Smooth, autodiff-friendly replacements for piecewise functions used across GridKit component models. See CommonMath.hpp for implementation details.
Primitives
The scale \(\mu=4\cdot f_{\text{sync}}=240\) is chosen so \(\sigma\) behaves like a step on inputs while keeping derivatives finite. As \(\mu \to \infty\), these functions approach their exact targets.
Name |
Description |
Usage |
|---|
sigmoid
|
Step function |
GENSAL, GENROU, REGCA, REECA
|
ramp
|
Smooth one-sided ramp |
REGCA, REECA, REPCA
|
qramp
|
Exact one-sided quadratic ramp |
IEEET1
|
\(\sigma\) - sigmoid
The sigmoid is also known as the logistic function. The equivalent tanh form is used for numerical stability because the exponential form can divide by a very large value.
\[\begin{split}\begin{aligned}
\sigma(x)
&=
\begin{cases}
0 & x\le 0 \\[0pt]
1 & x\gt 0
\end{cases} \\[0pt]
&\approx \dfrac{1}{2}\left(1+\tanh\left(\dfrac{\mu x}{2}\right)\right)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/sigmoid.svg
```
\(\rho\) - ramp
ramp is the softplus approximation to the one-sided ramp. We do not use \(x\sigma(x)\) directly because it introduces a negative tail for \(x \lt 0\), while softplus stays nonnegative and approaches \(\max(x, 0)\) as the smoothing becomes sharp.
\[\begin{split}\begin{aligned}
\rho(x)
&= x\,\sigma(x) \\[0pt]
&\approx \dfrac{x+\lvert x\rvert}{2}+\dfrac{\ln\!\left(1+e^{-\mu\lvert x\rvert}\right)}{\mu}
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/ramp.svg
```
\(q\) - qramp
Note: the implementation of the quadratic ramp q(x) could be optimized with Enzyme features down the road so that we don’t need the smooth approximation.
\[q(x)=x^2\,\sigma(x)\]
```{image} ../../Figures/CommonMath/qramp.svg
```
Derived Functions
Name |
Description |
Usage |
|---|
max
|
Smooth binary maximum |
REECA, REECB
|
min
|
Smooth binary minimum |
REECA
|
clamp
|
Bounded saturation |
IEEEST, REECA, REECB, REPCA
|
deadband1
|
Type 1 no-offset signed two-sided deadband |
- |
deadband2
|
Type 2 offset signed two-sided deadband |
REECA, REECB, REPCA
|
slew
|
Symmetric slew-rate limiter |
- |
linseg
|
Saturated linear segment contribution |
REGCA, REECA
|
above
|
Above-lower-limit indicator |
REPCA
|
below
|
Below-upper-limit indicator |
- |
inside
|
Interior pulse indicator |
- |
outside
|
Outside-band indicator |
REECA, REECB
|
antiwindup
|
Anti-windup limited derivative |
IEEET1, SEXS-PTI, TGOV1, REECA, REECB, REPCA
|
max
\[\begin{split}\begin{aligned}
\text{max}(x,y)
&=
\begin{cases}
x & x\gt y \\[0pt]
y & x\le y
\end{cases} \\[0pt]
&\approx y+\rho(x-y)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/max.svg
```
min
\[\begin{split}\begin{aligned}
\text{min}(x,y)
&=
\begin{cases}
x & x\lt y \\[0pt]
y & x\ge y
\end{cases} \\[0pt]
&\approx x-\rho(x-y)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/min.svg
```
clamp
\[\begin{split}\begin{aligned}
\text{clamp}(x;\ell,u)
&=
\begin{cases}
\ell & x\lt \ell \\[0pt]
x & \ell\le x\le u \\[0pt]
u & x\gt u
\end{cases} \\[0pt]
&\approx \ell+\rho(x-\ell)-\rho(x-u)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/clamp.svg
```
deadband1
\[\begin{split}\begin{aligned}
\text{deadband1}(x;\ell,u)
&=
\begin{cases}
x & x\lt \ell \\[0pt]
0 & \ell\le x\le u \\[0pt]
x & x\gt u
\end{cases} \\[0pt]
&\approx x\left[\sigma(\ell-x)+\sigma(x-u)\right]
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/deadband1.svg
```
deadband2
\[\begin{split}\begin{aligned}
\text{deadband2}(x;\ell,u)
&=
\begin{cases}
x-\ell & x\lt \ell \\[0pt]
0 & \ell\le x\le u \\[0pt]
x-u & x\gt u
\end{cases} \\[0pt]
&\approx \rho(x-u)-\rho(\ell-x)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/deadband2.svg
```
slew
\[\begin{split}\begin{aligned}
\text{slew}(f;r)
&=
\begin{cases}
-r & f\lt -r \\[0pt]
f & -r\le f\le r \\[0pt]
r & f\gt r
\end{cases} \\[0pt]
&\approx -r+\rho(f+r)-\rho(f-r)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/slew.svg
```
linseg
\[\begin{split}\begin{aligned}
\text{linseg}(x;a,b,h)
&=
\begin{cases}
0 & x\lt a \\[0pt]
\dfrac{h}{b-a}(x-a) & a\le x\le b \\[0pt]
h & x\gt b
\end{cases} \\[0pt]
&\approx \dfrac{h}{b-a}\left[\rho(x-a)-\rho(x-b)\right]
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/linseg.svg
```
above
\[\begin{split}\begin{aligned}
\text{above}(x;\ell)
&=
\begin{cases}
0 & x\le \ell \\[0pt]
1 & x\gt \ell
\end{cases} \\[0pt]
&\approx \sigma(x-\ell)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/above.svg
```
below
\[\begin{split}\begin{aligned}
\text{below}(x;u)
&=
\begin{cases}
1 & x\lt u \\[0pt]
0 & x\ge u
\end{cases} \\[0pt]
&\approx \sigma(u-x)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/below.svg
```
inside
\[\begin{split}\begin{aligned}
\text{inside}(x;\ell,u)
&=
\begin{cases}
1 & \ell\lt x\lt u \\[0pt]
0 & \text{else}
\end{cases} \\[0pt]
&\approx \sigma(x-\ell)+\sigma(u-x)-1
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/inside.svg
```
outside
\[\begin{split}\begin{aligned}
\text{outside}(x;\ell,u)
&=
\begin{cases}
1 & x\lt \ell\ \lor\ x\gt u \\[0pt]
0 & \text{else}
\end{cases} \\[0pt]
&\approx \sigma(\ell-x)+\sigma(x-u)
\end{aligned}\end{split}\]
```{image} ../../Figures/CommonMath/outside.svg
```
antiwindup
\[\begin{split}\begin{aligned}
\text{antiwindup}(x,f;\ell,u)
&=
\begin{cases}
f & \ell\lt x\lt u \\[0pt]
f & x\le\ell\ \land\ f\gt 0 \\[0pt]
f & x\ge u\ \land\ f\lt 0 \\[0pt]
0 & \text{otherwise}
\end{cases} \\[0pt]
\phi_L
&= \text{above}(x;\ell) \\[0pt]
\phi_U
&= \text{below}(x;u) \\[0pt]
\phi(x,f)
&= \phi_L\phi_U+(1-\phi_U)\sigma(-f)+(1-\phi_L)\sigma(f) \\[0pt]
\text{antiwindup}(x,f;\ell,u)
&\approx \phi(x,f)\,f
\end{aligned}\end{split}\]