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%$Header$ |
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\chapter[\csoconeshorttitle{}]{\csoconelongtitle{}} |
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\label{csoc1} |
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\beginchapterquote{``Anything is possible if you don't know what you're talking |
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about.''} |
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{Unknown} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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%Section tag: INT0 |
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\label{csoc1:sint0} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Potentiometers} |
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%Section tag: pot0 |
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\label{csoc1:spot0} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Ratiometric Conversion And Measurement Systems} |
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%Section tag: RCS0 |
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\label{csoc1:srcs0} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Introduction} |
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%Section tag: INT0 |
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\label{csoc1:srcs0:sint0} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Observation Error Due To A/D Quantization} |
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%Subsection tag: OEQ0 |
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\label{csoc1:srcs0:soeq0} |
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\index{quantization error} |
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The software which executes on a microcontroller is inherently digital |
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and can accept as input only digital data. Analog signals must first be |
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converted to integers using an \index{A/D converter}A/D converter, and |
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such a conversion always introduces \index{quantization error}quantization |
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error as a voltage which is conceptually real is mapped to |
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$\vworkintsetnonneg{}$. Any such quantization errors are compounded when |
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more than one quantized value is used to attempt to infer potentiometer |
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position. |
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The chief emphasis of this section is the analysis of error due to the |
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use of multiple quantized inputs in an attempt to infer a potentiometer |
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position. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsubsection{Prototype System I} |
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%Subsubsection tag: OEQ0 |
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\label{csoc1:srcs0:soeq0:spsa0} |
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In this section we consider the system shown in |
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Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}. |
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The figure represents |
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the simplest system which may present quantization |
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analysis difficulties. Subsequent sections will analyze |
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the quantization error of more difficult systems. |
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Table~\ref{tbl:csoc1:srcs0:soeq0:spsa0:01} defines the |
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variables used in |
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Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01} and for |
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analysis in this section. |
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\begin{figure} |
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\centering |
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\includegraphics[width=4.6in]{c_soc1/prcs001.eps} |
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\caption{Prototype Ratiometric Conversion System I} |
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\label{fig:csoc1:srcs0:soeq0:spsa0:01} |
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\end{figure} |
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\begin{table} |
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\begin{center} |
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\begin{tabular}{|c|l|} |
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\hline |
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Variable & Description \\ |
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\hline |
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\hline |
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$\alpha$ & Potentiometer position, parameterized through $0\leq\alpha\leq 1$. \\ |
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& $\alpha=0$ is defined to be the potentiometer position \\ |
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& that produces the lowest voltage at the A/D pin, and \\ |
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& $\alpha=1$ is defined to be the potentiometer position \\ |
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& that produces the highest voltage at the A/D pin. \\ |
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\hline |
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$\overline{\alpha}$ |
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& The estimate of potentiometer position, which may \\ |
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& contain error because of quantization error introduced \\ |
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& by A/D conversion. \\ |
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\hline |
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$N_{ADP}$ & The number of A/D counts (supplied to software by an \\ |
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& A/D converter) corresponding to the sensing of the \\ |
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& potentiometer position $\alpha$. \\ |
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\hline |
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$\overline{N_{ADP}}$ |
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& $N_{ADP}$ with no quantization error. \\ |
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\hline |
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$N_{ADS}$ & The number of A/D counts (supplied to software by an \\ |
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& A/D converter) corresponding to the sensing of the \\ |
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& supply voltage $V_S$. \\ |
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\hline |
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$\overline{N_{ADS}}$ |
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& $N_{ADS}$ with no quantization error. \\ |
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\hline |
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$R_P$ & The resistance (in Ohms) of the variable potentiometer \\ |
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& whose wiper position $\alpha$ is to be sensed (see \\ |
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& Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}). Note that this resistance does not appear \\ |
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& in the analysis of the circuit of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, as only \\ |
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& the potentiometer wiper position $\alpha$ affects $V_{ADP}$. \\ |
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\hline |
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$r_{ADP}$ & The A/D converter ratio (from volts to counts) \\ |
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& implemented by A/D converter monitoring the variable \\ |
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& potentiometer input. Note that $N_P = r_{ADP} V_{ADP}$. \\ |
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\hline |
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$r_{ADS}$ & The A/D converter ratio (from volts to counts) \\ |
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& implemented by A/D converter monitoring the $V_S$ input. \\ |
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& Note that $N_S = r_{ADS} V_{ADS}$. \\ |
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\hline |
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$V_{ADP}$ & The voltage supplied to the microcontroller or A/D \\ |
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& converter corresponding to the variable potentiometer\\ |
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& position $\alpha$. In the system of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, $V_{ADP} = \alpha V_S$.\\ |
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\hline |
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$V_{ADS}$ & The voltage supplied to the microcontroller or A/D \\ |
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& converter corresponding to supply voltage $V_S$. In \\ |
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& the system of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, $V_{ADS}=V_S$. \\ |
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\hline |
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$V_S$ & The supply voltage, presumed variable, which can be \\ |
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& sensed by the microcontroller software, and also \\ |
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& drives the high side of the variable potentiometer. \\ |
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\hline |
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\end{tabular} |
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\end{center} |
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\caption{Variables Used In Analysis Of Prototype System I |
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(Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01})} |
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\label{tbl:csoc1:srcs0:soeq0:spsa0:01} |
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\end{table} |
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To analyze the system depicted in Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, |
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first note that |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:01} |
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V_{ADS} = V_S |
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\end{equation} |
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\noindent{}and |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:02} |
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V_{ADP} = \alpha V_S. |
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\end{equation} |
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For simplicity of analysis, we assume that A/D converters quantize by |
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truncation, so that |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:03} |
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N_{ADS} = \lfloor V_S r_{ADS} \rfloor |
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\end{equation} |
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\noindent{}and |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:04} |
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N_{ADP} = \lfloor \alpha V_S r_{ADP} \rfloor. |
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\end{equation} |
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\noindent{}The assumption that A/D converters quantize by truncation |
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has little effect on the error analysis of practical systems. (Need |
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to include an exercise to show this.) |
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To aid in symbolic manipulation, we also introduce |
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$\overline{N_{ADS}}$ and $\overline{N_{ADP}}$, which are |
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$N_{ADS}$ and $N_{ADP}$, respectively, |
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without quantization error: |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:03b} |
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\overline{N_{ADS}} = V_S r_{ADS} |
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\end{equation} |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:04b} |
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\overline{N_{ADP}} = \alpha V_S r_{ADP} |
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\end{equation} |
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If quantization were not present (i.e. if Eqns. |
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\ref{eq:csoc1:srcs0:soeq0:spsa0:03} |
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and |
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\ref{eq:csoc1:srcs0:soeq0:spsa0:04} |
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did not include floor functions), the potentiometer wiper position |
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$\alpha$ could be calculated exactly as: |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:05} |
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\alpha = \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}} r_{ADP}}. |
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\end{equation} |
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Based on (\ref{eq:csoc1:srcs0:soeq0:spsa0:05}), |
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it makes sense to calculate $\overline{\alpha}$, the |
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estimate of $\alpha$, as: |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:05b} |
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\overline{\alpha} |
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= |
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\frac{N_{ADP} r_{ADS}}{N_{ADS} r_{ADP}}. |
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\end{equation} |
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The question that must be posed is: |
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\emph{How different from $\alpha$ can $\overline{\alpha}$ be?}. |
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We seek to bound $|\overline{\alpha}-\alpha|$. |
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Quantization can be treated by noting that applying the floor function |
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to an input introduces an error $\in (-1,0]$, i.e. |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:06} |
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\lfloor x \rfloor = x + \varepsilon; \;\; \varepsilon \in (-1,0]. |
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\end{equation} |
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Using this observation, we can rewrite |
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(\ref{eq:csoc1:srcs0:soeq0:spsa0:05b}) as |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:07} |
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\overline{\alpha} |
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= |
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\frac{N_{ADP} r_{ADS}}{N_{ADS} r_{ADP}} |
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= |
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\frac{(\overline{N_{ADP}} + \varepsilon_1) r_{ADS}}{(\overline{N_{ADS}} + \varepsilon_2) r_{ADP}} ; |
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\;\; |
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\varepsilon_1, \varepsilon_2 \in (-1, 0]. |
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\end{equation} |
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\noindent{}It can be seen from |
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(\ref{eq:csoc1:srcs0:soeq0:spsa0:07}) |
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that the minimum value of the estimate |
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$\overline{\alpha}$ occurs when $\varepsilon_1$ is minimized and |
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$\varepsilon_2$ is maximized. Similarly, the maximum |
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value occurs when $\varepsilon_1$ is maximized and |
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$\varepsilon_2$ is minimized. These observations lead to this |
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inequality: |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:08} |
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\frac{N_{ADP} r_{ADS}}{(N_{ADS}+1) r_{ADP}} |
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< |
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\alpha |
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< |
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\frac{(N_{ADP}+1) r_{ADS}}{r_{ADP}} . |
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\end{equation} |
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Intuitively, the form of (\ref{eq:csoc1:srcs0:soeq0:spsa0:08}) |
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makes sense. The smallest possible value of $\alpha$ will correspond |
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to the case |
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when $N_{ADP}$ contains no quantization error but $N_{ADS}$ contains |
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a quantization error of nearly 1. |
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The largest possible value of $\alpha$ will correspond |
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to the case |
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when $N_{ADS}$ contains no quantization error but $N_{ADP}$ contains |
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a quantization error of nearly 1. |
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It can also be seen from (\ref{eq:csoc1:srcs0:soeq0:spsa0:08}) that |
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the interval to which $\alpha$ is confined will be larger |
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when $V_S$ is smaller (implying a smaller $N_{ADS}$ and $N_{ADP}$). |
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This is also intuitively plausible, since the quantization error |
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of up to one count in $N_{ADS}$ or $N_{ADP}$ will have a larger |
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relative significance when $N_{ADS}$ and $N_{ADP}$ are smaller. |
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The inequality provided in (\ref{eq:csoc1:srcs0:soeq0:spsa0:08}) |
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is certainly useful, and gives insight into quantization error. However, |
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we seek an inequality that is more friendly to engineering work, i.e. |
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one that involves voltages rather than A/D counts. |
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Assume it is known for an application that $V_S$ can vary only over the |
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interval |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:08b} |
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V_S \in [V_{SMIN}, V_{SMAX}], |
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\end{equation} |
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\noindent{}and that $\alpha$ can very only over |
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the interval |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:08c} |
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\alpha \in [\alpha_{MIN}, \alpha_{MAX}]. |
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\end{equation} |
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\noindent{}Furthermore, we |
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seek useful inequalities where is it \emph{not} required |
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that\footnote{(\ref{eq:csoc1:srcs0:soeq0:spsa0:09}) represents the |
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requirement that $V_{SMIN}$ or $V_{SMAX}$ represent a precisely |
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integral number of A/D |
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counts. This requirement is almost never met in practical engineering |
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work.} |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:09} |
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r_{ADS} \vworkdivides \{V_{SMIN},V_{SMAX}\}, |
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\end{equation} |
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\noindent{}as if (\ref{eq:csoc1:srcs0:soeq0:spsa0:09}) |
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were required, it would introduce extra complexity |
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in applying the inequality. |
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To develop the desired type of inequality, we can use |
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a different analysis method. Assume that |
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$r_{ADS}$ and $r_{ADP}$ are fixed. Then, introduce |
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the function |
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\begin{equation} |
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\label{eq:csoc1:srcs0:soeq0:spsa0:10} |
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F(\overline{N_{ADP}}, \overline{N_{ADS}}) |
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= |
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\frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}} r_{ADP}}, |
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\end{equation} |
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\noindent{}which, in accordance with (\ref{eq:csoc1:srcs0:soeq0:spsa0:05}), |
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is a perfect estimate of $\alpha$. |
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\begin{figure} |
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\centering |
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\Huge{TBD} |
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\caption{Sample Staircase Pattern Of Estimate $\overline{\alpha}$ Of |
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Prototype Ratiometric Conversion System I} |
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\label{fig:csoc1:srcs0:soeq0:spsa0:02} |
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\end{figure} |
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We can examine the staircase pattern of $\overline{\alpha}$ as |
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$V_S$ is increased (see Figure \ref{fig:csoc1:srcs0:soeq0:spsa0:02}, |
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which provides an example staircase pattern). Note that the staircase |
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pattern may contain four qualitatively |
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distinct types of vertical discontinuities. In the descriptions below, |
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assume that $V_D \in [V_{SMIN}, V_{SMAX}]$ is the value of |
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$V_S$ at the discontinuity. |
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|
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\begin{itemize} |
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\item \textbf{Downward overshoot discontinuities (DOD):} |
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As $V_S$ is increased, these correspond to the values |
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of $V_S$ where $V_S r_{ADS} \in \vworkintset$ but |
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$V_S \alpha r_{ADS} \notin \vworkintset$. At such |
367 |
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discontinuities, |
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$\lim_{V_S \rightarrow V_D^-}\overline{\alpha} > \alpha$, |
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but |
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$\lim_{V_S \rightarrow V_D^+}\overline{\alpha} < \alpha$. |
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\item \textbf{Upward overshoot discontinuities (UOD):} |
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As $V_S$ is increased, these correspond to the values |
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of $V_S$ where $V_S r_{ADS} \notin \vworkintset$ but |
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$V_S \alpha r_{ADS} \in \vworkintset$. At such |
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discontinuities, |
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$\lim_{V_S \rightarrow V_D^-}\overline{\alpha} < \alpha$, |
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but |
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$\lim_{V_S \rightarrow V_D^+}\overline{\alpha} > \alpha$. |
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\item \textbf{Downward exact discontinuities (DED):} |
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As $V_S$ is increased, these correspond to the values |
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of $V_S$ where $V_S r_{ADS} \in \vworkintset$ and |
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$V_S \alpha r_{ADS} \in \vworkintset$. At such |
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discontinuities, |
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$\lim_{V_S \rightarrow V_D^-}\overline{\alpha} > \alpha$, |
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but |
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$\lim_{V_S \rightarrow V_D^+}\overline{\alpha} = \alpha$. |
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\item \textbf{Upward exact discontinuities (UED):} |
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As $V_S$ is increased, these correspond to the values |
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of $V_S$ where $V_S r_{ADS} \in \vworkintset$ and |
390 |
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$V_S \alpha r_{ADS} \in \vworkintset$. At such |
391 |
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discontinuities, |
392 |
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$\lim_{V_S \rightarrow V_D^-}\overline{\alpha} < \alpha$, |
393 |
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but |
394 |
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$\lim_{V_S \rightarrow V_D^+}\overline{\alpha} = \alpha$. |
395 |
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\end{itemize} |
396 |
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|
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We can place upper bounds on the magnitudes of the discontinuities |
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by examining the partial derivatives of |
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$F(\overline{N_{ADP}}, \overline{N_{ADS}})$ as specified in |
400 |
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(\ref{eq:csoc1:srcs0:soeq0:spsa0:10}), specifically: |
401 |
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|
402 |
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\begin{equation} |
403 |
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\label{eq:csoc1:srcs0:soeq0:spsa0:11} |
404 |
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\frac{\partial{}F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}} |
405 |
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= |
406 |
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- |
407 |
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\frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^2} r_{ADP}} |
408 |
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\end{equation} |
409 |
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|
410 |
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\begin{equation} |
411 |
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\label{eq:csoc1:srcs0:soeq0:spsa0:11b} |
412 |
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\frac{\partial{}^2 F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}^2} |
413 |
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= |
414 |
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2 |
415 |
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\frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^3} r_{ADP}} |
416 |
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\end{equation} |
417 |
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|
418 |
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\begin{equation} |
419 |
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\label{eq:csoc1:srcs0:soeq0:spsa0:11c} |
420 |
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\frac{\partial{}^i F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}^i} |
421 |
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= |
422 |
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(-1)^i (i!) |
423 |
|
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\frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^{i+1}} r_{ADP}} |
424 |
|
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\end{equation} |
425 |
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|
426 |
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|
427 |
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\begin{equation} |
428 |
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\label{eq:csoc1:srcs0:soeq0:spsa0:12} |
429 |
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\frac{\partial{}F(\cdot{},\cdot{})}{\partial \overline{N_{ADP}}} |
430 |
|
|
= |
431 |
|
|
- |
432 |
|
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\frac{r_{ADS}}{\overline{N_{ADS}} r_{ADP}} |
433 |
|
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\end{equation} |
434 |
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|
435 |
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\begin{equation} |
436 |
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\label{eq:csoc1:srcs0:soeq0:spsa0:12b} |
437 |
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\frac{\partial{}^i F(\cdot{},\cdot{})}{\partial \overline{N_{ADP}}^i} |
438 |
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= |
439 |
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0, \; i \geq 2 |
440 |
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\end{equation} |
441 |
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|
442 |
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A \emph{DOD} (discussed above) corresponds to the case where |
443 |
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$N_{ADS}$ increases by one count at $V_S=V_D$ without an increase in |
444 |
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$N_{ADP}$. |
445 |
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|
446 |
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|
447 |
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A \emph{UOD} (discussed above) corresponds to the case where |
448 |
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$N_{ADP}$ increases by one count at $V_S=V_D$ without an increase in |
449 |
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$N_{ADS}$. |
450 |
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|
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|
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Motion Control Systems} |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\vfill |
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\noindent\begin{figure}[!b] |
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\noindent\rule[-0.25in]{\textwidth}{1pt} |
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\begin{tiny} |
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\begin{verbatim} |
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$RCSfile: c_soc1.tex,v $ |
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$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_soc1/c_soc1.tex,v $ |
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$Revision: 1.7 $ |
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$Author: dtashley $ |
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$Date: 2002/04/21 22:50:04 $ |
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\end{verbatim} |
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\end{tiny} |
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\noindent\rule[0.25in]{\textwidth}{1pt} |
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\end{figure} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% $Log: c_soc1.tex,v $ |
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% Revision 1.7 2002/04/21 22:50:04 dtashley |
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% Safety checkin before working on laptop. |
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% |
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% Revision 1.6 2002/04/13 08:18:43 dtashley |
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% Edits from laptop. |
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% |
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% Revision 1.5 2002/04/13 06:04:32 dtashley |
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% Safety checkin before resuming work on laptop. |
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% |
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% Revision 1.4 2002/04/12 23:41:50 dtashley |
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% Safety checkin before major editorial additions. |
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% |
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% Revision 1.3 2002/04/10 06:12:33 dtashley |
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% Evening safety checkin. |
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% |
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% Revision 1.2 2002/04/09 23:32:20 dtashley |
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% Edits, preparing for addition of lists of tables and figures. |
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% |
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% Revision 1.1 2001/08/25 22:51:26 dtashley |
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% Complex re-organization of book. |
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% |
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%End of file C_PCO0.TEX |