Signed-filter current-sign certificate ($|i(t)-\hat{i}|\le ϵ+L_i{\sum }_j|a_j||t-t_j|$)

A signed-filter current-sign certificate guarantees the sign of an actual load current from a filtered estimate whose coefficients can be negative. Such a filter may extrapolate or sharpen a waveform, but its output is not necessarily a convex average of past currents. The sign must hold over the intended switching interval before it can justify current-polarity dead-time compensation; the relevant recirculation paths depend on current direction.^[ref]

Suppose $\hat{i}={\sum }_j{a}_{j}i(t_j)+\eta$, with ${\sum }_j{a}_j=1$, $|\eta |\le ϵ$ and the Lipschitz bound $|i(t)-i(s)|\le L_i|t-s|$ over all relevant times. ThenThe absolute coefficients are essential. To certify an interval, maximize the right-hand side over its times. A positive sign follows if the estimate exceeds that bound; a negative sign follows if it is less than the negative bound. If individual sample errors are bounded by ${ϵ}_j$, their contribution is at most ${\sum }_j|a_j|{ϵ}_j$ rather than a signed cancellation.

The generic bound can be conservative because all samples come from the same current history. For the two-tap predictor $\hat{i}=2i(0)-i(-T_a)+\eta$ and future time $\delta \ge 0$, rewrite the error asThis gives the sharper resultinstead of the direct absolute-weight bound $ϵ+L_i(3\delta +T_a)$. It is sharp under only a Lipschitz current assumption: a history proportional to $|t|$ changes slope at the central sample and attains the positive error limit. The Lipschitz assumption permits that corner even though a derivative need not exist at the corner itself.

For an original example, take $L_i=5000A/s$, $T_a=6\mu \mathrm{s}$, a target interval $0\le t\le 2\mu \mathrm{s}$, and aggregate estimate error $ϵ=0.020\mathrm{A}$. The sharper interval bound is $0.060\mathrm{A}$; the direct absolute-weight bound is $0.080\mathrm{A}$. A reading $\hat{i}=0.065\mathrm{A}$ therefore certifies at least $0.005\mathrm{A}$ throughout the interval using the sharper proof. A signed timestamp average would misleadingly place the predictor at time $+T_a$; it is an extrapolation, not a measured future sample.

A current-sign certificate with a curvature bound can exploit stronger information. For the target time $T_a$ and a twice differentiable current with $|i^{\prime \prime }|\le M_i$ on $[-T_a,T_a]$,Taylor’s theorem, or integrating the second derivative against its triangular weight, proves the inequality. At $T_a=6\mu \mathrm{s}$ and stipulated $M_i=10^{8}A/{\mathrm{s}}^2$, the model error bound is $0.0036\mathrm{A}$ before sample errors. This smaller number is justified only if the curvature bound also covers switching transitions; an affine-ramp fit alone does not establish it.

A proposed validation injects known ramps and waveforms with changing slope, records the actual filter coefficients and sampling times, and checks the resulting enclosures against an independent current trace. Account for sensor saturation, timing uncertainty and the history used by the filter. A sharper mathematical bound and a stronger physical assumption are different ways of obtaining a smaller certificate margin.

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.^[ref]

Let samples at $-T$ and $0$ produce measurements $y_{-}=i(-T)+e_{-}$ and $y_0=i(0)+e_0$, with $|e_{-}|\le {ϵ}_{-}$ and $|e_0|\le {ϵ}_0$. For a future time $t\ge 0$, use the linear extrapolationIf $i$ is twice continuously differentiable and $|i^{\prime \prime }(s)|\le M$ throughout $[-T,t]$, the polynomial interpolation error givesThe model term follows by applying the remainder to the two-node interpolant: $i(t)-[(1+t/T)i(0)-(t/T)i(-T)]=i^{\prime \prime }(\xi )t(t+T)/2$ 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 $M{T}^2$ and equal sample bounds contribute $3ϵ$.

For an original example, take $T=6\mu \mathrm{s}$, $M=10^{8}A/{\mathrm{s}}^2$ and each sample error bounded by $0.002\mathrm{A}$. At $t=T$,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 $\hat{i}(t)-B(t)>0$ 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.

Timing uncertainty in a curvature-based current-sign certificate adds the effect of uncertain sample and decision times to a bound on extrapolated electric current. Two accurate readings do not identify a future current precisely unless their timing relative to that future instant is also bounded. The physical purpose is to establish the current direction during the intended switching interval, whose recirculation paths depend on that direction.^[ref]

Let nominal sample times be $-T$ and $0$, and a nominal target time be $t\ge 0$. Actual sample times are $-T+{\delta }_{-}$ and ${\delta }_0$, and actual target time is $t+{\delta }_t$, with absolute timing bounds ${\Delta }_{-},{\Delta }_0,{\Delta }_t$. Measurements areAssume $|e_j|\le {ϵ}_j$, $|i^{\prime }|\le S$ and $|i^{\prime \prime }|\le M$ over all nominal and possible actual times and the intervals connecting them. Comparing each actual sample with its nominal-time value using the first-derivative bound, then applying the current-sign certificate with a curvature bound, givesThe triangle inequality proves this conservative enclosure without assuming independent or random timing errors. Its units are current: a timing error multiplied by a current-slew bound gives an amplitude error. A small curvature bound alone is insufficient, because a steep affine ramp has zero curvature but remains sensitive to relative timing.

For an original example, use $T=t=6\mu \mathrm{s}$, $M=10^{8}A/{\mathrm{s}}^2$, $S=5000A/s$ and ${ϵ}_0={ϵ}_{-}=0.002\mathrm{A}$. The amplitude-plus-curvature allowance is $0.0096\mathrm{A}$. With ${\Delta }_0={\Delta }_{-}=50\mathrm{ns}$ and ${\Delta }_t=100\mathrm{ns}$, timing addsThe full bound is $0.01085\mathrm{A}$. A prediction $0.020\mathrm{A}$ therefore certifies a positive target current of at least $0.00915\mathrm{A}$. To certify a whole switching interval, minimize $\hat{i}(t)-B_{\mathrm{time}}(t)$ over that interval rather than checking only one instant.

Shared-clock cancellation in current extrapolation can make the separate timing allowances conservative. If both samples and the target undergo exactly the same time translation, an affine ramp is still extrapolated exactly. The relevant error is relative timing; a common shift of the acquisition clock and a separately timed switching command need not cancel. Report which clocks, triggers and transport delays define the target.

A proposed check uses a known current ramp and an independently timed reference trace, deliberately perturbs sample and command timing, and tests whether the resulting errors stay inside the enclosure. Repeat near switching transitions. Slew and curvature assumptions must cover those transitions, not only the quiet portion used to fit a smooth waveform.