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

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