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EEL5173 Fall 2009 Lecture _12

# EEL5173 Fall 2009 Lecture _12 - 2 Stability of DT...

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ii. Asymptotic (Internal, Zero-input) Stability Theorem. The system is asymptotic stable if and only if the roots of the characteristic equation are all inside the unit circle centered at z = 0. i. BIBO Stability For BIBO stability, poles of the transfer function must be all inside the unit circle centered at z = 0. 2. Stability of DT Systems (continue)

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Theorem. Asymptotic stability implies BIBO stability. A zI A zI D B A zI adj C z H + = ) ( ) ( iii. Marginal Stability Definition. If the zero-input state vector, subject to any finite initial state, remains bounded as k goes to infinity, the system is said to be marginal stable. Theorem. The system is marginal stable if and only if the roots of the characteristic equation are all inside the unit circle centered at z = 0 or on the circle. When on the circle, roots must be simple. Theorem. Asymptotic stability implies marginal stability.
A zI A zI D B A zI adj C z H + = ) ( ) ( iv. Methods to Determine Stability. the location of the poles σ j ω 1 -1 j -j

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Three Methods: Jury’s Test Nyquist Criterion Root-Locus Technique (1) Jury’s Test -
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EEL5173 Fall 2009 Lecture _12 - 2 Stability of DT...

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