6. Dsp Pendulum Feedback System

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A high-resolution wideband digital feedback system for seismometers

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R J Adrian et a1 Erdmann J C 1979 Application of recurrence rate techniques to turbulence analysis Phys. Scr. 19 39&401 Erdmann J C and Gellert R P 1978 Recurrence rate correlation in scattered light intensity J . Opt. Soc. Ani. 68 787-95 Papoulis A 1984 Probability, Random Variables and Stochastic Processes 2nd edn (New York: McGraw-Hill) p 345 Rice S 0 1944 Mathemetical analysis of random noise Bell S y s f . Tech. J . 23 282-332 Smart A E and Mayo W T Jr 1979 Applications of laser anemometry to high Reynolds number flows Phys. Scr. 19 426-40 TSI Inc. 1987 Model 75 Correlex Signal Processor (St Paul. MN: TSI Inc.) Vehrenkamp R S. Schatzel K , Pfister G and Schulz-DuBois E 0 1979 Direct measurement of velocity correlation functions using the Erdmann-Gellert rate correlation method J . Phys. E: Sci. Instrum. 12 119-25

J . Phys. E: Sci. Instrum. 21 (1988) 748-752. Printed in the UK

A high-resolution wideband digital feedback system for seismometers Z Yint and M J Usher Department of Cybernetics. University of Reading, 3 Earley Gate, Whiteknights. Reading RG6 2AL. UK Received 21 December 1987, in final form 10 March 1988 Abstract. A versatile on-line digital feedback system for seismometers has been developed by combining an Amstrad PC 1512 computer with a wideband seismometer developed in the Cybernetics Department, Reading University. The PC 1512 computer is used for implementing A-D and D-.A conversion, proportional-integral-derivative ( P I , ) calculations, average filter and band selection, automatic scale control and data communication tasks. The system is controlled by a BASIC program with assembler subroutines, which makes it much faster than if only a high level language was used. This makes it possible to add more calculation and control tasks in a fixed sample period. The sampling rate can be up to 5000 per second. The digital system can operate in three-axis multiplex control in the whole of the wideband range (&25 Hz). The relative resolution of the system is 16 bits; by means of automatic floating-point scale control the dynamic range is 20 bits (120 dB). Four pass bands are pre-set and can be selected from the keyboard. 1. Introduction Seismometers employ the principle of inertial mass, whereby the relative motion between a suspended mass and its supporting frame is measured to provide information on ground motion. The frequency range of interest is from about 0.01 to l00Hz, and ground accelerations may range from lo-"' to lo-' m s-' (at sites remote from a centre of disturbance). Conventional instruments have suspended masses of several kilograms, with periods of about 1 s (short-period instruments) or 20 s (long-period instruments). They use a magneticoil velocity transducer for sensing relative motion, and provide a response flat to input velocity above their natural period (i.e. above 1 Hz or above 0.05 Hz). Modern instruments use much smaller masses of only a few hundred grams with natural period typically 1 s. but measure the relative displacement of the mass (instead of relative velocity), usually with a capacitive sensor. The response is controlled by means of force-feedback, which maintains the mass stationary with respect to its supporting frame, and the resulting response is flat to input acceleration from 0 to about 50 Hz (depending on the loop gain). A high detectivity is achieved by means of the capacitive sensorhow-noise electronics and by a high-Q suspension to reduce Brownian noise, and a single small instrument can cover the whole of the seismic frequency range. The most serious problems in modern instruments arise from drift in the suspension system (since very small relative Permanent address: Department of Instrumentation, Changchun College of Geology, China.

748

0022-3735/88i080748+05 $02.50 @ 1988 I O P Publishing Ltd

Digital feedback system f o r seismometers I - - - - - - -

- -1

Sensor

8

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

displacements are to be measured) due to creep, or temperature or pressure changes, and overloading due to the very high dynamic range of seismic signals. Even with a dynamic range of 120dB an instrument can tolerate a maximum of the acceleration of graground acceleration of only vity! It is therefore advantageous to control the instrument by an on-line microprocessor, providing immediate gain ranging or zero adjustment with full control of response at all times. However, several other advantages also become available. For example, the incoming data can be partially processed and appropriate action taken to avoid sudden gain adjustment or zero-adjustment at inconvenient future times, or the response can be optimised (in terms of frequency. gain or detectivity) to the nature of the particular event being recorded.

2. System design The block diagram of the wideband feedback seismometer developed by Usher (1979) is shown in figure 1. T h e sensor comprises a mass of about 0.2 kg in a horizontal boom arrangement with a period of about 1 s. The mass forms the centred plate of a capacitive displacement transducer, the output of which is suitably amplified and rectified by a phase-sensitive detector (PSD) before being applied to a magnet/coil force-feedback transducer. The instrument responds to vertical acceleration but similar instruments responding to horizontal ground acceleration have also been designed. The instrument was designed for a detection level of better than ms-' Hz-l'', which corresponds to the minimum observed seismic noise, and this was achieved using a variable-separation capacitive transducer of high response of 3000 V m - l (excitation 3 V RMS, plate separation 1 mm) followed by a low-noise amplifier (Usher et a1 1978). The noise level corresponds to an input acceleration of lo-'' m s-? Hz-"', and the frequency response is flat from 0 to 25 Hz; the dynamic range is about 140 dB. The digitisation process and the introduction of the digital feedback controller should not reduce these values (see S: 3). The modulation-PsD-demodulation arrangement in the previous analogue instrument is retained in our sampled-data digital system because this arrangement is very efficient in reducing DC drift and in amplifying the signal to a high level, so the effect of the noise introduced by the A-D convertor and other digital circuits can be reduced. The A-D convertor and the digital system are interconnected after the PSD. The block diagram is shown in figure 2. The block enclosed by the broken lines in figure 1 is totally removed; instead, the digital system is also controlled by the computer. The feedback network B/Rfis also controlled by the computer using a multiplexer to select a suitable feedback constant automatically to change the scale, and the data are recorded in floating-point representation when the signal

A-C

PSC

T

I

I

L

Digital

computer

'I 0

I

I

i 5lR,

C-2

Figure 2. The block diagram of the digital system

exceeds the range of levels that can be represented by a 16-bit number. Two methods were used for designing the sampled-data digital system, as described below.

2.1. Pole-zero cancellation design The transfer function of the sensor and

PSD

in figure 1 is

By test w p 1 5 . 3 , 6=0.15, so there is a pair of complex conjugate poles in the transfer function of the sensor. If the transfer function of the digital controller in the s domain is arranged to be G(s'+ as + b ) H,(s)= (2) and a = EO,,, b = U ; , then the open-loop ransfer function of the whole system in the s domain will be (3) We can therefore design a digital controller to cancel the pair of complex conjugate poles and implement an integral operation to reduce the low-frequency error. Substituting the tested values into a and b in equation (2), we have

which is a typical P I D controller. The integral operation k , / s can be approximated by the z form on the polygonal integration, k,T(z+ 1)/2(z-l), and k(z-1)iTz can be used for the derivative operation (Kuo 1981). Here T is the sampling period. So the transfer function of the digital controller in the z domain is Tz

(5)

where k,=4.63. k,=238. k d = 1. 749

Z Yin and

M

T. Usher

2.2.

PCT ipseudo-coratinuoiiIs-tir?ie) method The transfer function of the block surrounded by the broken lines in figure 1 is

1' - S T ?1+ S T , HA(s) = -S T ~ 1 +ST,

(6)

An expansion of equation (6) in partial fractions yields k kds H*(s) = k , 4- 1+s s + UT4'

(7)

For T2=0.68>T3=0.68, T4=0.02s, T 5 = 0 . 2 s , then k,= 1.26, ki= 1.47, k j z 8 . 7 4 . The analogue system had already been designed by C A D methods so we use the bilinear transformation on equation (7) directly to get the z-form transfer function

H ( z )= k, +

k,T(z + 1) ak,(z - 1) + z+b 2(z-l)

'

T is the sampling period, a = ( T / T 4 + 2 ) . b= ( T / T 4 - 2 ) / ( T / T q +2 ) . There is an extra time delay in the sampled-data system caused by the sample-and-hold: (9) That is to say, the sample-and-hold can be approximated as a first-order low-pass filter with a pole at 2/T (Houpis and Lamont 1985). When the frequency of sampling is considerably higher than the signal frequency this approximation is valid, and we can design a sampled-data system in the s domain, taking account of the delay of the sample-and-hold, known as the PCT (pseudo-continuous-time) method. In our case the delay of the sample-and-hold was considered together with the pole in the second term on the right-hand side of equation ( 6 ) . Because of the flexibility of the software controller it is easy to readjust the three parameters k,, ki and kd around the value calculated in equation (8) to get the best dynamic response.

3. Programming The block diagram 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-"

--

(10)

The inverse =-transform of equation (10) is ili(k)=

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

(11)

For the derivative operation u d ( k )= k d [ e ( k ) - e ( k - l)]/T.

Sensor

(12)

A

For the lead-and-lag compensation network in equation (8) u,(k)= ak,[e(k) - e ( k - l)]

4. Experimental results An ICL 7115 14-bit A-D chip was used in this sampled-data digital system; the conversion time is 4 0 ~ ~The s . ~ - . 4convertor is a 16-bit AD 7546. The digital system provides four pass bands by presetting parameters which can be selected from the keyboard. In the 0-25 Hz bandwidth version the system was designed by the method of pole-zero cancellation as described above and the parameters were adjusted to get the best frequency response. Figure 5 ( a ) is the output response of the seismometer for a 1 Hz square-wave input signal. The overshoot in time domain is a little larger than in the analogue case but the frequency response curve is flat over a wide range. Figure 5 ( b ) is the magnitude-frequency characteristic curve. The 0-1 Hz output was obtained by slight adjustment of the parameters, taking the output signal from the integrator. Figures 6 ( a ) and ( b ) show the unit-step response and the magnitude curve. The 0-20 Hz and 0-10 Hz versions were designed by the PCT method directly using the parameters given in the analo-

PS?

0-A

750

PID

controller

(13)

The program is run by using BASIC to call assembler subroutines. The BASIC facilitates person-to-computer dialogue and the assembler subroutines are used for the tasks of + . data acquisition, calculation and control at high speed. The program flow schematic is shown in figure 4. The highest signal frequency on one channel is 25 Hz. In order to multiplex three channels it is effectively 75 Hz, and if we demand ten times the highest signal frequency for the sampled-data system then the sampling frequency should be 750 Hz, so the sampling period should be less than 1.33 ms. The assembler subroutine runs very fast and the period of time available for processing one data sample is 150,~s. The conversion time of the A-D is 4 0 , ~ sso . a sampling frequency as high as 5000 Hz can be achieved. This sampling frequency is far greater than ten times the highest signal frequency. T o make full use of the extra time it is convenient to arrange an averaging filter after the A-D convertor for reducing highfrequency noise. This averaging filter is not insignificant because the time of the sample-and-hold circuits before the A-D is relatively short so that unwanted interference pulses can be sampled. A data output averaging filter was also arranged. It is not only for filtering high-frequency noise but also for data contraction. The forward gain preceding the A--D should be arranged so that the inherent sensor noise is amplified to a level a little higher than the least significant bit of the A-D. In fact the noise behaves like a dither signal, which helps to reduce limit cylce oscillation and increases the resolution by making the A-D quantisation errors equivalent to white noise (Carley 1987).

A- 0

i Figure 3. The digital system with

+ bu,(k - 1).

Digital feedback system f o r seismometers

0

0 Start

entr

Change

(47) initiation

read

A-0

Overflow? treatment Average f i Iter

10 and 20 Hz

1 Hz

Proportional

Proportional

Derivative

Derivative

D a t a aut

'

Feed back

Figure 4. Program flow schematic

I

LU I1

. 02

I

(Hzl

0.L

06

08

1

2

(Hz)

Figure 5. 0-25 Hz output; G = 0.07, k, = 6, k,T/2 = 0.96. k , / T = 200. ( a ) The output response for a 1 Hz square-wave input signal; ( b ) the magnitude-frequency characteristic curve.

Figure6. 0-1 Hz output; G=0.07, k,=5.7, k,T/2=0.107, k d / T = 133. ( a ) The unit-step response, ( b ) the magnitude-frequency curve.

gue system. The signal output was taken from the integrator in each case. By slightly changing the parameters a different frequency response was achieved. The seismometer output response in the time domain and the frequency domain for

the 0-20 Hz and the 0-10 Hz outputs are shown in figures 7 and 8 respectively. The instrument is now being operated in a seismic vault and its noise level is being analysed and compared with that of 75 1

Z Yin and M J Usher Programming in BASIC combined with assembler for highspeed control is very efficient. It seems that the digital system can achieve a higher level of accuracy. controllability and versatility and it has good potential prospects in digital and intelligent seismometer systems.

/----

Acknowledgment The authors wish to thank Mr L Comley of the Cybernetics Department, Reading Univesity, for his help in assembling and setting up the equipment. References Carley L R 1987 A n oversampling analog-to-digital converter topology for high-resolution signal acquisition systems IEEE Trans. Circuits and Systems CAS-34 83-90

(61

2

a i 0

6

4

4

Houpis C H and Lamont G B 1985 Digital Control Systems (New York: McGraw-Hill) Kuo B C 1981 Digital Control Systems (New York: Holt. Rinehart and Winston) Usher M J 1979 Wideband feedback seismometers Phys. Earth Planet. Inter. 18 38-50 Usher M J. Guralp C and Burch R F 1978 The design of miniature wideband seismometers Geophys. J.R. Astron. Soc. 55 605-13

6

8

1

0

20

(Hz! Figures. 0-10 Hz output; G = l . k,=1.29. k,T/2=0.008, akd=8 b = 0.77. ( a ) The output response for a 1 Hz square-wave input signal: ( b ) the magnitude-frequency curve.

a similar analogue instrument. Noise level measurements on seismometers are difficult because at most sites the seismic noise level is much higher than the inherent instrumental noise, and two adjacent similar instruments have to be compared. 5. Summary An Amstrad PC 1512 16-bit microcomputer with an A-D and D-A expansion card connected to a wideband seismometer forms a very versatile digital feedback control system. Two methods were used to design the system. Since the sensor has a pair of complex conjugate poles the pole-zero cancellation method seems more attractive than the PCT method. Both methods are valid when the sample rate is relatively high.

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