961ls08DesignMIMO2 - Fall 2007 Linear Systems Chapter 08...

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Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (MIMO) 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-LS8-DesignMIMO-2 Feng-Li Lian © 2007 ± Introduction ± State Feedback (8.2) ± Regulation & Tracking (8.3) ± State Estimator (8.4) ± Feedback from Estimated States (8.5) ± State Feedback – Multivariable Case (8.6) ± State Estimators – Multivariable Case (8.7) ± Feedback from Estimated States – Multivariable Case (8.8) Outline
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NTUEE-LS8-DesignMIMO-3 Feng-Li Lian © 2007 State Feedback (MIMO Systems) (8.6) n -state, p -input, q -output ¨ NTUEE-LS8-DesignMIMO-4 Feng-Li Lian © 2007 Theorem 8.M1 (8.6) Proof : rank rank [ ] [ ] [ ] an λλ λ rk ⎛⎞ ⎡⎤ −− = = ⎜⎟ ⎢⎥ ⎣⎦ ⎝⎠ I0 A B KI B AI B B for all λ , including eigenvalues of A BK and A As in the SISO case, after state feedback observability may change
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NTUEE-LS8-DesignMIMO-5 Feng-Li Lian © 2007 Theorem 8.M3 (8.6) ¨ Proof : “necessity” recall the single-input case For multi-input case only change b to B and k to K “sufficiency” see discussions below NTUEE-LS8-DesignMIMO-6 Feng-Li Lian © 2007 Cyclic Design: Theorem 8.7 (8.6.1) A has distinct eigenvalues ¨
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NTUEE-LS8-DesignMIMO-7 Feng-Li Lian © 2007 Cyclic Design – 2 ( A , Bv ) is controllable if v R 2 and v { v 1 +2 v 2 = 0} { v 1 = 0} for a randomly selected v , Probability ( v { v 1 +2 v 2 = 0} { v 1 = 0 }) = 1 .
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961ls08DesignMIMO2 - Fall 2007 Linear Systems Chapter 08...

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