= Supply rate
{title2=$w(u,y)$}
A <supply rate> specifies the instantaneous quantity whose time integral is compared with a change of <storage function> in <dissipativity (control theory)>. With <control input> $u$, <plant output> $y$ and state $x$, a chosen function $w(u,y)$ enters the balance
$$
S(x(t_1))-S(x(t_0))\leq\int_{t_0}^{t_1}w(u(t),y(t))\,dt.
$$
The <supply rate> can be negative: it then represents net withdrawal in the chosen accounting. Its physical interpretation depends on the variables and sign convention, rather than on the word supply alone.<Dissipative Dynamical Systems, Part I: General Theory by Jan C. Willems>[General Theory, definition of supply and storage inequality]{ref}
For a mechanical terminal with applied <force> $F$ and inward-convention <velocity> $v$, choose $w=Fv$, the entering <power>. For an electrical terminal with <electric current> entering its positive-potential terminal, the corresponding choice is $w=Vi$, with $V$ an <electric potential difference>. Both have units of watts when $S$ is <energy>. Reversing one terminal sign without changing the other reverses the apparent power flow.
An original <capacitor> example illustrates the sign. Let $C=2\,\mathrm F$, terminal <electric potential difference> $V=3\,\mathrm V$, and charging <electric current> $i=1\,\mathrm A$. Its stored <electrostatic energy> is $S=CV^2/2=9\,\mathrm J$, and at that instant $\dot S=Vi=3\,\mathrm W$. With the same potential and $i=-1\,\mathrm A$, the <supply rate> is $-3\,\mathrm W$ and storage decreases. This is consistent with nonnegative stored <energy>; negative supply does not mean negative storage.
A different <supply rate> can express a signal-performance requirement:
$$
w(u,y)=\gamma^2u^Tu-y^Ty.
$$
Nonnegative final storage and zero initial storage then bound accumulated output-square by $\gamma^2$ times accumulated input-square. These squares need not be physical <power>. Scaling or changing the units of the signals requires corresponding changes in the coefficients. A general real quadratic choice can be written as $w=y^TQy+2y^TLu+u^TRu$, with symmetric $Q,R$; the selected <quadratic form> states which input–output relation is being tested.
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