961ls06CtrbObsv2 - Fall 2007 Linear Systems Chapter 06...

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Fall 2007 線性系統 Linear Systems Chapter 06 Controllability & Observability 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-LS6-CtrbObsv-2 Feng-Li Lian © 2007 Outline ± Introduction ± Controllability (6.2) ± Observability (6.3) ± Canonical Decomposition (6.4) ± Conditions in Jordan-Form Equations (6.5) ± Discrete-Time State Equations (6.6) ± Controllability after Sampling (6.7)
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NTUEE-LS6-CtrbObsv-3 Feng-Li Lian © 2007 Introduction (6.1) ¨ Controllability and observability reveal the internal structure of the system ( model ) 1 input : u 2 states : x 1 , x 2 1 output : y The state x 2 is NOT controllable ” by the input u The state x 1 is NOT observable ” at the output y = x 2 +2 u NTUEE-LS6-CtrbObsv-4 Feng-Li Lian © 2007 Controllability (6.2)
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NTUEE-LS6-CtrbObsv-5 Feng-Li Lian © 2007 Controllability – 2 if x (0) = 0 , then x ( t ) = 0 , t 0, no matter what u ( t ) is if x 1 (0) = x 2 (0) , then x 1 ( t ) = x 2 ( t ) , t 0, no matter what u ( t ) is Un-Controllable Examples: NTUEE-LS6-CtrbObsv-6 Feng-Li Lian © 2007 Controllability – 3 Controllable Example:
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NTUEE-LS6-CtrbObsv-7 Feng-Li Lian © 2007 Controllability – 4: Diagonal Representation NTUEE-LS6-CtrbObsv-8 Feng-Li Lian © 2007 Controllability – 5: Controllable Representation
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NTUEE-LS6-CtrbObsv-9 Feng-Li Lian © 2007 Controllability – 6: Physical Representation NTUEE-LS6-CtrbObsv-10 Feng-Li Lian © 2007 Theorem 6.1 (6.2)
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NTUEE-LS6-CtrbObsv-11 Feng-Li Lian © 2007 Theorem 6.2 – 2 Proof : “1. 2.” ( A , B ) controllable W c ( t ) nonsingular , t > 0” NTUEE-LS6-CtrbObsv-12 Feng-Li Lian © 2007 Theorem 6.2 – 3 Proof :
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NTUEE-LS6-CtrbObsv-13 Feng-Li Lian © 2007 Theorem 6.2 – 4 Proof : NTUEE-LS6-CtrbObsv-14 Feng-Li Lian © 2007 Theorem 6.2 – 5 Proof : “2. 3.” W c ( t ) nonsingular , t > 0 C has rank n
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NTUEE-LS6-CtrbObsv-15 Feng-Li Lian © 2007 Theorem 6.2 – 6 Proof : NTUEE-LS6-CtrbObsv-16 Feng-Li Lian © 2007 Theorem 6.2 – 7 Proof : “3. 4.” C has rank n rank [ A −λ IB ] = n , e-value λ of A
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NTUEE-LS6-CtrbObsv-17 Feng-Li Lian © 2007 Theorem 6.2 – 8 Proof : NTUEE-LS6-CtrbObsv-18 Feng-Li Lian © 2007 Theorem 6.2 – 9 Proof :
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NTUEE-LS6-CtrbObsv-19 Feng-Li Lian © 2007 Theorem 6.2 – 10 Proof : “2. 5.” “For stable A , W c ( t ) nonsingular , t > 0 AW c + W c A = BB has a unique P.D. sol. W c ( ) NTUEE-LS6-CtrbObsv-20 Feng-Li Lian © 2007 Example 6.2 (6.2) rank = 4
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NTUEE-LS6-CtrbObsv-21 Feng-Li Lian © 2007 Example 6.3 (6.2) Mass = 0 NTUEE-LS6-CtrbObsv-22 Feng-Li Lian © 2007 Example 6.3 – 2
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NTUEE-LS6-CtrbObsv-23 Feng-Li Lian © 2007 Example 6.3 – 2 Larger u 1 transfers x (0) = [10 1] to x (2) = 0 in 2 seconds, & Smaller u 2 transfers x (0) = [10 1] to x (4) = 0 in 4 seconds. Note: Given the same x (0) , t 1 , and x ( t 1 ) , the formula in Theorem 6.1 for u (·) gives the minimal energy control than other u (·): NTUEE-LS6-CtrbObsv-24 Feng-Li Lian © 2007 Controllability Index (6.2.1)
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NTUEE-LS6-CtrbObsv-25 Feng-Li Lian © 2007 Controllability Index – 2 Given a controllable pair ( A , B ) R n × n × R n × p and rank B = p
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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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961ls06CtrbObsv2 - Fall 2007 Linear Systems Chapter 06...

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