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Asymptotic Stability/Zero Input Stability
Zero Input Response, I(t)
It is the response of the system due to initial conditions alone

Zero State Response
It is the response of the system due to system’s input alone with initial conditions set to zero

Response of system = Zero input Response + Zero State response

Zero Input Stability is defined as
If zero input response of a system with finite initial conditions reaches to zero as time approaches infinity, the system is said to be zero input stable otherwise zero input unstable.

Mathematically, if c(t) = zero input response of system
then there exist a positive number M such that

|𝐂(𝐭)|smaller than 𝐌 smaller than ∞ 〖𝐟𝐨𝐫 𝐚𝐥𝐥 〗⁡𝐭≥𝐭_𝟎

lim(𝒕→∞) |𝐂(𝐭)|=𝟎

Important points related to Asymptotic Stability

It depends on roots of Characteristic equations i.e. Closed loop poles of system.

If System is BIBO Stable then there must be zero input/asymptotic stability.

All the poles should lie on Left half of s plane.


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