input response will be bounded for all t If Re( λ i ) 0 for all i , but some λ j with Re( λ j ) = 0 has index 2 or higher , then part of zero-input response may grow unbounded as t
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NTUEE-LS4-Solution-21
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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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961ls04Solution2 - Fall 2007 Linear Systems Chapter 04...

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