EECE 301 Note Set 30 CT System Stability

EECE 301 Note Set 30 CT System Stability - EECE 301 Signals...

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1/30 EECE 301 Signals & Systems Prof. Mark Fowler Note Set #30 • C-T Systems: Laplace Transform… and System Stability • Reading Assignment: Section 8.1 of Kamen and Heck
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2/30 Ch. 1 Intro C-T Signal Model Functions on Real Line D-T Signal Model Functions on Integers System Properties LTI Causal Etc Ch. 2 Diff Eqs C-T System Model Differential Equations D-T Signal Model Difference Equations Zero-State Response Zero-Input Response Characteristic Eq. Ch. 2 Convolution C-T System Model Convolution Integral D-T System Model Convolution Sum Ch. 3: CT Fourier Signal Models Fourier Series Periodic Signals Fourier Transform (CTFT) Non-Periodic Signals New System Model New Signal Models Ch. 5: CT Fourier System Models Frequency Response Based on Fourier Transform New System Model Ch. 4: DT Fourier Signal Models DTFT (for “Hand” Analysis) DFT & FFT (for Computer Analysis) New Signal Model Powerful Analysis Tool Ch. 6 & 8: Laplace Models for CT Signals & Systems Transfer Function New System Model Ch. 7: Z Trans. Models for DT Signals & Systems Transfer Function New System Model Ch. 5: DT Fourier System Models Freq. Response for DT Based on DTFT New System Model Course Flow Diagram The arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).
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3/30 Ch. 8: System analysis and design using the transfer function We have seen that the system transfer function H ( s ) plays an important role in the analysis of a system’s output for a given input. e.g. for the zero-state case: { } ) ( ) ( ) ( ) ( ) ( ) ( 1 s H s X t y s H s X s Y - L = = Much insight can be gained by looking at H ( s ) and understanding how its structure will affect the form of y ( t ). Section 8.1: First we’ll look at how H ( s ) can tell us about a system’s “stability” Section 8.4: Then we’ll see how the form of H ( s ) can tell us about how the system output should behave Section 8.5: Then we’ll see how to design an H ( s ) to give the desired frequency response.
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4/30 Roughly speaking: A “stable” system is one whose output does not keep getting bigger and bigger in response to an input that does not keep getting bigger. We can state this mathematically and then use our math models (e.g. h ( t ) or H ( s )) to determine if a system will be stable. Mathematical checks for stability : The following are given without proof. They are all equivalent checks so you only need to test for one of them. 1. 2. All poles are in the “open left-half of the s-plane” 3. Routh-Hurwitz test (section 8.2, we’ll skip it) " Integrable Absolutely " ) ( 0 < dt t h Section 8.1 System Stability Math definition of stability “Bounded-Input, Bounded-Output” (BIBO) stability: A system is said to be BIBO stable if for any bounded input : |x(t ) | C 1 < ∞∀ t 0 ( x ( t ) = 0, t< 0) the output remains bounded : |y(t ) | C 2 < For SOME C 1 & C 2
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5/30 can only happen if |h(t ) | decays fast enough.
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This note was uploaded on 02/29/2012 for the course EECE 301 taught by Professor Fowler during the Fall '08 term at Binghamton University.

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EECE 301 Note Set 30 CT System Stability - EECE 301 Signals...

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