Talk:Auto-zero/Auto-calibration

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I have been working on creating a robust design structure for the design of Auto-Zero/Auto-calibration implementations. I have a lot of moving parts in my head; but I believe I need outside viewpoints and knowledge in order to construct a general approach. If anybody is interested please respond here. It is a bit more complicated than it would seem on the surface IMHO. I somewhat think it falls within convex optimization. On the other hand I sometimes think it doesn't. I do have a particular example that illustrates the various problems that can arise. Although the ideas should be applicable to Scientific measurements; the applications I have in mind relate to autonomous embedded software and hardware implementations.

Ray

Note on the examples: I think that due to the physically meaningful restrictions, R>0 and errors less than 100%, on the problem; a conversion process using logs and affine transforms will generate posynomial equations for optimization and constraints. I tried Geometric programing before but didn't put the proper (I hope) restrictions in place. Perhaps I gave up too early? They might not exactly fit geometric programing but they might fit convex programing. Ray

Trial: Poysnomial expressions

LaTeX: e^{\psi_{x}}=\left(1-\frac{v_{off}}{V_{x}}\right),e^{\psi_{c}}=\left(1-\frac{v_{off}}{V_{c}}\right),e^{\psi_{t}}=\left(1-\frac{v_{off}}{V_{t}}\right),e^{\mathcal{V}_{x}}=V_{x},e^{\mathcal{V}_{c}}=V_{c},e^{\mathcal{V}_{t}}=V_{t}

LaTeX: e^{\mathcal{R}_{d}}=R_{x}+e_{com}+R_{b}+e_{b},e^{\mathcal{R}_{x}}=R_{x},e^{\epsilon_{com}}=e_{com},e^{\mathcal{R}_{b}}=R_{b},e^{\epsilon_{b}}=\left(1-\frac{e_{b}}{R_{t}}\right)

LaTeX: e^{\mathcal{R}_{t}}=R_{t}

LaTeX: e^{\mathcal{V}_{ref}}=V_{ref},e^{\psi_{ref}}=\left(1-\frac{v_{off}}{V_{ref}}\right)

Thus the expression for LaTeX: V_{x} is

LaTeX: e^{\mathcal{V}_{x}}e^{\psi_{x}}=e^{\mathcal{V}_{ref}}e^{\psi_{ref}}\cdot\left(e^{\mathcal{R}_{x}}+e^{\epsilon_{com}}\right)\cdot e^{-\mathcal{R}_{d}}

Keeping the new variable LaTeX: e^{\mathcal{R}_{d}} we have the following constraint

LaTeX: e^{\mathcal{R}_{d}}=e^{\mathcal{R}_{x}}+e^{\epsilon_{com}}+e^{\mathcal{R}_{b}}e^{\epsilon_{b}}

The denominator of LaTeX: R_{t} can be expressed as

LaTeX: e^{\mathcal{\eta}_{e}}=V_{ref}+e_{ref}-V_{t}+v_{off}

Note a sign change this is complimented in the denominator.

Note that due to the circuit physics LaTeX: V_{ref}>V_{x} for all errors

The expression for LaTeX: R_{t} is

LaTeX: e^{\mathcal{R}_{t}}=\left(\left(e^{\mathcal{V}_{t}}e^{\psi_{t}}\right)\left(e^{\epsilon_{com}}+e^{\epsilon_{b}}e^{\mathcal{R}_{b}}\right)+e^{\epsilon_{com}}e^{\mathcal{V}_{ref}}e^{\mathcal{\psi}_{ref}}\right)e^{-\mathcal{\eta}_{e}}

With the constraint

LaTeX: e^{\mathcal{\eta}_{e}}=e^{\mathcal{V}_{t}}e^{\psi_{t}}+e^{\mathcal{V}_{ref}}e^{\mathcal{\psi}_{ref}}

Unfortunately applying the constraint algebraically leads to some negative terms. Perhaps collecting the positive and negative terms into separate conditions and placing the sum constraint would avoid this?

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