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exponential stability
:''See Lyapunov stability, which gives a definition of asymptotic stability for more general dynamical systems. All exponentially stable systems are also asymptotically stable.''
In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative real parts. (i.e., in the left half of the complex plane).〔David N. Cheban (2004), ''Global Attractors Of Non-autonomous Dissipative Dynamical Systems''. p. 47〕 A discrete-time input-to-output LTI system is exponentially stable if and only if the poles of its transfer function lie strictly within the unit circle centered on the origin of the complex plane. Exponential stability is a form of asymptotic stability. Systems that are not LTI are exponentially stable if their convergence is bounded by exponential decay.
==Practical consequences==
An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition. Moreover, if the system is given a fixed, finite input (i.e., a step), then any resulting oscillations in the output will decay at an exponential rate, and the output will tend asymptotically to a new final, steady-state value. If the system is instead given a Dirac delta impulse as input, then induced oscillations will die away and the system will return to its previous value. If oscillations do not die away, or the system does not return to its original output when an impulse is applied, the system is instead marginally stable.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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