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I am looking for a smooth (continuous differentiable) approximation of the following five-phased linear function: $$ P(U, R, P_r) = \begin{cases} R(U_{max} - U) + R(U_{up} - U_{max}) + P_r; & U \geq U_{max}\\ R(U_{up} - U) + P_r; & U_{up} < U < U_{max}\\ P_r; & U_{low} \leq U \leq U_{up}\\ R(U_{low} - U) + P_r; & U_{min} < U < U_{low}\\ R(U_{min} - U) + R(U_{low} - U_{min}) + P_r; & U \leq U_{min}\\ \end{cases}, $$ Here $U_{max} \ge U_{up} \ge U_{low} \ge U_{min} $ are the breakpoints and bounds of the variable $U$. $P_r$ and $R$ are a given setpoint value and slope of the line respectively. Also, $P$ has bounds as $P_{min} \leq P \leq P_{max}$.

I am trying to define these constraints as nested max functions and then apply the smooth approximation. Such that: $$F(x) = \max\{0,x\}, \quad for \quad x \in \mathbb{R}$$

By picking any positive number $\epsilon > 0$:

$$F^\epsilon(x) = \epsilon \ln \left(1+ \exp(x/\epsilon)\right), \quad for \quad x \in \mathbb{R}$$

with bound verification:

$$F^\epsilon(x) - \epsilon \ln2 \leq F(x) \leq F^\epsilon(x)$$

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  • $\begingroup$ If I further simplify the function as: $$ P(U, R, P_r) = \begin{cases} R(U_{up} - U) + P_r; & U_{up} < U \leq U_{max}\\ P_r; & U_{low} \leq U \leq U_{up}\\ R(U_{low} - U) + P_r; & U_{min} \leq U < U_{low}\\ \end{cases}, $$ $\endgroup$ Feb 2, 2023 at 0:30

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