In Passivity-Based Control and Estimation in Networked Robotics (among others), a passive system is defined as one that:

$$\int_0^t y^T(t)u(t) dt \geq -\beta $$

for output $y$ and input $u$.

I have never understood how this describes a system where

the total internal energy stored in the system is bounded above by the energy externally supplied to the system

This comes down to $y$ and $u$ being under defined.

  • One way to interpret it, is since we are discussing energy, that $u$ and $y$ have units that result in their product being a measure of energy? And implicitly that the integral defines some energy reservoir or dissipation?

  • Alternatively, the only way that the above is violated, is if $y$ is negatively proportional to $u$? But what does that mean mechanically? That the robot is compliant?

How should I imagine $y$ and $u$ in order to make sense of the passivity criteria above?

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