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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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