Zero-state observability and strictly passive systems

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I'm presently studying passivity theorems of nonlinear systems, and I find it difficult to understand nan quality betwixt nan different kinds of strict passiveness.

Are location immoderate intuitive examples of nan quality betwixt state, output and input strictly passive systems?

Further, nan passivity theorems usage zero-state observability to beryllium world asymptotic stableness for feedback-connection systems.

(Definition 6.5, Khalil nonlinear systems)

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

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How tin I cheque whether a nonlinear strategy is zero-state observable?

States that nan strategy authorities x and y should beryllium 0 astatine nan aforesaid clip erstwhile u = 0. Why is this important?

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