 Dissipativity (control theory)

Dissipativity expresses an input–output balance for a dynamical system: the increase of a nonnegative storage function cannot exceed the accumulated supply rate. For state $x$, control input $u$, plant output $y$ and chosen supply rate $w(u,y)$, the requirement isfor every admissible trajectory and time interval, with $S\ge 0$. The storage function and supply rate need not be literal physical energy and power; choosing them makes a particular performance statement precise. Jan C. Willems's1972paper develops this general storage–supply formulation.^[ref]

An original mechanical example is a mass with viscous damping, driven by an applied force $u$:Choose the kinetic energy $S(v)=m{v}^2/2$ and entering power $w(u,y)=uy$. ThenIntegration gives the required inequality. The nonnegative integral $\int b{v}^{2}dt$ accounts for the mechanical energy dissipated by the damper. With no applied force, the stored kinetic energy cannot increase. A dashpot driven at constant velocity provides an experimental interpretation: supplied power is dissipated even when the mass's stored kinetic energy remains constant.

A different choice tests an input–output gain. If a storage function establishes the inequality forthen $S(x(0))=0$ and nonnegative final storage implyThe physical power balance and this squared-signal bound use the same logical structure. Neither inequality, by itself, proves that every internal state converges to zero: a hidden lossless motion can retain storage.