6. DSP Pendulum Feedback System

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A high-resolution wideband digital feedback system for seismometers

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1988 J. Phys. E: Sci. Instrum. 21 748

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R J Adrian et a1 J . Phys . E: Sci. Instrum. 21 (1988) 748-752. Printed in the UKErd ma nn J C 1979 Applicat ion of recurrence ratetechniques to turbulence analysisPhys. Scr. 19 39&401E r d m a n n J C and Gel lert R P 1978 Recur rence ratecorrelation in scattered light intensityJ . Opt. Soc. A n i . 68 787-95Papoulis A 1984 Probability, Ran dom Variables andStochastic Processes 2nd edn (New York: McGraw-Hil l )p 345Rice S 0 1944 Mathe metical analysis of random noiseBell S y s f . Tech. J . 23 282-332Smar t A E and Mayo W T Jr 1979 Applications of laseranem om etry to high Reynolds num ber f lowsPhys . Scr. 19 426-40TSI Inc. 1987 Model 75 Correlex Signal Processor (S t Paul .M N : TS I Inc . )Vehrenkam p R S . Schatze l K , Pfister G and Schulz-DuBoisE 0 1979 Direct measurement of velocity correlationfunctions using the Erdmann-G ellert rate correlationmethodJ . Phys. E : Sci. Instrum. 12 119-25

A high-resolution widebanddigital feedback system forseismometersZ Yint an d M J UsherD e p a r t m e n t of Cybernetics. University of Reading ,3 Ear ley Gat e , Whi teknight s . Reading RG 6 2AL. UKReceived 21 De cem ber 19 87, in f inal form 10 March 1988Abstract. A versatile on-line digital feedback system forseismometers has been developed by combining an Amst radPC 1512 com puter with a wideband seismometer developedin the Cybernet ics Department , Reading Univers i ty. The PC1512 com puter is used for implementing A-D an d D-.Aconversion, proportional-integral-derivative ( P I , )calculations, average filter and band selection, automaticscale control and data com municat ion tasks . The sys tem iscontrolled by a BASIC program with assembler subroutines,which makes it much faster than if only a high levellanguage was used . This makes i t poss ible to ad d mo recalculation and control tasks in a fixed sample period.Th e sampling rate can be up to 5000 per second. Thedigital system can operate in three-axis multiplex control inthe whole of the wideband range (&25 Hz) . The relativeresolution of the system is 16 bits; by mean s of autom aticfloating-point scale control the dynamic range is 20 bits(120 dB) . Four pass bands are pre-set and can be selectedf rom the keyboard .1. IntroductionSeismometers employ the principle of inertial mass, wherebythe relative motion between a suspended mass and its sup-port ing fram e is measured to provide information on groundmotio n. The frequency range of interest is from abou t 0.01 tol00Hz, and ground accelerat ions may range from lo-" ' tolo-' m s- ' (a t s i tes remote from a centre of dis turbance).Convent ional ins truments have suspended masses ofseveral kilograms, with periods of about 1 s (short-periodinstrumen ts) or 20 s (long-period instrume nts). The y use amagneticoil velocity transducer for sensing relative motion,and pro vide a respo nse flat to input velocity ab ov e theirnatural period ( i .e . above 1Hz or above 0.05 H z ) .Modern ins truments use much smaller masses of only afew hundred grams with natural period typically 1 s . butmeasure the relat ive displacement of the mass (instead ofrelative velocity), usually with a capacitive sen sor . Th e res-ponse is controlled by means of force-feedback, which main-tains the mass stationary with respect to its supporting fra me ,and th e resulting response is flat to input acceleration from 0to about 50 Hz (depending on the loop gain). A high detecti-vity is achieved by means of the capacitive sensorhow-noiseelectronics and by a high-Q suspension to red uce Browniannoise, and a single small instrument can cover the whole ofthe seismic frequency ran ge.The most serious problems in modern ins truments arisefrom drift in the susp ension system (since very small relative

Permanent address : Depar tment of Instrumentation, ChangchunCollege of Geology, China .74 8 0022-3735/88i080748+05 $02.50 @ 1988 IO P Publ ish ing Ltd


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Digital feedback system fo r seismometers

5 R ,

I - - - - - - - - 1Sensor

8

C-2

Figure 1. The diagram of the analogue wideband feedback seismometer.

5 R ,

displacements a re to be measured) due to c reep , or t empera -ture or pressure changes , and overloading due to the veryhigh dynamic range of seismic signals. Even with a dynamicrange of 1 20 dB an ins trument can tolera te a maximumground acceleration of only of the acceleration of gra-vi ty! I t i s therefore advantageous to control the ins trument byan on-line microprocessor, providing immediate gain rangingor zero a dju stm ent with full control of response at all times.However, several other advantages also become avai l -ab le . Fo r example, the incoming data can be part ia l ly pro-cessed and appropriate act ion taken to avoid sudden gainadjus tment or zero-ad jus tmen t at inconvenient future t imes ,or the respo nse can be optim ised (in terms of frequency. gain

or detect ivi ty) to the nature of the part icular event beingrecorded .

C-2

2. System designThe block diagram of the wideband feedback seismometerdeveloped by Usher (1979) is shown in figure 1.Th e sensor comprises a mass of a b o u t 0.2 kg in a horizon-tal boom arrangement with a period of about 1 s. The massforms the centred plate of a capacitive displacement trans-du cer , the ou tpu t of which is suitably amplified and rectifiedby a phase-sensitive detector (PSD) before being appl ied to amagnet/coil force-feedback transducer. The instrumentresponds to vertical acceleration but similar instruments res-ponding to horizontal ground accelerat ion have also beendesigned.The instrument was designed for a detection level ofbett er than m s-' Hz-l '' , which correspon ds to theminimum observ ed seismic noise, and this was achieved usinga variable-separation capacitive transducer of high responseof 3000 V m - l (excitation 3 V RMS, pla te s epara t ion 1m m )followed by a low-noise amplifier (Usher et a1 1978). T h enoise level corresponds to an in put accelerat ion of lo- ' ' m s- ?Hz-"', an d the frequency response is flat from 0 to 25 Hz; thedynamic range is abo ut 140d B .Th e digitisation process and the introduction of the digitalfeedback control ler should not reduce these values (see S: 3) .T h e modulation-PsD-demodulation arrangement in the pre-vious analogue instrument is retained in our sampled-datadigital system because this arrangement is very efficient inreducing DC drift and in amplifying the signal to a high level,so the effect of the noise introduced by the A-D convertor andother digital c i rcui ts can be re duced . The A-D conver tor andthe digital system are interconnected after the PSD. The b lockdiagram is shown in figure 2.

The block enclosed by the bro ken lines in figure 1 s totallyremoved; instead, the digital system is also controlled by thecomputer . The feedback network B/Rfs also controlled bythe compute r us ing a multiplexer to select a suitable feedbackconstant automatical ly to change the scale , and the data a rerecorded in floating-point representation when the signal

P S C A - C

I TI LDig i ta l'I 0ompu te rI Ii

Figure 2 . The block diagram of the digital system

exceeds the range of levels that can be represe nted by a 16-bitn u m b e r .Two methods were used fo r des igning the sample d-datadigital system, as described below.

2.1 . Pole-zero cancellation designThe t ransfer funct ion of the sensor and PSD in figure 1 is

By test w p 1 5 . 3 , 6 = 0. 1 5 , s o there is a pair of complexconjug ate poles in the t ransfer funct ion of the sen sor. If t hetransfer function of the digital controller in the s domain i sa r ranged to b eG(s'+ as + b)H,(s)=

and a = EO,,, b = U ; , then the open- loopthe whole system in the s domain will be

(2 )ransfer function of

(3 )We can th erefore design a digital controller to cancel the p airof complex conju gate poles and implement an integral ope-rat ion to reduce the low-frequency error . Subst i tut ing thetested values into a and b in equat ion (2 ) , we have

which is a typical P ID control ler . The integral operat ion k , / scan be approximated by the z form on the polygonal integ-ra t ion , k,T (z+ 1) /2(z- l) , and k(z-1)iTz can be used for thederivat ive operat ion (Kuo 1981). H e r e T is the samplingper iod . So the transfer function of the digital controller in thez domain is

( 5 )Tzwhere k , = 4 . 6 3 . k , = 2 3 8 . k d = 1.

74 9


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Z Yin an d M T Usher2.2. PCT ipseudo-coratinuoiiIs-tir?ie) method For the lead-and-lag compensat ion network in equat ion (8)

u , ( k )= ak ,[e(k)- ( k- )] +bu,(k - ) . (13)Th e t ransfer funct ion of the block surrou nded by the brokenlines in figure 1 isThe program is run by using BASIC to call assemblersubrout ines . The BASIC facilitates person-to-computer dialo-gue and the assembler subroutines are used for th e tasks of( 6 )

1 - S T ?1+ S T ,H A ( s )=--T ~ +ST,An expansion of equat ion (6 ) in partial fractions yields

k k d ss s + UT 4 '*(s) = k , 4- 1 - (7 )For T 2 = 0 .6 8 > 3 = 0 .6 8 , T 4 = 0 .0 2 s , T 5 = 0 . 2 s , t h en k,= 1.26,

The analogue sys tem had already been des igned by CADm e t h o d s so we use the bilinear transformation on equation(7) directly to get the z-form transfer function

ki = 1.47, k j z 8 . 7 4 .

k , T ( z + 1) ak,(z - 1)H ( z )= k , + +2 ( z - l ) z + b 'T is the sampling period, a = ( T / T 4 + 2 ) . b=( T / T 4 - 2 ) / ( T / T q + ) . There is an extra time delay in thesampled-data sys tem caused by the sample-and-hold:

(9)That i s to say, the sample-and-hold can be approximated as afirst-order low-pass filter with a pole at 2/T (Houpis andLam ont 1985). When the frequency of sampling is considera-bly higher than the signal frequency this approximation isvalid, and we can design a sampled-data system in the sdom ain, taking account of the delay of the sam ple-and-hold,known as the PCT (pseudo-cont inuous-t ime) method. In ourcase the delay of the sample-and-hold was consideredtogether with the pole in the second ter m o n the r ight-handside of equat io n ( 6 ) . Because of the flexibility of th e softwarecontrol ler it i s easy to readjus t the th ree para meter s k, , ki andkd aro und the value calculated in equat ion (8) to get the bes tdynamic response.3. ProgrammingTh e block diagra m of the digital system is redrawn in figure 3.For programming the integrator

U(Z) k , T ( z+ l ) k , T l + z - 'E ( = ) - 2 ( z - 1 ) 2 1 - z - "

i l i ( k )= k , T [ r ( k ) e ( k - 1)]/2 + u , ( k - 1).

(10)- =--The inverse =-transform of equat ion (10) is

(11)For the derivat ive operat ionu d ( k )= k d [ e ( k ) - e ( k - l ) ] /T . (12)

.+data acquisi t ion, calculation and control a t high speed . Theprogram flow schem atic is shown in figure 4.

The highest signal frequency on on e channel is 25 Hz. Inord er to m ultiplex three ch annels it is effectively 75 Hz, an d ifwe demand ten t imes the highest s ignal frequency for thesamp led-da ta sys tem then the sampling frequency should be750 Hz, so the sampling period should be less than 1.33 m s.The assembler subrou t ine runs very fas t and the period oft ime avai lable for process ing o ne d ata sample is 150,~s.h econversion time of the A-D is 4 0 , ~ s .o a sampling frequencyas high as 5000 Hz can be achieved. This sampling frequencyis far greater than ten t imes th e highest signal frequency. T omake full use of the extra time it is convenient to ar ran ge anaveraging filter after the A-D convertor for reducing high-frequency noise. This averaging filter is not insignificantbecause the time of the sample-and-hold circuits before theA-D is relatively short so that unwanted interference pulsescan be sampled. A data output averaging filter was alsoarranged. It is not only for filtering high-frequency noise butalso for data contract ion.The forward gain preceding the A--D should be arrangedso tha t the inheren t senso r noise is amplified to a level a littlehigher than the least significant bit of the A-D. In fact thenoise behaves like a dither signal, which helps to reduce limitcylce oscillation and increases the resolution by making theA-D quantisation errors equivalent to white noise (Carley1987).4 . Experimental resultsAn ICL 7115 14-bi t A-D chip was used in this sampled-datadigital system; the conversion time is 4 0 ~ ~ s .h e ~ - . 4 onver-to r is a 16-bit AD 7546.The digital system provides four pass bands by presettingparam eters which can be selected from the keyboard. In the0-25 Hz bandwidth version the system was designed by themethod of pole-zero cancellation as described above and theparam eters were adjus ted to get the bes t frequency response.Figure 5 ( a ) is the output response of the seismometer for a1Hz square-wave inpu t s ignal . The overshoot in t ime domainis a l i tt le larger than in the analogue case but th e frequencyresponse curve is flat over a wide range. Figure 5 ( b ) is themagnitude-frequency characteris t ic curve. The 0-1 Hz out-put was obtained by s l ight adjus tment of the pa ramete rs ,taking the ou tput s ignal from th e integrator . Figures 6 ( a )an d(b) how the uni t -s tep response and the magnitude curve.The 0-20 H z and 0-10 Hz vers ions were des igned by thePCT method direct ly using the param eters given in the analo-

Sensor A P S? A- 0

i0 - A

Figure 3. The digital system with P I D controller75 0


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Digital feed back system fo r seismometers0ntr(47)nit iat ion

r e a d A - 0

S t a r t0Change

Over f low?treatmentAverage

f i te r

1 Hz 10 an d 20 H z

Proport ional Proport ional

Derivative Derivative

D a t a a ut' eed backFigure 4. Program flow schemat ic

IL-

I ( H z lFigure 5. 0-25 Hz o u tp u t ; G = 0.07, k , = 6 , k,T/2= 0.96. k , / T = 200.( a )Th e ou tpu t response fo r a 1 Hz square-wav e input signal; (b )the magnitude-frequen cy characteristic curv e.

U I1 .0 2 0 . L 0 6 0 8 1 2

(Hz)Figure6. 0-1 Hz output; G = 0 . 0 7 , k , = 5 . 7 , k ,T/2=0.107 , k d / T =133. ( a ) The un i t -s tep response, ( b ) he magn i tude-frequencycurve.

gue sys tem. Th e s ignal output was taken from th e integrato rin each case. By slightly changing the parameters a differentfrequency response was achieved. The seismometer outputresponse in the t ime domain and the f requency dom ain for

the 0-20 Hz a n d t h e 0-10 Hz outputs are shown in f igures 7an d 8 respectively.The ins trument is now being op erated in a seismic vaul tand its noise level is being analysed and com pared with that of751


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Z Yin and M J Usher

2 4 6 a i 0

/----

(61

4 6 8 1 0 20(Hz!

Figures . 0-10 Hz output; G = l . k,=1.29. k ,T / 2=0 .008 ,a k d = 8b = 0 .77 . ( a ) The output response for a 1 Hz square-wave inputsignal: ( b ) he magnitude-frequency curve.

a s imilar analogue ins trum ent . Noise level m easure men ts onseismometers are difficult because at most sites the seismicnoise level is much higher than the inherent ins trumentalnoise, an d two adjac ent s imilar ins truments have to be com-pared .

Programming in BASIC combined with assembler for high-speed con trol is very efficient. It seems th at th e digital systemcan achieve a higher level of accuracy. controllability andversatility and it has good potential prospects in digital andintelligent seismometer systems.AcknowledgmentThe authors wish to thank Mr L Comley of the Cybernet icsDepartment , Reading Unives i ty, for his help in assemblingand se t ting up the equ ipment .ReferencesCarley L R 1987 A n oversampling analog-to-digitalconverter topology for high-resolution signal acquisitionsystemsIE EE Trans. Circuits and Systems CAS-34 83-90Houpis C H and Lamon t G B 1985 Digital Control Systems(New York: McGraw-Hil l )Kuo B C 1981 Digital Co ntrol Systems (New York: Hol t .R inehar t and Wins ton)U s h e r M J 1979 Wideban d feedback seismometersPhys. Earth Planet. Inter. 18 38-50U s h e r M J . Gura lp C and Burch R F 1978 Th e design ofminiature wideband seismometersGeophys. J .R . Astron. Soc. 55 605-13

5. SummaryAn Amst rad PC 1512 16-bit microc omp uter w ith an A-D an dD-A expansion card connected to a wideband seismometerforms a very versatile digital feedback control system. Twomethods were used to design th e system. Since the sensor hasa pair of complex conjugate poles the pole-zero cancellationmethod seems more at t ract ive than the PCT method. Bothmethods are valid when the sample rate is relatively high.75 2

6. DSP Pendulum Feedback System

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