Data acquisition for real-time process control systems

Copyright 0 IFAC Real Time Programming, Lake Constance, Germany, 1994

DATA ACQUISITION SYSTEMS Marten

FOR REAL-TIME

PROCESS CONTROL

D. van der LAAN

of Groningen, Dept. ofComputing Science, P.O. Boz 800, 9700 AV Groningen, The Netherlands. e-mail: marten&r.rug.nl

University

A general scheme for rignal sampling is presented, which is based on requirements of real-time Abstract. systemr. Instead of the ideal low-pass filter in Shannon’s sampling theorem, alternative filters are derived from wavelet theory. To evaluate the filters, digital simulations have been carried out. The results of these simulations are briefly discussed in this paper. Finally, some remarks are made on the implementation of the alternative sampling methods and their usage for process control systems.

Key Words. ximation.

Analogue-to-digital

conversion; data acquisition;

are always defined on a finite time-interval, we restrict our attention to signals defined on the interval [0, T]. Spaces to consider are L’[O, T] I, the space of squareintegrable signals, or CI,[O,T], the space of continuous and bounded signals.

1. INTRODUCTION

In real-time computing systems employed for process control the communication with the controlled processes is mainly taking place via analogue signal lines. In this situation, the quality of data acquired by A/D converters and the generation of analogue control signals by D/A converters is of major importance for the overall performance of the processes.

Sampling can be general&d to an operation extracting coefficients from the signal, in such a way that the information contents are preserved. To ensure this, we demand the existence of the reverse operation which (approximately) recovers the signal from its samples. In the sequel, this reverse operation will be denoted as reconrtruction. Signals defined on [O,Tj should be represented by a finite number, say N, of coefficients (the samples).

Standard A/D converters are based on Shannon’s sampling theorem (Shannon, 1949; Jerri, 1977), which states that if a bandlimited signal, i.e., a signal for which the frequency spectrum has a maximum fn+ quency fN, is sampled during iniinite time, at a fre+ quency 2 2fN using an ideal sampler, i.e., a device that can take true point measurements, then it is possible to reconstruct the signal from its samples without error. The reconstruction can be obtained by filtering with an ideal low-pass filter.

A signal can be reconstructed by selecting N linearly independent basis elements wn out of the signal space, and using the coefficients as weights. The reconstruction 4 is then defined as: N-l

However, in practical situations these conditions cannot be met: signals are measured during a finite tim&nterval and can, therefore, never be bandlimited. FWhermore, a physical A/D converter is not able to take point measurements but will always le tally integrate the signal. Finally, the ideal low-pass filter cannot be real&d in a practical system.

i(t) =

PROBLEM

c

Cn

wn, cn E

R

(1)

nro The N basis elements {wn} span an N-dimensional, linear subspace of the signal space. The cascade of sampling and reconstruction, considered as single operator, can be regarded as a projection on this subspace. In order to obtain a reconstruction, the following steps must be taken:

Moreover, practical signals are corrupted by noise and, typically in an industrial environment, with spikes and distortions from high-power currents, which cannot be handled by standard A/D converters. Therefore we aim for a device that covers the needs in a practical situation.

2. THE SAMPLING

digital signal processing; wavelets; appro-

0 Select a subspace to project on, l Choose a basis in it, l Select an operator for calculating the coefficients. The choice of a proper subspace depends heavily on the nature of the original signal. Therefore, it is not possible to choose a subspace which provides equal approximation quality for every signal in the space. Once a subspace ie chosen, approximation theory pro-

REVISITED

To gene&se the sampling problem, we start with a geometrical perspective. Let the signal s(t) be a vector in an infinitely dimensional vector space, the space of all possible signals. Since practical signals

’ The space L’[O, T] is exactly ddined gration theory.

109

within Lebesgue in&

vides methods to find an optimal set of coefficients. An optimal sampling operator minim&s the ‘distance’ between the two vectors rft) and 4(t). In a normed vector space the distance equaIs the norm of the error signal. If we consider the space L’[O,q, the natural norm is the mean square norm or L2-norm. However, in fault-tolerant systems it is better to consider the space C&, Tl and to use the s~p~murn norm or Loo-norm:

Using this norm it is poss~%leto predict the worst-case performance of the sampling operator. With this general framework, many sampling operators can be deduced. For instance, ‘sampli& by ta king the first N Fourier coefficients fits into it. However, to consider a general operator as a sampling operator, additional requirements must be fulfilled: 0 Linearity An operator should be linear (in the mathematical sense). Only in this case linear operations on an analogue signal can be approximated by linear operations on the sampled signal. l Time-invariance As it is not economical to have a different pm cedure for each of the N 8amp!es, it is recommended to implement the sampling with a single procedure, initiated at IV different times. o Equidistant evaluation In order to prevent additional timing data to be stored with the samples, the procedure should be repeated at equidistant times. It can be shown (van der Laan, 1992) that operators ful@ing the above requirements can be modelled by a linear filter followed by an ideal sampler. Analogize sig&s can be reconstructed by b-modulation, i.e., mu& tiplying with a train of Dirac &distributions, followed by interpolation with a second filter. The combination pre&ter~ sampler, and post-filter is referred to as the generai sampling scheme, see Figure f. Samp les are taken at multiples of the sample time At, in other words P[lc] = P(kAt).

Fig. 1. General sampling scheme, The snafogue signd r(t) is ii&red and sampled at t = kAt to obtain the sequence @k]. The reconstructionf(t) is obtaiaed by ~-modulation and filtering.

Shannon’s theorem fits perfectly in this scheme: both filters are ideal low-pass filters. The pr&ilter is commonly known as anti-alias filter and makes the signal 5(t) vomited. It ~~titut~ a projection on the space of bandlimited ~~~ytic) functions. Note that if the signal was &eady bandlimited, the projection operator act as identity, giving a perfect reconstruction.

However, in the above framework, Shannon’s samp ling is just one out of many L’-optimal schemes, We can select filters that are easier to implement and/or have other nice properties, e.g., reduced sensitivity to disturbances present in the signals.

3. MULTIRESOLUTI~N ANALYSIS WAVELETS

AND

Recent research topics in the field of signal processing are wavefet theory and m~t~lution analysis. Wavelet theory involves representing general fnnctions in terms of simpler, fixed building blocks (‘wavelets’) at different scales and positions. Multiresolution analysis (MRA) is a method for &‘-approximation of functions with arbitrary precision, MRA gives approximates on different scales in such a way, that an approximate on a fine scale can be obtained by adding the ‘details’ to an approximate on a coarse scale. A function (1(t) may be called a wavelet if it is 15calion the time-rutis, i.e., the support of the &m&on is concentrated on a certain part of the axis, and if its frequency spectrum has a bandpass character. l+om this wavelet we can extract a whole family of equally shaped functions by shifts in time (tr~mn) and scaling (dilation): d

For waveIets in a NRA, additional conditions hold. A MRA is a co&&ion of closed subspaces 5 of L’(R), such that (Chui, 1992): (o)c...cv-icvoc%

c..-cP(Rf

j(t) E vo * r(t - k) E 6,

Vk E z

f(t)EQ*f(2t)EQ+lt

jEZ

(4)

34 E Vo such that

Vi is spanned by integer translations of the same function 4, and the basis functions on other scales are obtained by so-called dyadic dilations of 4. AJl basis f&uctions are thus derived from 4 by:

(b is called the @her waueletor sc&ng function. fn addition, the wavelet spaces Wi are introduced which are the orthogonal complements of Vi in V&l, and contain the ‘details’ mentioned before. Every Wj is spanned by integer translations of a single function, too: the (mother) wavelets $, which can be derived from the father wavelet #. A function E L2(R) can be approximated with a certain precision in one of the Vj by orthogonal pmjection. The approximation can be taken to a higher level by combining it with approximates in W;, 12 j. Schematically:

...

+

v-1

+

/” w-1

KJ

wo

-+

Vl

2

+

splines, other combinations of B-splines are not (bi-)orthogonal.

**-

/” WI

5. SIMULATION

We have carried out simulations of the general samp ling scheme deerribed above. The A/D conversion is simulated by filtering a digital signal and sub sampling at a lower rate. The D/A conversion is simulated by upsampling and filtering, see Figure 2. The question arises whether a digitalsimulation resemblea the original problem, where the interface between the analogue and the digital world is the principal matter. Fortunately, there is a strong relation between the sampling scheme and the sub-sampling scheme. In both cases the ‘samples’ represent a projection on a subspace of the original signal space - only this signal space is dilferent. We therefore assume the results of the simulations to be representative for the analogue case.

Only recently it was shown that the sampling theorem is an incomplete formulation of a MRA (Unser and Aldroubi, 1992). The impulse response of the ideal low-pass filter, which is a sine-function, is the father wavelet 4. The corresponding wavelet JI is the ideal bandpass filter. On the other hand, the MRA framework meets our requirements for sampling operators, which implies that every d in a MRA can be used in our general sampling scheme. The efforts spent on these topics resulted in the development of wavelet-Alters with both good time and frequency localisation. Daubechies (Daubechies, 1988) revealed the existence of orthonormol waveleta with compact support by a constructive proof. For the corresponding MRA this implies, that we can easily ap proximate a signal at arbitrary precision with compactly supported functions.

4. ALTERNATIVE

SAMPLING

RESULTS

“PI -

Post

i[nk]

Fig. 2. General sub-sampling scheme. The sequence z(k) is filtered and down-nampled to obtain the sequence Z[nk]. The reconstruction 4[k] is obtained by upsampling and filtering.

OPERATORS

Since all MRA-filters can be used in the general samp ling scheme, we obtain a large class of alternative lilters. tirn the practical point of view, filters should also have (some of) the following properties:

Knowing that the general sampling scheme yiekls a projection on a subspace of the signal space, we would expect that the performance of a certain scheme totally depends on the subspace that ia implicitly chosen. If the original signal is ckxre to this a&space, the approximation error will be small. It is therefore more interesting to investigate the approximation quality of a certain scheme for a larger set of signals which are in different parts of the signal space. Moreover, by checking all schemes for physical signals, we can measure the ‘location’ of the signal in the signal space.

0 Compact support It is desirable, but not necessary, for a samp ling operator to have compact support. Hence, the inlluence of erroneous samples remains local. If the support cannot be compact, the support must damp out rapidly. 0 Causality For real-time applications, filters can only be implemented if the impulse response is causal, i.e., future values are not to be used. If the time conditions are weak, also semi-causal filters are allowed. These filters use only a Ilnite part of the future input signal, which can be realised by delaying the input. l Orthogonality In the case that the integer translations are mutually orthogonal, the pre- and post-filter can be identical. If this is not the case, either an ortho gonal filter can be derived, or a joint filter can be derived in such a way that the two flltenr are bi-orthogonal (Unser and Aldroubi, 1992). Such a joint filter does not have compact support. l Suppression of noise and other distortions An operator should possibly be insensitive to other distortions commonly present in practical systems, such as noise or spikes. It was noted before that a locally supported filter can amure this.

Our simulations were carried out in MATLAB, originally a package for matrix algebra, but now also widely used for signal processing applications. In addition an environment was developed to implement and test the subsampling scheme. The scheme was tested for different filters: l l l l

FIR truncation of the ideal low-pass filter, Daubechies’ wavelets, bi-orthogonal splines, B-splines.

All filters within the MRA framework are L’-optimal. For the combinations of B-splin<ers, this is not the case, but they are included because of our interest in their Lw-performance. The bi-orthogonal spline filters were derived from B-splines using the method described in (Unaer and Aldroubi, 1992).

Daubechies’ orthogonal wavelets meet the first three properties and are, therefore, good candidates. Other promising alternatives are B-spline functions, which have already been used in a sampling scheme (Halang, 1984; Hummel, 1988). However, except for the Be-

The schemes were tested with synthetic test signals and physical signals. For every filter class a synthetic signal was generated in such a way that it was close to the corresponding subspace. Moreover, these si-

111

gnals were combined, attempting to generate a signal far from all subspaces. The synthetic signal5 were oversampled by a factor 32. For the physical data, we used the output of a g~~rnato~~h and data from a physiological experiment. Both signals were already digitised, but can still be re-sampled to determine which of the subspaces fits best. The signals were down-sampled by factor5 2,4 and 8.

ception, all filter combination5 showing a good L2 performance were highly sensitive to disturbances, and the reverse was observed as well. It was therefore not possible to favour any of the ~ter-~mbinatio~ over others.

To eliminate the boundary effects in the result, the signal5 were extended at the left and right side by repeating the llrst and last value, respectively. For every combination of filters and signals, the experiment was repeated 10 times and the L’, L2, and L” norms of the error signal z[k] - g[k] were saved. Afterward5 the norms were normalised, i.e., divided by 11z][and averaged for the 10 experiments.

In order to compete with traditional techniques, alternative A/D and D/A converter5 based on the general sampling scheme must be implemented as low-cost mtegrated circuits. Only for spline filters of low order, this can be done diredly in an analogue IC (Wang et al., 1989). In all other cases digital technique5 have to be used, but then the filters need to be surrounded by traditional converter5 operating at high rates. Both spline and wavelet filter5 can be easily implemented as digital FIR filters or, alternatively, by the so-called Pyramid algorithm (Mallat, 1889), which is recursive. For both method5 VLSI designs are available (Parhi and Nishitani, 1993).

6. IMPLEMENTATION

The values for the L2 and L” case can be found in Table 1 and Table 2, respectively, Horizontally we see the different synthetic test signals, vertically the filter combinations. Combination5 in boldface are L2-optimal. For each signal the filter combination giving the smallest error is typeset in boldface and the second-best is underlined.

The digital input for the pre-filter is obtained by sampling the analogue signal using a standard A/D converter. The sampling frequency should be sufliciently high to obtain the required precision. After filtering, the signal is suhsampled, giving the digital output of the whole device. The high frequency is only used intemally, it does not aSect the overall system performance. The same hold5 for D/A conversion.

From Table 1 the following can be observed: Filter combination5 perform best for the signal they correspond to, e.g., the combination spO-sp0 performs best for the splineO-signal, vv2-vv2 for the vavelet2-signal, etc. In general, the approximation errors of the biorthogonal spline and wavelet filters are comparable to those of the sine filters. In general, (bi-)orthogonal combination5 performed significantly better than non-orthogonal combinations. There is one exception: the nonorthogonal combination spO-snc performs remarkably good. For the combined signal (nixed) the smooth orthogonal combination5 ant, 1112, ~20, cs2, ca3 had almost equal performance, but again the combination spO-snc does well.

7. DISCUSSION We can conclude that the results achieved in MRA and wavelet theory have led to a better insight into sampling theory. Wavelet and spline filters m be used instead of the ideal low-pa55 filter. Simulations showed results comparable with the standard filters. Especially the B-spline filter5 are easy to implement, yielding robust device5 for A/D and D/A conversion. The experiment5 to test the sensitivity to disturbances seem to have been failed. Giving this problem a second thought, this is not very surprising. If the noise is wide-band noise, it will always afhzct the digital data, and consequently increase the approximation error of the general sampling scheme. But this is not the question to be asked: in fault-tolerant systems it is more important that the effect of a disturbance remain5 local, which is not studied in the simulation5 so far. This certainly is a topic for further research.

For the LoD-case (table 2) the results are similar, only the ranking order of performance is slightly different than in the L2-case. Moreover, all differences are smaller. Table 3 give5 the results for the gas-chromatographic data. LFYorn this table we see that: The ~12, ~20% cs2, ~53, and snc filter5 show comparable approximation quality. 5 Results for the different norms are highly correlated. l The ranking order change5 slightly for the different down-sampling factors. The difference5 are most significant at dsf = 2. l

Incorporating the sampling scheme in a process control system require5 extra care. So far, the issue of sampling quality was not addressed in control systems theory, because the extra delay caused by longer filters had opposite effect5 on the overall etabiity. In practical systems, a zero-order hold is used as D/Aconverter, which is equivalent to a Bs-spline postfilter. Taking this for true, including a Bo-spline prefilter would give an L2-optimal scheme, and might improve the performance of the controi system. Other (short) filters like the B-splines could further increase the quality of the sampled data. The exact effect of

The results of the physiological data were similar, and therefore not included in this paper. In addition, data were corrupted by white noise and spikes, in order to test the sensitivity of the different filter combination5 to these disturbances. The results, however, disagreed with our expectations: without ex-

112

Filters snc-snc sllc-spa snc-spl snc-sp2 snc-sp3 spO-snc

spo-spo spa-SPl spasp2 spas# spl-spl sp2-sp2 sP3-sp3 csl-csl cs%csZ c53-es3 wv2-WV2 wv6-WV6 WC&W12 w20-w20 Table 1

sine

bspline0

0.071

0.484 0.192 0.472 0.529 0.583 0.510 0.038 0.422 0.479 0.545 0.534 0.624 0.686 0.391 0.377 0.410 0.529 0.421 0.558 0.544

0.466 0.329 0.379 0.440 0.177 0.464 0.392 0.444 0.495 0.450 0.536 0.602 0.279 0.218 0.187 0.3855 0.253 0.205 m

bsphel 0.217 0.330 0.225 0.312 0.370 0.221 0.322 0.246 0.331 0.388 0.323 0.436 0.501 0.043 o.100 0.118 0.281 0.169 0.297 0.281

0,147 0.337 0.213 0.274 0.330 0.162 0.315 0.246 0.307 0.360 0.314 0.407 0.473 0.128 0.069 0.087 0.266 0.144 0.215 0.203

0.047 0.257 0.118 0.144 0.176 0.072 0.245 0.153 0.186 0.210 0.163 0.236 0.264 0.091 0.054 0.052 0.140 0.072 0.055 0.051

I wavelet 0.385 0.606 0.478 0.515 0.551 0.455 0.559 0.499 0.549 0.587 0.537 0.615 0.665 0.416 0.413 0.400 0.100 0.279 0.448 0.419

wavelet 0.246 0.693 0.446 0.458 0.494 0.347 0.534 0.485 0.514 0.546 0.516 0.574 0.624 0.421 0.372 0.352 0.444 0.409 0.086 0.108

mixed 0.167 0.526 0.367 0.402 0.452 0.252 0.471 0.416 0.459 0.593 0.464 0.540 0.599 0.333 0.276 0.259 0.380 0.311 0.183 O.L67

SimulaGon results for s~t~ti~ signals. AU values refer to the relative L2-nom of the error signal aud have arbitrary units. Filter combinations are coded as follows: Pre-filter - Post-fdter, where snc: sine, sp: Espline, cs: b&orthogonal spline, and WVor u: Daubechies’ wavelet. Signal names are straightforward, except for the signal mixed which is a combination of sine, bapline2, and uavelet 12.

Signal Type

Filters tic

snc-snc snc*spO snc-spl snc-sp2 8nc-sp3 SpO-snc

spa-spo @+Pl spo-sP2 sPO-sp3 spl-spl sp2-sp2 sp3-sp3 Cal-csl cs2-cs2 ci+cs3 wv!&wv2 wv&wv6 w12-w12 w20-w20 Table 2

Signal Type bspline2 bsplhe6

0.271 0.813 0.433 0.447 0.496 m 0.743 0.481 0.500 0.545 0.526 0.590 0.652 0.369 0.304 0.294 0.519 0.346 0.336 0.322

bspline0 0.838 0.339 0.748 0.748 0.762 0.916 0.169 0.752 0.752 0.760 0.774 0.789 0.811 0.747 0.759 0.777 1.054 0.912 0.830 0.814

bsplinel 0.316 0.493 0.319 0.466 0.518 0.334 0.465 0.337 0.483 0.534 0.412 0.577 0.637 0.189 0.239 0.249 0.396 0.278 0,400 0.389

bspline2 0.343 0.607 0.346 0.380 0.414 0.353 0.552 0.362 0.402 0.436 0.389 0.470 0.533 0.407 0.326 0.355 0.497 0.364 0.346 0.350

bspline6 0.253 0.479 0.263 0.273 0.290 0.267 0,441 0.288 0.295 0.314 0.291 0.327 0.364 0.311 0.244 0.269 0.414 0.299 0.237 0.244

wavelet2 0.496 0.660 0.531 0.626 0.664 0.473 0.692 0.499 0.618 0.665 0.574 0.703 0.751 0.354 0.496 0.412 0.232 0.313 0.619 0.587

wavelet 0.413 0.814 0.501 0.509 0.539 0.410 0.734 0.545 0.553 0.576 0.571 0.663 0.641 0.536 0.403 0.435 0.516 0.440 0.422 0.412

mixed 0.440 0.761 0.479 0.499 0.524 0.436 0.668 0.508 0.528 0.551 0.540 0.579 0.619 0.551 0.415 0.460 0.485 0.412 0.456 0.449

Sim~tion results for synthetic signals. A.Il values refer to the relative Loo-nom of the error signal and have arbitrary units. For the code of the filter comb&&ions see Table 1.

113

Filters snc-snc snc-sp0 snc-spl snc-sp2 snc-sp3 sp@snc spa-spa +-SPl spkp2 spO-sp3 spl-spl sp2-sp2 sp3sp3 csl-csl cs2-cs2 cs3-cs3 wv2-WV2 wv6wv6 w12-w12 w20-w20 Table 3

Ls _dsf=2 0.0461 0.2450 0.0889 0.1287 0.1573 0.1763 0.1760 0.1883 0.2079 0.2243 0.1272 0.2057 0.2451 0.0683 0 0373 0.0354 O.OQOQ 0.0916 0.0764 0.0747

dsf=4 0.2209 0.3816 0.2556 0.3139 0.3580 0.2532 0.3606 0.3045 0.3563 0.3928 0.3197 0.4100 0.4558 0.1876 0.1706 0.1715 0.3147 0.3411 0.1857 0.2857

dsf=8 0.5289 0.5281 0.5178 0.5418 0.5506 0.5551 0.5043 0.5014 0.5335 0.5454 0.5151 0.5589 0.5783 0.4909 0.5231 0.5306 0 4937 0.5189 0.5240 0.5321

%f=2 0.0461 0.3489 0.1193 0.1675 0.2075 0.1971 0.1981 0.1981 0.2791 0.3061 0.1466 0.2603 0.3062 0.1008 0.0578 i0 0506 0.1196 0.1256 0.0745 0.0807

dsf=l 0.2448 0.5821 0.2747 0.3678 0.4081 0.3188 0.5084 0.3247 0.4114 0.4416 0.3332 0.4481 0.4895 0.2665 0.2360 0.2225 0.4123 0.3911 0.2187 0.3031

d&8 0.4878 0.4910 0.4910 0.5444 0.5621 0.5280 0.4830 0.4830 0.5401 0.5592 0.5051 0.5854 0.6131 0.4837 0.4909 0.4841 0.5043 0.4557 0.4577 0.5172

Simulation results for gas-chromatographic data. The table has both Lz and L” data in arbitrary units. Results are given for different down-sampling factors (dsf). For the code of the filter combinations see table 1.

the filter length on the balance between approximation quality and closed-loop stability needs to be studied in more detail.

Shannon, C. (1949). ‘Communication in the presence of noise’. Proc. of the I.R.E. pp. 10-21. Unser, M. and A. Aldroubi (1992). Polynomial splines and wavelets - a signal processing perspective. In C. Chui (Ed.). ‘Wavelets - A tutorial in Theory and Applications’. Vol. 2 of Wouefet Analysis ond ita Applications. Academic Press. Boston, MA. pp. Ql122. van der Laan, M. (1992). Towards alternative strategies for signal-sampling. In ‘Proc. of the 13th. Symposium on Information Theory in the Benelux’. pp. 81-88. Wang, Q., K. Toraichi, M. Kamada and R. Mori (1989). ‘Circuit design of a D/A converter using spline functions’. Signal Processing (16) 274-288.

ACKNOWLEDGMENTS The author would like to thank Z. Korendo for developing the simulation environment and carrying out major part of the simulations. Moreover, thanks to E. Barakova who completed the simulations.

8. REFERENCES Chui, C. K. (1992). An introduction to Wavelets. Vol. 1 of Wouefet Analysis and its Applications. Academic Press. Boston, MA. Daubechies, I. (1988). ‘Orthonormal bases of compactly supported wavelets’. Comm. Pure Appl. Moth. 41, QOQ-996. Halang, W. (1984). ‘Acquisition and representation of empirical time functions on the basis of directly measured local integrals’. Interfaces in Computing 2, 345-364. Hummel, R. (1983). ‘Sampling for sphne reconstruction’. SIAM Journal of Applied Mathematics 43(2), 278-288. Jerri, A. (1977). ‘The Shannon sampling theorem its various extensions and applications: A tutorial review’. Proc. of the IEEE 65( ll), 1565-1598. Mallat, S. (1889). ‘A theory for multiresolution signal decomposition: the wavelet representation’. IEEE Tmns. Pott. Anol. Moth. Intell. (ll), 674-693. Parhi, K. and T. Nishitani (1993). ‘VLSI architectures for discrete wavelet transforms’. IEEE VLSI l(2), 191-202. 114