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1  %$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_soc1/c_soc1.tex,v 1.7 2002/04/21 22:50:04 dtashley Exp $  %$Header$
2    
3  \chapter[\csoconeshorttitle{}]{\csoconelongtitle{}}  \chapter[\csoconeshorttitle{}]{\csoconelongtitle{}}
4    
5  \label{csoc1}  \label{csoc1}
6    
7  \beginchapterquote{``Anything is possible if you don't know what you're talking  \beginchapterquote{``Anything is possible if you don't know what you're talking
8                       about.''}                       about.''}
9                       {Unknown}                       {Unknown}
10    
11  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
12  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
14  \section{Introduction}  \section{Introduction}
15  %Section tag: INT0  %Section tag: INT0
16  \label{csoc1:sint0}  \label{csoc1:sint0}
17    
18    
19  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
20  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
21  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
22  \section{Potentiometers}  \section{Potentiometers}
23  %Section tag: pot0  %Section tag: pot0
24  \label{csoc1:spot0}  \label{csoc1:spot0}
25    
26    
27  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
28  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
29  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
30  \section{Ratiometric Conversion And Measurement Systems}  \section{Ratiometric Conversion And Measurement Systems}
31  %Section tag: RCS0  %Section tag: RCS0
32  \label{csoc1:srcs0}  \label{csoc1:srcs0}
33    
34    
35    
36  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
37  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
38  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
39  \subsection{Introduction}  \subsection{Introduction}
40  %Section tag: INT0  %Section tag: INT0
41  \label{csoc1:srcs0:sint0}  \label{csoc1:srcs0:sint0}
42    
43    
44    
45    
46    
47  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
48  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
49  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
50  \subsection{Observation Error Due To A/D Quantization}  \subsection{Observation Error Due To A/D Quantization}
51  %Subsection tag: OEQ0  %Subsection tag: OEQ0
52  \label{csoc1:srcs0:soeq0}  \label{csoc1:srcs0:soeq0}
53    
54  \index{quantization error}  \index{quantization error}
55  The software which executes on a microcontroller is inherently digital  The software which executes on a microcontroller is inherently digital
56  and can accept as input only digital data.  Analog signals must first be  and can accept as input only digital data.  Analog signals must first be
57  converted to integers using an \index{A/D converter}A/D converter, and  converted to integers using an \index{A/D converter}A/D converter, and
58  such a conversion always introduces \index{quantization error}quantization  such a conversion always introduces \index{quantization error}quantization
59  error as a voltage which is conceptually real is mapped to  error as a voltage which is conceptually real is mapped to
60  $\vworkintsetnonneg{}$.  Any such quantization errors are compounded when  $\vworkintsetnonneg{}$.  Any such quantization errors are compounded when
61  more than one quantized value is used to attempt to infer potentiometer  more than one quantized value is used to attempt to infer potentiometer
62  position.  position.
63    
64  The chief emphasis of this section is the analysis of error due to the  The chief emphasis of this section is the analysis of error due to the
65  use of multiple quantized inputs in an attempt to infer a potentiometer  use of multiple quantized inputs in an attempt to infer a potentiometer
66  position.  position.
67    
68    
69  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
70  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
71  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
72  \subsubsection{Prototype System I}  \subsubsection{Prototype System I}
73  %Subsubsection tag: OEQ0  %Subsubsection tag: OEQ0
74  \label{csoc1:srcs0:soeq0:spsa0}  \label{csoc1:srcs0:soeq0:spsa0}
75    
76  In this section we consider the system shown in  In this section we consider the system shown in
77  Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}.    Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}.  
78  The figure represents  The figure represents
79  the simplest system which may present quantization  the simplest system which may present quantization
80  analysis difficulties.  Subsequent sections will analyze  analysis difficulties.  Subsequent sections will analyze
81  the quantization error of more difficult systems.  the quantization error of more difficult systems.
82  Table~\ref{tbl:csoc1:srcs0:soeq0:spsa0:01} defines the  Table~\ref{tbl:csoc1:srcs0:soeq0:spsa0:01} defines the
83  variables used in  variables used in
84  Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01} and for  Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01} and for
85  analysis in this section.  analysis in this section.
86    
87  \begin{figure}  \begin{figure}
88  \centering  \centering
89  \includegraphics[width=4.6in]{c_soc1/prcs001.eps}  \includegraphics[width=4.6in]{c_soc1/prcs001.eps}
90  \caption{Prototype Ratiometric Conversion System I}  \caption{Prototype Ratiometric Conversion System I}
91  \label{fig:csoc1:srcs0:soeq0:spsa0:01}  \label{fig:csoc1:srcs0:soeq0:spsa0:01}
92  \end{figure}  \end{figure}
93    
94  \begin{table}  \begin{table}
95  \begin{center}  \begin{center}
96  \begin{tabular}{|c|l|}  \begin{tabular}{|c|l|}
97  \hline  \hline
98  Variable           & Description \\  Variable           & Description \\
99  \hline  \hline
100  \hline  \hline
101  $\alpha$           & Potentiometer position, parameterized through $0\leq\alpha\leq 1$. \\  $\alpha$           & Potentiometer position, parameterized through $0\leq\alpha\leq 1$. \\
102                     & $\alpha=0$ is defined to be the potentiometer position \\                     & $\alpha=0$ is defined to be the potentiometer position \\
103                     & that produces the lowest voltage at the A/D pin, and \\                     & that produces the lowest voltage at the A/D pin, and \\
104                     & $\alpha=1$ is defined to be the potentiometer position \\                     & $\alpha=1$ is defined to be the potentiometer position \\
105                     & that produces the highest voltage at the A/D pin.    \\                     & that produces the highest voltage at the A/D pin.    \\
106  \hline  \hline
107  $\overline{\alpha}$      $\overline{\alpha}$    
108                     & The estimate of potentiometer position, which may    \\                     & The estimate of potentiometer position, which may    \\
109                     & contain error because of quantization error introduced \\                     & contain error because of quantization error introduced \\
110                     & by A/D conversion.                                   \\                     & by A/D conversion.                                   \\
111  \hline  \hline
112  $N_{ADP}$          & The number of A/D counts (supplied to software by an \\  $N_{ADP}$          & The number of A/D counts (supplied to software by an \\
113                     & A/D converter) corresponding to the sensing of the   \\                     & A/D converter) corresponding to the sensing of the   \\
114                     & potentiometer position $\alpha$.                     \\                     & potentiometer position $\alpha$.                     \\
115  \hline  \hline
116  $\overline{N_{ADP}}$            $\overline{N_{ADP}}$          
117                     & $N_{ADP}$ with no quantization error.                \\                     & $N_{ADP}$ with no quantization error.                \\
118  \hline  \hline
119  $N_{ADS}$          & The number of A/D counts (supplied to software by an \\  $N_{ADS}$          & The number of A/D counts (supplied to software by an \\
120                     & A/D converter) corresponding to the sensing of the   \\                     & A/D converter) corresponding to the sensing of the   \\
121                     & supply voltage $V_S$.                                \\                     & supply voltage $V_S$.                                \\
122  \hline  \hline
123  $\overline{N_{ADS}}$            $\overline{N_{ADS}}$          
124                     & $N_{ADS}$ with no quantization error.                \\                     & $N_{ADS}$ with no quantization error.                \\
125  \hline  \hline
126  $R_P$              & The resistance (in Ohms) of the variable potentiometer \\  $R_P$              & The resistance (in Ohms) of the variable potentiometer \\
127                     & whose wiper position $\alpha$ is to be sensed (see   \\                     & whose wiper position $\alpha$ is to be sensed (see   \\
128                     & Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}).  Note that this resistance does not appear \\                     & Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}).  Note that this resistance does not appear \\
129                     & in the analysis of the circuit of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, as only  \\                     & in the analysis of the circuit of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, as only  \\
130                     & the potentiometer wiper position $\alpha$ affects $V_{ADP}$. \\                     & the potentiometer wiper position $\alpha$ affects $V_{ADP}$. \\
131  \hline  \hline
132  $r_{ADP}$          & The A/D converter ratio (from volts to counts)       \\  $r_{ADP}$          & The A/D converter ratio (from volts to counts)       \\
133                     & implemented by A/D converter monitoring the variable \\                     & implemented by A/D converter monitoring the variable \\
134                     & potentiometer input.  Note that $N_P = r_{ADP} V_{ADP}$. \\                     & potentiometer input.  Note that $N_P = r_{ADP} V_{ADP}$. \\
135  \hline  \hline
136  $r_{ADS}$          & The A/D converter ratio (from volts to counts)       \\  $r_{ADS}$          & The A/D converter ratio (from volts to counts)       \\
137                     & implemented by A/D converter monitoring the $V_S$ input. \\                     & implemented by A/D converter monitoring the $V_S$ input. \\
138                     & Note that $N_S = r_{ADS} V_{ADS}$.                   \\                     & Note that $N_S = r_{ADS} V_{ADS}$.                   \\
139  \hline  \hline
140  $V_{ADP}$          & The voltage supplied to the microcontroller or A/D   \\  $V_{ADP}$          & The voltage supplied to the microcontroller or A/D   \\
141                     & converter corresponding to the variable potentiometer\\                     & converter corresponding to the variable potentiometer\\
142                     & position $\alpha$.  In the system of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, $V_{ADP} = \alpha V_S$.\\                     & position $\alpha$.  In the system of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, $V_{ADP} = \alpha V_S$.\\
143  \hline  \hline
144  $V_{ADS}$          & The voltage supplied to the microcontroller or A/D   \\  $V_{ADS}$          & The voltage supplied to the microcontroller or A/D   \\
145                     & converter corresponding to supply voltage $V_S$.  In \\                     & converter corresponding to supply voltage $V_S$.  In \\
146                     & the system of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, $V_{ADS}=V_S$.  \\                     & the system of Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01}, $V_{ADS}=V_S$.  \\
147  \hline  \hline
148  $V_S$              & The supply voltage, presumed variable, which can be  \\  $V_S$              & The supply voltage, presumed variable, which can be  \\
149                     & sensed by the microcontroller software, and also     \\                     & sensed by the microcontroller software, and also     \\
150                     & drives the high side of the variable potentiometer.  \\                     & drives the high side of the variable potentiometer.  \\
151  \hline  \hline
152  \end{tabular}  \end{tabular}
153  \end{center}  \end{center}
154  \caption{Variables Used In Analysis Of Prototype System I  \caption{Variables Used In Analysis Of Prototype System I
155           (Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01})}           (Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01})}
156  \label{tbl:csoc1:srcs0:soeq0:spsa0:01}  \label{tbl:csoc1:srcs0:soeq0:spsa0:01}
157  \end{table}  \end{table}
158    
159  To analyze the system depicted in Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01},  To analyze the system depicted in Figure~\ref{fig:csoc1:srcs0:soeq0:spsa0:01},
160  first note that  first note that
161    
162  \begin{equation}  \begin{equation}
163  \label{eq:csoc1:srcs0:soeq0:spsa0:01}  \label{eq:csoc1:srcs0:soeq0:spsa0:01}
164  V_{ADS}  =  V_S  V_{ADS}  =  V_S
165  \end{equation}  \end{equation}
166    
167  \noindent{}and  \noindent{}and
168    
169  \begin{equation}  \begin{equation}
170  \label{eq:csoc1:srcs0:soeq0:spsa0:02}  \label{eq:csoc1:srcs0:soeq0:spsa0:02}
171  V_{ADP}  =  \alpha V_S.  V_{ADP}  =  \alpha V_S.
172  \end{equation}  \end{equation}
173    
174  For simplicity of analysis, we assume that A/D converters quantize by  For simplicity of analysis, we assume that A/D converters quantize by
175  truncation, so that  truncation, so that
176    
177  \begin{equation}  \begin{equation}
178  \label{eq:csoc1:srcs0:soeq0:spsa0:03}  \label{eq:csoc1:srcs0:soeq0:spsa0:03}
179  N_{ADS}  =  \lfloor V_S r_{ADS} \rfloor  N_{ADS}  =  \lfloor V_S r_{ADS} \rfloor
180  \end{equation}  \end{equation}
181    
182  \noindent{}and  \noindent{}and
183    
184  \begin{equation}  \begin{equation}
185  \label{eq:csoc1:srcs0:soeq0:spsa0:04}  \label{eq:csoc1:srcs0:soeq0:spsa0:04}
186  N_{ADP}  =  \lfloor \alpha V_S r_{ADP} \rfloor.  N_{ADP}  =  \lfloor \alpha V_S r_{ADP} \rfloor.
187  \end{equation}  \end{equation}
188    
189  \noindent{}The assumption that A/D converters quantize by truncation  \noindent{}The assumption that A/D converters quantize by truncation
190  has little effect on the error analysis of practical systems.  (Need  has little effect on the error analysis of practical systems.  (Need
191  to include an exercise to show this.)  to include an exercise to show this.)
192    
193  To aid in symbolic manipulation, we also introduce  To aid in symbolic manipulation, we also introduce
194  $\overline{N_{ADS}}$ and $\overline{N_{ADP}}$, which are  $\overline{N_{ADS}}$ and $\overline{N_{ADP}}$, which are
195  $N_{ADS}$ and $N_{ADP}$, respectively,  $N_{ADS}$ and $N_{ADP}$, respectively,
196  without quantization error:  without quantization error:
197    
198  \begin{equation}  \begin{equation}
199  \label{eq:csoc1:srcs0:soeq0:spsa0:03b}  \label{eq:csoc1:srcs0:soeq0:spsa0:03b}
200  \overline{N_{ADS}}  =  V_S r_{ADS}  \overline{N_{ADS}}  =  V_S r_{ADS}
201  \end{equation}  \end{equation}
202    
203  \begin{equation}  \begin{equation}
204  \label{eq:csoc1:srcs0:soeq0:spsa0:04b}  \label{eq:csoc1:srcs0:soeq0:spsa0:04b}
205  \overline{N_{ADP}}  =  \alpha V_S r_{ADP}  \overline{N_{ADP}}  =  \alpha V_S r_{ADP}
206  \end{equation}  \end{equation}
207    
208  If quantization were not present (i.e. if Eqns.  If quantization were not present (i.e. if Eqns.
209  \ref{eq:csoc1:srcs0:soeq0:spsa0:03}  \ref{eq:csoc1:srcs0:soeq0:spsa0:03}
210  and  and
211  \ref{eq:csoc1:srcs0:soeq0:spsa0:04}  \ref{eq:csoc1:srcs0:soeq0:spsa0:04}
212  did not include floor functions), the potentiometer wiper position  did not include floor functions), the potentiometer wiper position
213  $\alpha$ could be calculated exactly as:  $\alpha$ could be calculated exactly as:
214    
215  \begin{equation}  \begin{equation}
216  \label{eq:csoc1:srcs0:soeq0:spsa0:05}  \label{eq:csoc1:srcs0:soeq0:spsa0:05}
217  \alpha = \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}} r_{ADP}}.  \alpha = \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}} r_{ADP}}.
218  \end{equation}  \end{equation}
219    
220  Based on (\ref{eq:csoc1:srcs0:soeq0:spsa0:05}),  Based on (\ref{eq:csoc1:srcs0:soeq0:spsa0:05}),
221  it makes sense to calculate $\overline{\alpha}$, the  it makes sense to calculate $\overline{\alpha}$, the
222  estimate of $\alpha$, as:  estimate of $\alpha$, as:
223    
224  \begin{equation}  \begin{equation}
225  \label{eq:csoc1:srcs0:soeq0:spsa0:05b}  \label{eq:csoc1:srcs0:soeq0:spsa0:05b}
226  \overline{\alpha}  \overline{\alpha}
227  =  =
228  \frac{N_{ADP} r_{ADS}}{N_{ADS} r_{ADP}}.  \frac{N_{ADP} r_{ADS}}{N_{ADS} r_{ADP}}.
229  \end{equation}  \end{equation}
230    
231  The question that must be posed is:  The question that must be posed is:
232  \emph{How different from $\alpha$ can $\overline{\alpha}$ be?}.  \emph{How different from $\alpha$ can $\overline{\alpha}$ be?}.
233  We seek to bound $|\overline{\alpha}-\alpha|$.  We seek to bound $|\overline{\alpha}-\alpha|$.
234    
235  Quantization can be treated by noting that applying the floor function  Quantization can be treated by noting that applying the floor function
236  to an input introduces an error $\in (-1,0]$, i.e.  to an input introduces an error $\in (-1,0]$, i.e.
237    
238  \begin{equation}  \begin{equation}
239  \label{eq:csoc1:srcs0:soeq0:spsa0:06}  \label{eq:csoc1:srcs0:soeq0:spsa0:06}
240  \lfloor x \rfloor = x + \varepsilon; \;\; \varepsilon \in  (-1,0].  \lfloor x \rfloor = x + \varepsilon; \;\; \varepsilon \in  (-1,0].
241  \end{equation}  \end{equation}
242    
243  Using this observation, we can rewrite  Using this observation, we can rewrite
244  (\ref{eq:csoc1:srcs0:soeq0:spsa0:05b}) as  (\ref{eq:csoc1:srcs0:soeq0:spsa0:05b}) as
245    
246  \begin{equation}  \begin{equation}
247  \label{eq:csoc1:srcs0:soeq0:spsa0:07}  \label{eq:csoc1:srcs0:soeq0:spsa0:07}
248  \overline{\alpha}  \overline{\alpha}
249  =  =
250  \frac{N_{ADP} r_{ADS}}{N_{ADS} r_{ADP}}  \frac{N_{ADP} r_{ADS}}{N_{ADS} r_{ADP}}
251  =  =
252  \frac{(\overline{N_{ADP}} + \varepsilon_1) r_{ADS}}{(\overline{N_{ADS}} + \varepsilon_2) r_{ADP}} ;  \frac{(\overline{N_{ADP}} + \varepsilon_1) r_{ADS}}{(\overline{N_{ADS}} + \varepsilon_2) r_{ADP}} ;
253  \;\;  \;\;
254  \varepsilon_1, \varepsilon_2 \in (-1, 0].  \varepsilon_1, \varepsilon_2 \in (-1, 0].
255  \end{equation}  \end{equation}
256    
257  \noindent{}It can be seen from  \noindent{}It can be seen from
258  (\ref{eq:csoc1:srcs0:soeq0:spsa0:07})  (\ref{eq:csoc1:srcs0:soeq0:spsa0:07})
259  that the minimum value of the estimate  that the minimum value of the estimate
260  $\overline{\alpha}$ occurs when $\varepsilon_1$ is minimized and  $\overline{\alpha}$ occurs when $\varepsilon_1$ is minimized and
261  $\varepsilon_2$ is maximized.  Similarly, the maximum  $\varepsilon_2$ is maximized.  Similarly, the maximum
262  value occurs when $\varepsilon_1$ is maximized and  value occurs when $\varepsilon_1$ is maximized and
263  $\varepsilon_2$ is minimized.  These observations lead to this  $\varepsilon_2$ is minimized.  These observations lead to this
264  inequality:  inequality:
265    
266  \begin{equation}  \begin{equation}
267  \label{eq:csoc1:srcs0:soeq0:spsa0:08}  \label{eq:csoc1:srcs0:soeq0:spsa0:08}
268  \frac{N_{ADP} r_{ADS}}{(N_{ADS}+1) r_{ADP}}  \frac{N_{ADP} r_{ADS}}{(N_{ADS}+1) r_{ADP}}
269  <  <
270  \alpha  \alpha
271  <  <
272  \frac{(N_{ADP}+1) r_{ADS}}{r_{ADP}} .  \frac{(N_{ADP}+1) r_{ADS}}{r_{ADP}} .
273  \end{equation}  \end{equation}
274    
275  Intuitively, the form of (\ref{eq:csoc1:srcs0:soeq0:spsa0:08})  Intuitively, the form of (\ref{eq:csoc1:srcs0:soeq0:spsa0:08})
276  makes sense.  The smallest possible value of $\alpha$ will correspond  makes sense.  The smallest possible value of $\alpha$ will correspond
277  to the case  to the case
278  when $N_{ADP}$ contains no quantization error but $N_{ADS}$ contains  when $N_{ADP}$ contains no quantization error but $N_{ADS}$ contains
279  a quantization error of nearly 1.  a quantization error of nearly 1.
280  The largest possible value of $\alpha$ will correspond  The largest possible value of $\alpha$ will correspond
281  to the case  to the case
282  when $N_{ADS}$ contains no quantization error but $N_{ADP}$ contains  when $N_{ADS}$ contains no quantization error but $N_{ADP}$ contains
283  a quantization error of nearly 1.  a quantization error of nearly 1.
284    
285  It can also be seen from (\ref{eq:csoc1:srcs0:soeq0:spsa0:08}) that  It can also be seen from (\ref{eq:csoc1:srcs0:soeq0:spsa0:08}) that
286  the interval to which $\alpha$ is confined will be larger  the interval to which $\alpha$ is confined will be larger
287  when $V_S$ is smaller (implying a smaller $N_{ADS}$ and $N_{ADP}$).  when $V_S$ is smaller (implying a smaller $N_{ADS}$ and $N_{ADP}$).
288  This is also intuitively plausible, since the quantization error  This is also intuitively plausible, since the quantization error
289  of up to one count in $N_{ADS}$ or $N_{ADP}$ will have a larger  of up to one count in $N_{ADS}$ or $N_{ADP}$ will have a larger
290  relative significance when $N_{ADS}$ and $N_{ADP}$ are smaller.  relative significance when $N_{ADS}$ and $N_{ADP}$ are smaller.
291    
292  The inequality provided in (\ref{eq:csoc1:srcs0:soeq0:spsa0:08})  The inequality provided in (\ref{eq:csoc1:srcs0:soeq0:spsa0:08})
293  is certainly useful, and gives insight into quantization error.  However,  is certainly useful, and gives insight into quantization error.  However,
294  we seek an inequality that is more friendly to engineering work, i.e.  we seek an inequality that is more friendly to engineering work, i.e.
295  one that involves voltages rather than A/D counts.  one that involves voltages rather than A/D counts.
296    
297  Assume it is known for an application that $V_S$ can vary only over the  Assume it is known for an application that $V_S$ can vary only over the
298  interval  interval
299    
300  \begin{equation}  \begin{equation}
301  \label{eq:csoc1:srcs0:soeq0:spsa0:08b}  \label{eq:csoc1:srcs0:soeq0:spsa0:08b}
302  V_S \in [V_{SMIN}, V_{SMAX}],  V_S \in [V_{SMIN}, V_{SMAX}],
303  \end{equation}  \end{equation}
304    
305    
306  \noindent{}and that $\alpha$ can very only over  \noindent{}and that $\alpha$ can very only over
307  the interval  the interval
308    
309  \begin{equation}  \begin{equation}
310  \label{eq:csoc1:srcs0:soeq0:spsa0:08c}  \label{eq:csoc1:srcs0:soeq0:spsa0:08c}
311  \alpha \in [\alpha_{MIN}, \alpha_{MAX}].  \alpha \in [\alpha_{MIN}, \alpha_{MAX}].
312  \end{equation}  \end{equation}
313    
314  \noindent{}Furthermore, we  \noindent{}Furthermore, we
315  seek useful inequalities where is it \emph{not} required  seek useful inequalities where is it \emph{not} required
316  that\footnote{(\ref{eq:csoc1:srcs0:soeq0:spsa0:09}) represents the  that\footnote{(\ref{eq:csoc1:srcs0:soeq0:spsa0:09}) represents the
317  requirement that $V_{SMIN}$ or $V_{SMAX}$ represent a precisely  requirement that $V_{SMIN}$ or $V_{SMAX}$ represent a precisely
318  integral number of A/D  integral number of A/D
319  counts.  This requirement is almost never met in practical engineering  counts.  This requirement is almost never met in practical engineering
320  work.}  work.}
321    
322  \begin{equation}  \begin{equation}
323  \label{eq:csoc1:srcs0:soeq0:spsa0:09}  \label{eq:csoc1:srcs0:soeq0:spsa0:09}
324  r_{ADS} \vworkdivides \{V_{SMIN},V_{SMAX}\},  r_{ADS} \vworkdivides \{V_{SMIN},V_{SMAX}\},
325  \end{equation}  \end{equation}
326    
327  \noindent{}as if (\ref{eq:csoc1:srcs0:soeq0:spsa0:09})  \noindent{}as if (\ref{eq:csoc1:srcs0:soeq0:spsa0:09})
328  were required, it would introduce extra complexity  were required, it would introduce extra complexity
329  in applying the inequality.  in applying the inequality.
330    
331  To develop the desired type of inequality, we can use  To develop the desired type of inequality, we can use
332  a different analysis method.  Assume that  a different analysis method.  Assume that
333  $r_{ADS}$ and $r_{ADP}$ are fixed.  Then, introduce  $r_{ADS}$ and $r_{ADP}$ are fixed.  Then, introduce
334  the function  the function
335    
336  \begin{equation}  \begin{equation}
337  \label{eq:csoc1:srcs0:soeq0:spsa0:10}  \label{eq:csoc1:srcs0:soeq0:spsa0:10}
338  F(\overline{N_{ADP}}, \overline{N_{ADS}})  F(\overline{N_{ADP}}, \overline{N_{ADS}})
339  =  =
340  \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}} r_{ADP}},  \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}} r_{ADP}},
341  \end{equation}  \end{equation}
342    
343  \noindent{}which, in accordance with (\ref{eq:csoc1:srcs0:soeq0:spsa0:05}),  \noindent{}which, in accordance with (\ref{eq:csoc1:srcs0:soeq0:spsa0:05}),
344  is a perfect estimate of $\alpha$.  is a perfect estimate of $\alpha$.
345    
346  \begin{figure}  \begin{figure}
347  \centering  \centering
348  \Huge{TBD}  \Huge{TBD}
349  \caption{Sample Staircase Pattern Of Estimate $\overline{\alpha}$ Of  \caption{Sample Staircase Pattern Of Estimate $\overline{\alpha}$ Of
350           Prototype Ratiometric Conversion System I}           Prototype Ratiometric Conversion System I}
351  \label{fig:csoc1:srcs0:soeq0:spsa0:02}  \label{fig:csoc1:srcs0:soeq0:spsa0:02}
352  \end{figure}  \end{figure}
353    
354  We can examine the staircase pattern of $\overline{\alpha}$ as  We can examine the staircase pattern of $\overline{\alpha}$ as
355  $V_S$ is increased (see Figure \ref{fig:csoc1:srcs0:soeq0:spsa0:02},  $V_S$ is increased (see Figure \ref{fig:csoc1:srcs0:soeq0:spsa0:02},
356  which provides an example staircase pattern).  Note that the staircase  which provides an example staircase pattern).  Note that the staircase
357  pattern may contain four qualitatively  pattern may contain four qualitatively
358  distinct types of vertical discontinuities.  In the descriptions below,  distinct types of vertical discontinuities.  In the descriptions below,
359  assume that $V_D \in [V_{SMIN}, V_{SMAX}]$ is the value of  assume that $V_D \in [V_{SMIN}, V_{SMAX}]$ is the value of
360  $V_S$ at the discontinuity.  $V_S$ at the discontinuity.
361    
362  \begin{itemize}  \begin{itemize}
363  \item \textbf{Downward overshoot discontinuities (DOD):}  \item \textbf{Downward overshoot discontinuities (DOD):}
364        As $V_S$ is increased, these correspond to the values        As $V_S$ is increased, these correspond to the values
365        of $V_S$ where $V_S r_{ADS} \in \vworkintset$ but        of $V_S$ where $V_S r_{ADS} \in \vworkintset$ but
366        $V_S \alpha r_{ADS} \notin \vworkintset$.  At such        $V_S \alpha r_{ADS} \notin \vworkintset$.  At such
367        discontinuities,        discontinuities,
368        $\lim_{V_S \rightarrow V_D^-}\overline{\alpha} > \alpha$,        $\lim_{V_S \rightarrow V_D^-}\overline{\alpha} > \alpha$,
369        but        but
370        $\lim_{V_S \rightarrow V_D^+}\overline{\alpha} < \alpha$.        $\lim_{V_S \rightarrow V_D^+}\overline{\alpha} < \alpha$.
371  \item \textbf{Upward overshoot discontinuities (UOD):}  \item \textbf{Upward overshoot discontinuities (UOD):}
372        As $V_S$ is increased, these correspond to the values        As $V_S$ is increased, these correspond to the values
373        of $V_S$ where $V_S r_{ADS} \notin \vworkintset$ but        of $V_S$ where $V_S r_{ADS} \notin \vworkintset$ but
374        $V_S \alpha r_{ADS} \in \vworkintset$.  At such        $V_S \alpha r_{ADS} \in \vworkintset$.  At such
375        discontinuities,        discontinuities,
376        $\lim_{V_S \rightarrow V_D^-}\overline{\alpha} < \alpha$,        $\lim_{V_S \rightarrow V_D^-}\overline{\alpha} < \alpha$,
377        but        but
378        $\lim_{V_S \rightarrow V_D^+}\overline{\alpha} > \alpha$.        $\lim_{V_S \rightarrow V_D^+}\overline{\alpha} > \alpha$.
379  \item \textbf{Downward exact discontinuities (DED):}  \item \textbf{Downward exact discontinuities (DED):}
380        As $V_S$ is increased, these correspond to the values        As $V_S$ is increased, these correspond to the values
381        of $V_S$ where $V_S r_{ADS} \in \vworkintset$ and        of $V_S$ where $V_S r_{ADS} \in \vworkintset$ and
382        $V_S \alpha r_{ADS} \in \vworkintset$.  At such        $V_S \alpha r_{ADS} \in \vworkintset$.  At such
383        discontinuities,        discontinuities,
384        $\lim_{V_S \rightarrow V_D^-}\overline{\alpha} > \alpha$,        $\lim_{V_S \rightarrow V_D^-}\overline{\alpha} > \alpha$,
385        but        but
386        $\lim_{V_S \rightarrow V_D^+}\overline{\alpha} = \alpha$.        $\lim_{V_S \rightarrow V_D^+}\overline{\alpha} = \alpha$.
387  \item \textbf{Upward exact discontinuities (UED):}  \item \textbf{Upward exact discontinuities (UED):}
388        As $V_S$ is increased, these correspond to the values        As $V_S$ is increased, these correspond to the values
389        of $V_S$ where $V_S r_{ADS} \in \vworkintset$ and        of $V_S$ where $V_S r_{ADS} \in \vworkintset$ and
390        $V_S \alpha r_{ADS} \in \vworkintset$.  At such        $V_S \alpha r_{ADS} \in \vworkintset$.  At such
391        discontinuities,        discontinuities,
392        $\lim_{V_S \rightarrow V_D^-}\overline{\alpha} < \alpha$,        $\lim_{V_S \rightarrow V_D^-}\overline{\alpha} < \alpha$,
393        but        but
394        $\lim_{V_S \rightarrow V_D^+}\overline{\alpha} = \alpha$.        $\lim_{V_S \rightarrow V_D^+}\overline{\alpha} = \alpha$.
395  \end{itemize}  \end{itemize}
396    
397  We can place upper bounds on the magnitudes of the discontinuities  We can place upper bounds on the magnitudes of the discontinuities
398  by examining the partial derivatives of  by examining the partial derivatives of
399  $F(\overline{N_{ADP}}, \overline{N_{ADS}})$ as specified in  $F(\overline{N_{ADP}}, \overline{N_{ADS}})$ as specified in
400  (\ref{eq:csoc1:srcs0:soeq0:spsa0:10}), specifically:  (\ref{eq:csoc1:srcs0:soeq0:spsa0:10}), specifically:
401    
402  \begin{equation}  \begin{equation}
403  \label{eq:csoc1:srcs0:soeq0:spsa0:11}  \label{eq:csoc1:srcs0:soeq0:spsa0:11}
404  \frac{\partial{}F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}}  \frac{\partial{}F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}}
405  =  =
406  -  -
407  \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^2} r_{ADP}}  \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^2} r_{ADP}}
408  \end{equation}  \end{equation}
409    
410  \begin{equation}  \begin{equation}
411  \label{eq:csoc1:srcs0:soeq0:spsa0:11b}  \label{eq:csoc1:srcs0:soeq0:spsa0:11b}
412  \frac{\partial{}^2 F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}^2}  \frac{\partial{}^2 F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}^2}
413  =  =
414  2  2
415  \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^3} r_{ADP}}  \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^3} r_{ADP}}
416  \end{equation}  \end{equation}
417    
418  \begin{equation}  \begin{equation}
419  \label{eq:csoc1:srcs0:soeq0:spsa0:11c}  \label{eq:csoc1:srcs0:soeq0:spsa0:11c}
420  \frac{\partial{}^i F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}^i}  \frac{\partial{}^i F(\cdot{},\cdot{})}{\partial \overline{N_{ADS}}^i}
421  =  =
422  (-1)^i (i!)  (-1)^i (i!)
423  \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^{i+1}} r_{ADP}}  \frac{\overline{N_{ADP}} r_{ADS}}{\overline{N_{ADS}^{i+1}} r_{ADP}}
424  \end{equation}  \end{equation}
425    
426    
427  \begin{equation}  \begin{equation}
428  \label{eq:csoc1:srcs0:soeq0:spsa0:12}  \label{eq:csoc1:srcs0:soeq0:spsa0:12}
429  \frac{\partial{}F(\cdot{},\cdot{})}{\partial \overline{N_{ADP}}}  \frac{\partial{}F(\cdot{},\cdot{})}{\partial \overline{N_{ADP}}}
430  =  =
431  -  -
432  \frac{r_{ADS}}{\overline{N_{ADS}} r_{ADP}}  \frac{r_{ADS}}{\overline{N_{ADS}} r_{ADP}}
433  \end{equation}  \end{equation}
434    
435  \begin{equation}  \begin{equation}
436  \label{eq:csoc1:srcs0:soeq0:spsa0:12b}  \label{eq:csoc1:srcs0:soeq0:spsa0:12b}
437  \frac{\partial{}^i F(\cdot{},\cdot{})}{\partial \overline{N_{ADP}}^i}  \frac{\partial{}^i F(\cdot{},\cdot{})}{\partial \overline{N_{ADP}}^i}
438  =  =
439  0, \; i \geq 2  0, \; i \geq 2
440  \end{equation}  \end{equation}
441    
442  A \emph{DOD} (discussed above) corresponds to the case where  A \emph{DOD} (discussed above) corresponds to the case where
443  $N_{ADS}$ increases by one count at $V_S=V_D$ without an increase in  $N_{ADS}$ increases by one count at $V_S=V_D$ without an increase in
444  $N_{ADP}$.      $N_{ADP}$.    
445    
446    
447  A \emph{UOD} (discussed above) corresponds to the case where  A \emph{UOD} (discussed above) corresponds to the case where
448  $N_{ADP}$ increases by one count at $V_S=V_D$ without an increase in  $N_{ADP}$ increases by one count at $V_S=V_D$ without an increase in
449  $N_{ADS}$.      $N_{ADS}$.    
450    
451    
452    
453    
454  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
455  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
456  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
457  \section{Motion Control Systems}  \section{Motion Control Systems}
458    
459    
460  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
461  \vfill  \vfill
462  \noindent\begin{figure}[!b]  \noindent\begin{figure}[!b]
463  \noindent\rule[-0.25in]{\textwidth}{1pt}  \noindent\rule[-0.25in]{\textwidth}{1pt}
464  \begin{tiny}  \begin{tiny}
465  \begin{verbatim}  \begin{verbatim}
466  $RCSfile: c_soc1.tex,v $  $RCSfile: c_soc1.tex,v $
467  $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_soc1/c_soc1.tex,v $  $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_soc1/c_soc1.tex,v $
468  $Revision: 1.7 $  $Revision: 1.7 $
469  $Author: dtashley $  $Author: dtashley $
470  $Date: 2002/04/21 22:50:04 $  $Date: 2002/04/21 22:50:04 $
471  \end{verbatim}  \end{verbatim}
472  \end{tiny}  \end{tiny}
473  \noindent\rule[0.25in]{\textwidth}{1pt}  \noindent\rule[0.25in]{\textwidth}{1pt}
474  \end{figure}  \end{figure}
475    
476  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
477  % $Log: c_soc1.tex,v $  % $Log: c_soc1.tex,v $
478  % Revision 1.7  2002/04/21 22:50:04  dtashley  % Revision 1.7  2002/04/21 22:50:04  dtashley
479  % Safety checkin before working on laptop.  % Safety checkin before working on laptop.
480  %  %
481  % Revision 1.6  2002/04/13 08:18:43  dtashley  % Revision 1.6  2002/04/13 08:18:43  dtashley
482  % Edits from laptop.  % Edits from laptop.
483  %  %
484  % Revision 1.5  2002/04/13 06:04:32  dtashley  % Revision 1.5  2002/04/13 06:04:32  dtashley
485  % Safety checkin before resuming work on laptop.  % Safety checkin before resuming work on laptop.
486  %  %
487  % Revision 1.4  2002/04/12 23:41:50  dtashley  % Revision 1.4  2002/04/12 23:41:50  dtashley
488  % Safety checkin before major editorial additions.  % Safety checkin before major editorial additions.
489  %  %
490  % Revision 1.3  2002/04/10 06:12:33  dtashley  % Revision 1.3  2002/04/10 06:12:33  dtashley
491  % Evening safety checkin.  % Evening safety checkin.
492  %  %
493  % Revision 1.2  2002/04/09 23:32:20  dtashley  % Revision 1.2  2002/04/09 23:32:20  dtashley
494  % Edits, preparing for addition of lists of tables and figures.  % Edits, preparing for addition of lists of tables and figures.
495  %  %
496  % Revision 1.1  2001/08/25 22:51:26  dtashley  % Revision 1.1  2001/08/25 22:51:26  dtashley
497  % Complex re-organization of book.  % Complex re-organization of book.
498  %  %
499  %End of file C_PCO0.TEX  %End of file C_PCO0.TEX

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