# What is Dissipitative Control?

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?

• 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 May 13 '20 at 18:35
• thanks but as usual, wikipedia is not that intuitive. What exactly is dissipative control? May 13 '20 at 18:53
• a controller such that the system is disspative, thus the "energy" in the system does increase faster than the supply rate. May 14 '20 at 14:00
• I see, makes sense. Any example of such controller? May 14 '20 at 15:00
• e.g. a P-controller that stabilizes a linear first-order system. May 14 '20 at 16:10

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. 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. $$V(x) = x^\top P x$$.