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1 dashley 140 %$Header$
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 dashley 279 $HeadURL$
467     $Revision$
468     $Date$
469     $Author$
470 dashley 140 \end{verbatim}
471     \end{tiny}
472     \noindent\rule[0.25in]{\textwidth}{1pt}
473     \end{figure}
474 dashley 279 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
475 dashley 140 %
476     %End of file C_PCO0.TEX

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