EE313_stability - EE 313 Linear Systems and Signals Fall...

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Stability EE 313 Linear Systems and Signals Fall 2008 Prof. Adnan Kavak Dept. of Electrical and Computer Engineering The University of Texas at Austin Courtesy of Prof. Brian Evans
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2 Stability Many possible definitions Two key issues for practical systems System response to zero input System response to non-zero but finite amplitude (bounded) input For zero-input response If a system remains in a particular state (or condition) indefinitely, then state is an equilibrium state of system System’s output due to nonzero initial conditions should approach 0 as t →∞ System’s output generated by initial conditions is made up of characteristic modes
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3 Stability Many possible definitions Two key issues for practical systems System response to zero input ( Internal stability ) System response to non- zero but finite amplitude (bounded) input ( Bounded Input Bounded Output- BIBO Stability ) For zero-input response If a system remains in a particular state (or condition) indefinitely, then state is an equilibrium state of system System’s output due to nonzero initial conditions should approach 0 as t →∞ System’s output generated by initial conditions is made up of characteristic modes
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4 Internal Stability Three cases for zero-input response A system is stable if and only if all characteristic modes 0 as t A system is unstable if and only if at least one of the characteristic modes grows without bound as t A system is marginally stable if and only if the zero- input response remains bounded (e.g. oscillates between lower and upper bounds) as t
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5 Characteristic Modes Distinct characteristic roots λ 1 , λ 2 , …, λ n ( 29 { } { } { }
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