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I'm currently studying passivity theorems of nonlinear systems, and I find it difficult to understand the difference between the different kinds of strict passiveness.

Are there any intuitive examples of the difference between state, output and input strictly passive systems?

Further, the passivity theorems use zero-state observability to prove global asymptotic stability for feedback-connection systems.

(Definition 6.5, Khalil nonlinear systems)

Σ is zero-state observable if no solution of x˙ = f(x,0) can stay identically in S = {x ∈ R n |h(x,0) = 0} other than the trivial solution x(t) = 0.

This thread:

How can I check whether a nonlinear system is zero-state observable?

States that the system state x and y should be 0 at the same time when u = 0. Why is this important?

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This is important because usually one is interested in the stability of $x=0$, where $x$ is the state. However, often the output $y=h(x)$ does not contain the whole state $x$. Then it is important to make conclusions about $x=0$, when only $y=0$ can be stabilized.

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