961ls05Stability2 - Fall 2007 Linear Systems Chapter 05...

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Fall 2007 線性系統 Linear Systems Chapter 05 Stability 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-LS5-Stability-2 Feng-Li Lian © 2007 Outline ± Introduction ± Input-Output Stability (5.2) ± Internal Stability (5.3) ± Lyapunov Theorem (5.4)
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NTUEE-LS5-Stability-3 Feng-Li Lian © 2007 Basic Concept (5.1) Stable Systems ¨ Bounded Input and Bounded Output ( BIBO ) Stability Unstable Systems ¨ Small Input generates Unbounded Output Response of Linear Systems ¨ Zero-State Response + Zero-Input Response ¨ BIBO Stability + Marginal/Asymptotic Stability ¨ Input-Output Stability + Internal Stability System Definition: NTUEE-LS5-Stability-4 Feng-Li Lian © 2007 Input-Output Stability (5.2) Bounded Signal s(t): ¨ a constant bound b < , s.t. | s ( t ) |≤ b , t 0 System Bounded-Input-Bounded-Output ( BIBO ) Stable Systems: ¨ EVERY bounded input excites a bounded output ¨ Bounded outputs in response to ALL bounded inputs ¨ (zero-state response only) An SISO causal LTI systems: Definition:
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NTUEE-LS5-Stability-5 Feng-Li Lian © 2007 Theorem 5.1 (5.2) unit-impulse response Proof : NTUEE-LS5-Stability-6 Feng-Li Lian © 2007 Theorem 5.1 – 2 Proof : The corresponding output is unbounded at t = t 1 , as Suppose Then for the bounded input u ( t ) with (Contradiction)
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NTUEE-LS5-Stability-7 Feng-Li Lian © 2007 Theorem 5.1 – 3 →∞ < ∞= 0 () l im () 0 t ft d t ft area = 1/ n 2 2 2 0 1 , but lim ( ) 0 n t t n = =< An absolutely integrable function may not approach zero! ¨ That is, For example, ? NTUEE-LS5-Stability-8 Feng-Li Lian © 2007 Theorem 5.2 (5.2) g(t)
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NTUEE-LS5-Stability-9 Feng-Li Lian © 2007 Theorem 5.2 – 2 Proof : 1. NTUEE-LS5-Stability-10 Feng-Li Lian © 2007 Theorem 5.2 – 3 Proof : 2.
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NTUEE-LS5-Stability-11 Feng-Li Lian © 2007 Theorem 5.2 – 4 NTUEE-LS5-Stability-12 Feng-Li Lian © 2007 Theorem 5.3 (5.2) Proof :
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NTUEE-LS5-Stability-13 Feng-Li Lian © 2007 Irrational Transfer Function (5.2) : irrational Example 5.1: (Fig. 2.5(a)) NTUEE-LS5-Stability-14 Feng-Li Lian © 2007 Input-Output Stability: MIMO Case (5.2)
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NTUEE-LS5-Stability-15 Feng-Li Lian © 2007 BIBO of State Equation (5.2) Poles and Eigenvalues: State Equations and Transfer Function: NTUEE-LS5-Stability-16 Feng-Li Lian © 2007
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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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961ls05Stability2 - Fall 2007 Linear Systems Chapter 05...

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