# Talk:Auto-zero/Auto-calibration

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?

## Abstract form of Likelihood case

Setup (obsolete?):

$LaTeX: P\,$ a n-dimensional collection of Gaussian probability distribution functions. This is a condition I want to liberalize to any convex PDF (and in some sense to a uniform PDF).

$LaTeX: p\in P\,$ ; $LaTeX: \bar{p}\,$ PDF of $LaTeX: p\,$

General formula: $LaTeX: y=f(x;p)\,$

Calibration pair $LaTeX: [y_{c},x_{c}]\,$ constraining $LaTeX: p\,$  : $LaTeX: y_{c}=f(x_{c};p)\,$ and consequently forming a new PDF $LaTeX: \bar{p'}\,$ with $LaTeX: p'\,$ the constrained $LaTeX: p\,$.

$LaTeX: y_{t}=f(x_{t};p')\,$

Clearly given $LaTeX: y_{t}\,$ fixed, $LaTeX: x_{t}\,$ has a PDF. $LaTeX: \bar{x_{t}}=\bar{x}_{t}\left(y_{t},y_{c},x_{c},\bar{p}\right)=\bar{x}_{t}\left(y_{t},\bar{p'}\right)\,$

Problem 1: mode

maximize: The most likely value of $LaTeX: \bar{x}_{t}\,$

(i.e. $LaTeX: \frac{dx_{t}}{dp'}=0\,$ for differentiable functions)

wrt : $LaTeX: p'\,$

given $LaTeX: y_{t},y_{c},x_{c},\bar{x_{t}},y=f(x;p)\,$ or alternately $LaTeX: y_t,\bar{x_{t}},y_t=f(x_t;p')\,$

## /* Optimization Format */

Since the example equations are not obviously deterministic, much less convex, here I will back track to the original problem and put it in matrix form. This is possible because the underlying physical problem is a linear electrical circuit. Some work is neccesary to disentangle the unknows which turn up not linear.

The fundamental equation is:$LaTeX: V=I\cdot R\,$ We introduce three new variables to produce coupled equations: $LaTeX: \widehat{I_{c}},\widehat{k_{1}}=\widehat{I_{c}}\widehat{e_{b}};\widehat{k_{2}}=\widehat{I_{c}}\widehat{e_{com}}\,$

This hides the nonlinearity.

Considering the calibration example where only the component errors are to be estimated.

Starting from the original circuit:

$LaTeX: \begin{array}{lcl} V_{ref}-\widehat{e_{ref}}-\widehat{I_{c}}R_{b}-\widehat{I_{c}}e_{b}-\widehat{I_{c}}\widehat{e_{com}}-\widehat{I_{c}}R_{c} & = & 0\\ V_{c}-\widehat{v_{off}}-\widehat{I_{c}}\widehat{e_{com}}-\widehat{I_{c}}R_{c} & = & 0\end{array}$

We start to seperate constants from variables.

$LaTeX: \begin{array}{lcl} \left[\begin{array}{c} V_{ref}\\ V_{c}\end{array}\right] & = & \left[\begin{array}{ccccc} 1 & (R_{b}+R_{c}) & 1 & 1 & 0\\ 0 & R_{c} & 0 & 1 & 1\end{array}\right]\left[\begin{array}{c} \widehat{e_{ref}}\\ \widehat{I_{c}}\\ \widehat{k_{1}}\\ \widehat{k_{2}}\\ \widehat{v_{off}}\end{array}\right]\end{array}$

$LaTeX: \begin{array}{lcl} \widehat{e_{ref}} & < & V_{ref}\\ \widehat{I_{c}} & > & 0\\ V_{c} & < & V_{ref}\end{array}$

Now if we have resolved this we can state:

$LaTeX: \begin{array}{lcl} \widehat{k_{1}} & =\widehat{I_{c}'} & \widehat{e_{b}}\\ \widehat{k_{2}} & =\widehat{I_{c}'} & \widehat{e_{com}}\end{array}$