Talk:Auto-zero/Auto-calibration - Revision history

Rrogers314: /* /* Optimization Format / /

Rrogers314 — Thu, 23 Sep 2010 19:29:38 GMT

- Optimization Format -

←Older revision		Revision as of 19:29, 23 September 2010
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	The fundamental equation is:<math>V=I\cdot R\,</math>		The fundamental equation is:<math>V=I\cdot R\,</math>
-	We introduce ~~two~~ new variables to produce coupled equations:	+	We introduce three new variables to produce coupled equations:
-	<math>\widehat{k_{1}}=\widehat{I_{c}}\widehat{e_{b}};\widehat{k_{2}}=\widehat{I_{c}}\widehat{e_{com}}\,</math>	+	<math>\widehat{I_{c}},\widehat{k_{1}}=\widehat{I_{c}}\widehat{e_{b}};\widehat{k_{2}}=\widehat{I_{c}}\widehat{e_{com}}\,</math>

	This hides the nonlinearity.		This hides the nonlinearity.

Rrogers314: New section: /* Optimization Format */

Rrogers314 — Thu, 23 Sep 2010 19:09:12 GMT

New section: - Optimization Format -

←Older revision		Revision as of 19:09, 23 September 2010
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	given <math>y_{t},y_{c},x_{c},\bar{x_{t}},y=f(x;p)\,</math>		given <math>y_{t},y_{c},x_{c},\bar{x_{t}},y=f(x;p)\,</math>
	or alternately <math>y_t,\bar{x_{t}},y_t=f(x_t;p')\,</math>		or alternately <math>y_t,\bar{x_{t}},y_t=f(x_t;p')\,</math>
		+
		+	== /* 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:<math>V=I\cdot R\,</math>
		+	We introduce two new variables to produce coupled equations:
		+	<math>\widehat{k_{1}}=\widehat{I_{c}}\widehat{e_{b}};\widehat{k_{2}}=\widehat{I_{c}}\widehat{e_{com}}\,</math>
		+
		+	This hides the nonlinearity.
		+
		+	Considering the calibration example where only the component errors
		+	are to be estimated.
		+
		+	Starting from the original circuit:
		+
		+	<math>
		+	\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}
		+	</math>
		+
		+	We start to seperate constants from variables.
		+
		+	<math>\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}
		+	</math>
		+
		+	<math>\begin{array}{lcl}
		+	\widehat{e_{ref}} & < & V_{ref}\\
		+	\widehat{I_{c}} & > & 0\\
		+	V_{c} & < & V_{ref}\end{array}</math>
		+
		+	Now if we have resolved this we can state:
		+
		+	<math>\begin{array}{lcl}
		+	\widehat{k_{1}} & =\widehat{I_{c}'} & \widehat{e_{b}}\\
		+	\widehat{k_{2}} & =\widehat{I_{c}'} & \widehat{e_{com}}\end{array}
		+	</math>

Rrogers314: /* Abstract form of Likelihood case */

Rrogers314 — Thu, 23 Sep 2010 18:24:34 GMT

Abstract form of Likelihood case

←Older revision		Revision as of 18:24, 23 September 2010
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	== Abstract form of Likelihood case ==		== Abstract form of Likelihood case ==
-	Setup:	+	Setup (obsolete?):

	<math>P\,</math> a n-dimensional collection of Gaussian probability distribution		<math>P\,</math> a n-dimensional collection of Gaussian probability distribution

Rrogers314 at 15:59, 12 September 2010

Rrogers314 — Sun, 12 Sep 2010 15:59:09 GMT

←Older revision		Revision as of 15:59, 12 September 2010
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	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?		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:
		+
		+	<math>P\,</math> 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).
		+
		+	<math>p\in P\,</math> ; <math>\bar{p}\,</math> PDF of <math>p\,</math>
		+
		+	General formula: <math>y=f(x;p)\,</math>
		+
		+	Calibration pair <math>[y_{c},x_{c}]\,</math> constraining <math>p\,</math> : <math>y_{c}=f(x_{c};p)\,</math>
		+	and consequently forming a new PDF <math>\bar{p'}\,</math> with <math>p'\,</math> the constrained <math>p\,</math>.
		+
		+	<math>y_{t}=f(x_{t};p')\,</math>
		+
		+	'''Clearly given <math>y_{t}\,</math> fixed, <math>x_{t}\,</math> has a PDF. <math>\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)\,</math>'''
		+
		+	Problem 1: mode
		+
		+	maximize: The most likely value of <math>\bar{x}_{t}\,</math>
		+
		+	(i.e. <math>\frac{dx_{t}}{dp'}=0\,</math> for differentiable functions)
		+
		+	wrt : <math>p'\,</math>
		+
		+	given <math>y_{t},y_{c},x_{c},\bar{x_{t}},y=f(x;p)\,</math>
		+	or alternately <math>y_t,\bar{x_{t}},y_t=f(x_t;p')\,</math>

Rrogers314: New section: Trial: Poysnomial expressions

Rrogers314 — Thu, 02 Sep 2010 17:12:53 GMT

New section: Trial: Poysnomial expressions

←Older revision		Revision as of 17:12, 2 September 2010
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	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.		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		Ray
		+
		+	== Trial: Poysnomial expressions ==
		+
		+	<math>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}</math>
		+
		+	<math>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)</math>
		+
		+	<math>e^{\mathcal{R}_{t}}=R_{t}</math>
		+
		+	<math>e^{\mathcal{V}_{ref}}=V_{ref},e^{\psi_{ref}}=\left(1-\frac{v_{off}}{V_{ref}}\right)</math>
		+
		+	'''Thus the expression for <math>V_{x}</math> is'''
		+
		+	<math>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}}</math>
		+
		+	Keeping the new variable <math>e^{\mathcal{R}_{d}}</math> we have the following
		+	constraint
		+
		+	<math>e^{\mathcal{R}_{d}}=e^{\mathcal{R}_{x}}+e^{\epsilon_{com}}+e^{\mathcal{R}_{b}}e^{\epsilon_{b}}</math>
		+
		+	The denominator of <math>R_{t}</math> can be expressed as
		+
		+	<math>e^{\mathcal{\eta}_{e}}=V_{ref}+e_{ref}-V_{t}+v_{off}</math>
		+
		+	Note a sign change this is complimented in the denominator.
		+
		+	Note that due to the circuit physics <math>V_{ref}>V_{x}</math> for all errors
		+
		+	'''The expression for <math>R_{t}</math> is'''
		+
		+	<math>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}}</math>
		+
		+	With the constraint
		+
		+	<math>e^{\mathcal{\eta}_{e}}=e^{\mathcal{V}_{t}}e^{\psi_{t}}+e^{\mathcal{V}_{ref}}e^{\mathcal{\psi}_{ref}}</math>
		+
		+	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?

Rrogers314 at 19:06, 1 September 2010

Rrogers314 — Wed, 01 Sep 2010 19:06:00 GMT

←Older revision		Revision as of 19:06, 1 September 2010
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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.		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		Ray

Rrogers314: Discussion & Motivation for Auto-Zero/Auto-Calibration

Rrogers314 — Sat, 14 Aug 2010 13:58:40 GMT

Discussion & Motivation for Auto-Zero/Auto-Calibration

New page

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