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The definition of passivity for control systems is: $\langle y,u \rangle _T \geq 0 \: \: \: \forall T >0 \: \: \: \forall u$. Considering signals $u,y :[0,T] \rightarrow R^m$.

As I understand it, passivity is when the system is still stable and does not change the direction and lose stability. If we look at the vectors I understand that in the scalar multiplication $\cos\theta \geq 0$. Does that mean that the direction or phase doesn't change from positive to negative and why?

Another definition using the storage function is:

$$V(x(T)) \leq \int_0^Ty(t)u(t)dt + V(x(0)), \: \: \forall T > 0 $$

Passivity idea: “Increase in stored energy ≤ Added energy”

How do I describe the passivity for a system in words without the storage function? I mean, what is passivity and why is it useful? In what way is it different from stability?

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A short answer for each of your questions in order:

  • How do I describe passivity without the storage function? Passivity analysis is based around the idea of energy flowing into and out of the system. When your system is passive it means it is not generating energy, only storing it or 'passing it through'.

  • Why is passivity useful? When dealing with nonlinear systems it can be difficult if not impossible to prove that they are stable. It is much easier to prove that they are passive, and to design controllers that ensure passivity.

  • In what way is passivity different from stability? It is important to understand that passivity is a sufficient condition for stability, in that a system that is passive is also stable. However, not all stable systems are passive.

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