= Current-sign certificate with a curvature bound
A <current-sign certificate with a curvature bound> uses a bound on the second <derivative> of an <electric current> to certify the sign of an extrapolated current. An affine ramp is predicted exactly by two correctly timed samples, so its uncertainty should depend on curvature rather than counting the entire ramp as error. Current direction matters physically because it selects recirculation paths during switching dead time.<Using power MOSFETs in DC motor control applications by Nexperia>[IAN50004, §4.1, Figures 5–6]{ref}
Let samples at $-T$ and $0$ produce measurements $y_-=i(-T)+e_-$ and $y_0=i(0)+e_0$, with $|e_-|\le\epsilon_-$ and $|e_0|\le\epsilon_0$. For a future time $t\ge0$, use the linear <extrapolation>
$$
\widehat i(t)=(1+t/T)y_0-(t/T)y_-,\qquad T>0.
$$
If $i$ is twice continuously differentiable and $|i''(s)|\le M$ throughout $[-T,t]$, the <polynomial interpolation error> gives
$$
\boxed{|i(t)-\widehat i(t)|\le B(t)=\frac M2t(t+T)+(1+t/T)\epsilon_0+(t/T)\epsilon_-.}
$$
The model term follows by applying the remainder to the two-node interpolant: $i(t)-[(1+t/T)i(0)-(t/T)i(-T)]=i''(\xi)t(t+T)/2$ for some $\xi\in(-T,t)$. The sample-error terms use absolute coefficients, as required by the <signed-filter current-sign certificate>. At $t=T$, the model contribution is $MT^2$ and equal sample bounds contribute $3\epsilon$.
For an original example, take $T=6\,\mathrm{\mu s}$, $M=10^8\,\mathrm{A/s^2}$ and each sample error bounded by $0.002\,\mathrm A$. At $t=T$,
$$
B(T)=0.0036+0.0060=0.0096\,\mathrm A.
$$
If $y_0=0.030\,\mathrm A$ and $y_-=0.040\,\mathrm A$, the predicted current is $0.020\,\mathrm A$, certifying actual current at least $0.0104\,\mathrm A$ at that future instant. To certify the entire interval $0\le t\le T$, check $\widehat i(t)-B(t)>0$ everywhere. In this example the prediction decreases linearly and $B$ increases, so the minimum margin occurs at $T$ and the same lower bound applies throughout.
The curvature limit is a physical assumption. A switching event, <sensor> clipping or a slope discontinuity can invalidate it even when a fitted ramp looks smooth at the sample times. A bounded first <derivative> alone does not imply any finite $M$. <Timing uncertainty in a curvature-based current-sign certificate> also matters: two accurate current values at incorrectly modeled times need not extrapolate an affine ramp correctly.
A proposed test supplies known smooth ramps and curved waveforms, records sample timestamps and checks the independently measured future waveform against the enclosure. Test transitions separately. The result is a deterministic sign guarantee only when the sample and curvature bounds are deterministic; substituting one <standard uncertainty> for an absolute error bound changes the claim.
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