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I am reading an article that says:

stabilize the multi-vehicle system to one of its local minima via dissipative control

And other that deals with dissipative system:

(PID) controllers is designed to make the closed-loop linear system asymptotically stable and strictly quadratic dissipative

Question: What exactly is dissipative control or quadratic dissipative?

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  • $\begingroup$ en.m.wikipedia.org/wiki/Dissipative_system, quadratic refers to the storage function, thus dissaptive for a quadratic storage function, e.g. x^T P x $\endgroup$ May 13 '20 at 18:35
  • $\begingroup$ thanks but as usual, wikipedia is not that intuitive. What exactly is dissipative control? $\endgroup$
    – gfdsal
    May 13 '20 at 18:53
  • $\begingroup$ a controller such that the system is disspative, thus the "energy" in the system does increase faster than the supply rate. $\endgroup$ May 14 '20 at 14:00
  • $\begingroup$ I see, makes sense. Any example of such controller? $\endgroup$
    – gfdsal
    May 14 '20 at 15:00
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    $\begingroup$ e.g. a P-controller that stabilizes a linear first-order system. $\endgroup$ May 14 '20 at 16:10
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The intuitive idea is that a dissipative system cannot store more energy than what was initially stored plus what is supplied during an experiment, which is schematically depicted below. Visualization of dissipativity This figure is adopted from: http://www.eeci-institute.eu/pdf/M012/lec2.pdf

So we write that a system $\dot{x} = f(x,u)$, $y = g(x,u)$ is dissipative with respect to the supply rate $s(u,y)$ if there exists a storage function $V:\mathbb{R}^n\to\mathbb{R}$ such that the dissipation inequality $$V\big(x(t_1)\big) \leq V\big(x(t_0)\big) + \int_{t_0}^{t_1} s\big(u(t),y(t)\big) \;\mathrm{d}t$$ hold for all system trajectories and for all $t_0< t_1$.

We call it quadratic dissipative if the storage function is a quadratic function, e.g. $V(x) = x^\top P x$.

Thus dissipative control is a controller such that the closed-loop system is dissipative with respect to the in- and output of the closed loop system.

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  • $\begingroup$ Well the visuals make it more understandable. Thanks $\endgroup$
    – gfdsal
    May 15 '20 at 14:32

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