- Email: [email protected]
Identification of a Large Water-Heated Crossflow Heat Exchanger with Binary Multifrequency Signals
rig-Ill © I FA( : Id t' lItifi(." ation alld Sys tem Pa rallle te r Est imation I (I~F' . Yor k. [ ' K. I ! I~F,
Cop~
IDENTIFICATION OF A LARGE WATERHEATED CROSSFLOW HEAT EXCHANGER WITH BINARY MULTIFREQUENCY SIGNALS G. Franck and H. Rake lnstitut .fiir Rl'gl'iulIgstl'rllllik. I?h l'illi.lrh-IVI'.lt/fi/isr!II' T n h11 i ll /II' IJo r!tsr!1II11' ,40 rhl'll . 1"1'1 /1'1'0/ RI'/iII"'ir of (;I'I'I/III1l)'
Abstract: The dynamic behaviour of a large water-heated crossflow heat e xchanger, the heatlng power of which can be influenced by varying the water flow, is investigated by experimental identification with binary multifrequency test signals. It will be shown that it is possible to describe this non-linear distributed parameter system using linear non-parametric and parametric models. Both correlation analysis and direct leastsquares parameter estimation are employed in order to do this, and the identification results are compared with each other. Furthermore, the necessary signal processing uSing antialiasing filters and digital highpass filters is described, as well as the evaluation of the qual ity of estimation results with the aid of the coherence function. It is shown that the identification errors of the non-linear heat exchanger decrease when the appropriate choice for characteristics is made for the binary multifrequency signal. Keywords. Identification; crossflow heat exchanger; non-linear distributed parameter system; binary multifrequency test signal; antial iasing filter; highpass filter; correlation analysis ; coherence function; direct least-squares parameter estimation.
INTRODUCTION Crossflow heat exchangers in air-conditioning systems are subject to frequently changing operation points and therefore it is necessary to adapt the heating power to the level requi red in each case. In general, heat exchangers can be described as non-linear systems with distributed parameters. Knowledge of their dynamic behaviour is therefore important e.g. for the design of effective automation concepts, for the determination of (sub-) optimal controller parameters, or for val idation of theoretically calculated models (Franck, 19B5). In thi s paper the dynamic behaviour of a 1arge water-heated crossfl ow heat exchanger during normal operation in an air-conditioning system for an office bui 1ding is invest igated. A few important technical data of the heat exchanger are summarized in Table 1. TABLE
1 iJata of the investigated crossflow heat exchanger
by correlation analysis for the non-parametric model and by direct least-squares estimation for a parametric model of the heat exchanger. The nonparametric model provides a particularly impressive picture of the process in graphic form, whereas the parametric model can give a very cl ear overall description of the heat exchanger at different operation pOints by means of a few parameters of the 1 inear discrete time process mOdel. It is shown that the coherence function presents a suitable criterion fol' eval uation of the qual ity of the estimated results. Thi s function is further used as an aid in order to optimize the characteristics of the test signal s used in prel iminary experiments with the purpose to achieve to good identification results. EXPERIMENTAL SET-UP For identification of the dynamic behaviour between the water flow and the average ai rout 1et temperature a measuring system was installed at the heat exchanger. This system is schematicall y illustrated in Fig. 1.
Width 1. 75 m 1.45 m Height 0,16 m Air flow path 19 . 00 m Water flow path Hea t i ng power 600,000 kJ/ h Volume flow rate of the air 27 , 000 m3/ h Nominal temperatures: -12 °C Air inlet 5°C Ai r outlet 100 °C Water inlet 70 °C Water outlet For identification purposes only binary multifrequency test signals were used, and the response of the local mean value of the air outlet temperature to changing water flow is investigated. At first the experimental set up and important c harac teristics of the test signal used are described. Finally, the data processing using antialiasing filters and digital highpass filters, whi c h is necessary because the measured signals are disturbed, is described. The evaluation of the data has been done
Fi g.
Ex perimental set-u p
IH60
(; . Franck a nd H . Rake
The test signal Y which activates the system is given out by a process computer as analogue voltage and transferred to the pneumatic valve via a voltage to pressure converter. In thi s way the water flow through the heat exchanger is changed. A built-in inductive magnetic flow meter produces an output voltage which is proportional to the velocity of the water Vw. The input signal referring to the maximum velocity VWN is chosen for the identification U
=~ VWN
(1)
: "' urnunmu I I Umi n i t . .
2 ••
n-
Fig. 2 One period of binary multi frequency test si gnal Table 2 seruences of Signs of the used binary mu tlfreguency test slgnal
Thus the resul ts become independent from the val ve characteristics and from the dynamic behaviour of the heating water supply. However the measured signal still contains the dynamics of the flow meter. The response of the distributed air outlet temperature to the test signal is approximated by the mean val ue of the temperatures measurea with the eight electronic temperature sensors
8 ALA = 1. • L ALAi • 8 i=1
(2 )
Therefore the sensors were placed in the air current at equidistant points along the flow path of the water. Further sensors for the control of boundary conditions were installed to measure the air inl et temperature ALE' and the water temperatures A and AWA • A process minicomputer MODC(»\P MODACS 11Wf was used for the test signal output, the data acquisition, and the monitoring of the experiment. For data storage and off-l ine analYsi s a master process computer MOOC(»\P CLASSIC 7870 was used, which is linked to the minicomputer. BINARY MULTIFREQUENCY TEST SIGNALS Due to some advantages over other test signal s it was decided to stimulate the investigated heat exchanger by using binary multifrequency test signals. It is particularly advantageous that the signal amplitude is limited to two extremal values i.e. the signal is binary, and that the number, pOSition, and power of the corresponding sinusoidal signal components can be chosen with few 1 imitations. While the bounded signal amplitude is particularly important for the identification of non-linear systems using linear methods, high test signal power has an influence on the available accuracy in the case of disturbed systems and therefore also on the identificaton time. Opposite to the application of mono frequency signals, the heat exchanger investigated is stimulated at several frequencies, however it is only necessary to wait for one transient. After preliminary experiments, the signal, one period of which is plotted in Fig. 2 as an example, was taken from a large signal 1 ibrary of about 40 binary multifrequency signals (Paehlike, 1980). The chronological course of the signal is included in Tab 1e 2 as sequences of si gns, whereby the number of elementary intervals (equal to the sampling time) between the change from max ima 1 to mi n ima 1 value is specified. Figure 3 shows the power density spectrum of the si gnal. The synthesi s and further c haracteristics of binary multi frequency signals were described by Paehlike and Rake, (1979).
"'''1 Su~JIIIIIJ •
I, .. 25
"""I,. '" "1,,,, 5B
Fig. 3 Power density spectrum of the test signal
lul
The signal amplitude was calculated for each operation point investigated with the relation empirically developed in preliminary experiments IUI = 0,27 • U~ + 0.022
(3 )
Finally, the analogue output voltage to the pneumatic val ve was determined by means of the experimentally measured static valve characteristic Yo = f(U o )· DATA PROCESSING In order to st imul ate the heat exchanger in an interesting frequency range an elementary interval of the binary multifrequency signal is required, and therefore a sampl ing time of T = 10s was used. To avoid aliasing errors by sampl ing the analogue measuring signal an antialiasing filter with very low cut-off frequency and hi gh dampi ng in the stop band is therefore necessary. Here the antial iasing filter was realised by an analogue and a digital part. At first the analogue measuring signals were sampled for the identification, admittedly with higher sampl ing rate than necessary, but nevertheless in such a way that Signal s components above the respective f'iyqui st frequency were damped by at 1 east 40 dB due to the lowpass behaviour of the measuring sensors and converters. Finally the data were filtered digitally. Because of its high signal dampi ng in the stop band and it s zero pha se shift, a finite impul se response digital filter with a non causal symmetric weight function was designed by use of a Blackmann window (Oppenheim, Schafer, 1975 ) g(n)=0.42+O.5 cos(2!.fl.)+O.08 cos(l!ill.) -M
M
M
for this purpose and realized with the window 1 ength N = 2M + 1 = 31 according to the convol ution
WattT-Heatccl ( :rossllo\\' I kat M
g(O)a(k)+ S(k)
2 g(n) (a(k+n)+a(k-n))
n=l
(5 )
The frequency response of thi s fi 1 ter is shown in Fig. 4. At the convolution length M = 15 and the short sampl ing time T = Is the first zero of the fi 1 ter frequency response was chosen to coinc ide with the ~qui st frequency of the enl arged sampl ing time on the digital computer at T = 10s. Thus the digital lowpass filter covers the frequency range betvleen the ~qui st frequency of short sampl ing time and the Nyqui st frequency necessary for analysis.
~
.,
..., 0.'"
~
...., • 1!In1
.. Ill
. . .1
1.1
,-'
I H()I
the transient response time to the highpass filter, which is the slowest part of the system, until it comes to a steady state. The steady state is thereby important for the development of reliable estimations. In order to el iminate the influences o f the digital filters in the evaluation both the input and output signals were filtered. The data thus produced contain therefore only those signal components, which have significant power in the frequency range stimulated by the periodic binary multifrequency test signal. IOENTIFICATIUN In order to evaluate the filtered test signals and average air outlet temperature signals a non-parametric and a parametric procedure were introduced. Thi s was done because we were not sure whether a distributed parameter system could be described precisely enough by a rational transfer function. Thus for small deviations from an operating point the 1 inear non-parametric model should yield the best possible description of the non-linear heat exchanger with distributed system parameters.
I
Correlation Analysis
w-
Fig. 4 Magnitude of the digital lowpass filter Low-frequency disturbance, steady-state values and drift ing were el iminated by means of a digital highpass filter. An infinite impul se response digita 1 fi 1ter of second order wa s introduced for thi s (Oppenheim, Schafer, 1975). The cut-off frequency was chosen to be equal to the basic frequency w1 = "/1000 s-l of the binary signal. The filter works according to the relation Y( k)=PY(k-l )+QY(k-2 )+R (S (k) -2 S(k-l)+ S (k-2))
(6)
with P Q R
E~thallgn
0.9780 -1. 9556 0.9566
The orthogonal correlation, identical to the Fourier analysis of the signals, was used to estimate the non-parametric model. For thi s purpose the Fourier transformation was executed with data segments of the signals. The segments were not shifted aga in st each other and had a 1ength equa 1 to one period of the test signal. The computational effort is thus reduced due to the fact that Fourier coefficients of every period only have to be calculated at n frequencies '"k which are activated by the test signal. The expected val ues of the Fourier coefficients are approximated by using the arithmetic mean va 1 ue of the coeffi c i ents of a 11 per i od s m measured so that the frequency response of the investigated heat exchanger can be defined as
(7)
The frequency response of this highpass filter with the sampl ing time T = 10s is plotted in Fig. 5. 11
Coherence Function
,
., L.,
I1
Since the measuring time is finite, errors arise from the disturbance Signal and affect the accuracy of the results just as the non-l inearity of the heat exchanger does.
/
...., [7
..
"'0 ,
..
-110
-,
The following coherence function was chosen as a measure for evaluating these errors :
'""
(9 )
w-
Fig. 5 Frequency response of the digital fi lter
highpass
In general a transient response time is necessary when working with filters, particularly when highpass fi lters with very low cut-off frequency are involved. As a rule this leads to a loss of data in the off-line evaluation. It is however possible to operate these digital filters on-line and to adapt
Thi s can be cal cuI ated from the expected val ues E of the estimated power spectral S( w). The coherence function for the frequencies wk considered here WhlCh are contained in the test si gnal is approx imately detenni ned by
m 1;11 (a0i+ jb Oi) (aUi -jb Ui ) 12
m
m
( 10)
i l l 1aUi +j b Ui 12 i 111 a0 i +j b0i 12 the deriviation of this is given in Paelike, Ra ke (1979). It is thus available as a c riterion whi c h
(;. Frallck alld I-I. Rake
IH(i~
not only allows to frequency response the characteristics large the measuring
determine the quality of the estimation but al so to change of the test signal, or to entime.
and z-transformation correspondences. This transfer function can then be plotted in the Bode diagram as frequency response F(jw).
In this way the test signal ampl itude, for example, can be enlarged if the coherence function has too small values for all frequencies. In doing this a comprom i se mus t be found between the mea sur i ng time and the test signal ampl itude because the process being investigated has non-linear characteristics. If the coherence function has a small value only for some specific frequencies, an improvement can be reached by choosing another binary multifrequency signal with high signal power of the correspondi ng frequency. Fig. 6 Model spl itted into parall el branches RESULTS Least-squares parameter estimation A parametric description is chosen for the determination of the dynamic behaviour of the investigated heat exchanger, which depends on the operating point. The few characteristic data of the 1 inear discrete-time model
(11 )
n 1+
L a. ·z-l
i=l
1
thereby give an excellent survey of the operating point dependance on the process. tt is guaranteed by the activating test signal that the linear range of the non-l inear heat exchanger is not or only insigni ficantly exceeded.
First the behaviour of the heating water supply was investigated. The steady state relation of the pneumatic valve and the pipes between the test signal Yo and the water flow Uo is depicted in Fig. 7 as a measured characteristic curve. An example of the dynamic behaviour of the water flow is shown for an operating pOint Uo = 0.54 as a frequency response in the Bode diagram, Fig. 8. It is recognised that the water flow responds to the test signal approximately proportionally with a dead time. Thus the dynamics of the water flow have a negligible influence on the overall behaviour of the heat exchanger, in the same way as the dynamics of the air temperature meas1lri"0 elements investigated in preliminary experiments.
U 0.8
The parameter estimation was excecuted using the comput i ng-t ime-sav i ng-procedure with the di rect I east-squares method desc ribed by Breddermann (19Bl) for a set of model s of 3rder n and dead time The systematical bias is small, because the disturbance part in the measured output signal is low due to the high test signal power and because of estimation with a model of enlarged order.
0,6 0,4
a·I.
The evaluation of the models with 1<;;<4 and time delay 02a~6 is done off-line. The bestmodel order and tllne delay were determined from these 28 model combinations by finding out the model with minimal equation error loss function. The thus resulting model order was = 3 and = 4 in all cases. A reduction of the order could be achieved by a decomposition of the model. The model is thereby spl it into parallel branches, Fig. b, by determining the partial fractions of the rational component of the z-transfer function (Eq. 11). Branches with a comparatively small steady state value
n
0,2
6
8 V 10
Y --
Fig. 7 Static relation between test signal Yo and water fl ow Uo
n
" 1' 1
0. 1
Ri
R·
=_1_ l-z Pi
(12 )
could be neglected 1n th1S way. The same appl1es for conjugated complex poles ZP1 and zPi+l' Thus a satisfactory description of the heat exchanger could normally be achieved by the model order = 1 with a dead time about 10s.
n
...
1. 21 \
o·
--.......
- 27111 -
111. 0 1
In order to be able to compare the estimated parameters of the discrete-time model with the results of the Fourier-analysi s in the Bode diagram, a transformation to a time-continuous model was executed. Starting from the description of a continuous process using a first order holding element, the Laplace-transfer function of the heat exchanger G(s), and a sampl ing element, the continuous transfer function G(s) can be determined by the Laplace-
111. 1
. -'
I
Fig. 8 Frequency response of the water flow In Fig. 9 one period of the test signal, the measured lowpass filtered air outlet temperature over 10 periods, and the high pass filtered air outlet temperature are plotted. The drift, which is due to a slowly rising air inlet temperature, is thereby el iminated.
lHfi3
Water-Heated ( :rossllol\' Heat Exch;llIger
~~~:gll Ila
I
2ae
n --
the coherence-function Y~O(~) was greater than U.Y for the majority of the actlvating frequency components. Generally thi s could be achieved by increasing the test signal ampl itude and keeping the identification time constant. Figure 12 shows the corresponding frequency responses of the di screte time models of 1st order to 4th order with the dead time between zero and 10s in comparison to the non-parametric model of the Fourier-analysis. Fourier-analysis
lowpass filtered 17.1
15. 1 '.I!II I---t-----f----'~..-I
21ea
1511
n -a. eel
~=~~=:::::==~
f _~:~--j---,----1 ~ -. u·~-~~~~~~~ -36.- L-_-.-J~.!.!.::.=_.J.~_>_-.J
Isaa
IDaa
•. 001
2Dli!la
n-
Fig. 9 Test signal and air outlet temperature at Uo = 0.29 The result of the Fourier-analysis is shown in the Bode diagram in Fig. 10 using four experiments as exampl es. In order to be able to eval uate the quality of the frequency response ~~surement, the corresponding coherence function Yeu(~) is depicted in Fig. 11. .1
I
IFI
UOOO,o"", 0,35 -
0" , -
0,111
L.
~
...
-
.........
I..
e. eD)
I
Fig. 12 Compari son of Fourier-analysis and parameter estimation of the models 1st order to 4th order The model of first order describes the investigated heat exchanger insufficiently in general, considerably better results could be achieved by choosing the order of the model s = 3 and 4 according to the procedure described above. Thus the process characteristics in dependance on the setpoint can be described by the dominatinq parameters Rand T1 of the respective s-transfer function (Fig. 13, 14). The dead time varies between 0 and 10s and is not prese!'teq separately. In non-linear systems the results K, T1 cannot be appointed to one value of h! water flow Uo due to the test signal ampl itude U and so the confidence area of the parameters In h diagram is valid.
n
n ..
4,0
K
~~ u.~
9LAN-9LEN 3 , 0
2,0 ' . 81
LI
"
I
,
.. Y0U
,' 1
w-
I
Fig. 10 Frequency response of air outlet temperature at several set points
1 '2 •
11. 1
I
"\
0.1 •
I·
e.ll l
1,0
O,O +-~~-~~-r-~~~~-'---'
0,0
I I a,al
"'" e, •
C,I
Fig. 13 Gain K in dependance on Uo for the crossflow heat exchanger
.-'
•
Fig. 11 Corresponding coherence-function Y~ (~) at several set points The parameter estimation showed that convenient and interpretable results, which could be compared with the Fourier-analysis, were only achieved, when
(;. FralHk and 1-1. Rake
IH(i4
O +-~~~~--.-~~-r~-,
0,0
0,5
1,0
uo
Fig. 14 Time constant T1 in dependance on Uo for the crossflow heat exchangr SUMMAR Y For the identification using binary multi frequency test signal s of the heat exchanger investigated a measured data processing with an antialiasing filter and a highpass filter was necessary. The limited signal ampl itude and process excitation in a wi de frequency rang e with hi g h si gna 1 power of the test signal made a satisfactory identification possible. The evaluation with the aid of the Fourier-analys is and the di rect 1east-squares parameter estimation yielded corresponding results in the stimul ated frequency range. The coherence function proved to be hel pful and expedient as criterion for the qual ity of the resul ts and choice of the test signal. The identification of a non-linear system using linear models is generally possible when important boundary conditions are taken into consideration.
REF ERE NC ES Breddermann, R. (1981). Ei nsparurlS von Rechenaufwand bei der off-l ine Parameterschiitzung mit Modellstrukturbestimmung. Regelungstechnik, 29, pp. 380-386. Franci<:" G. (1985). Zum dynamischen Verhalten wasserstrom estellter Kreustromwarmeubertra er 1 n K lmaan afen. Fortsc rltt-Berlc t Rel e 8, to be pub ished, VDI-Verlag, Diisseldorf. Oppenheim, A. and R. Schafer. (1975). Digital Signal Processing. Prentice-Hall Inc., Englewood Cliffs. Paehl ike, K.-D. (1980). Regelstreckenidentifikation mit binaren Mehrfreguenzslgnalen. Dlssertatlon, RWTH Aachen. Paehlike, K.-D. and H. Rake. (1979). Binary M.Jltifrequency Signals-Synthesis and Application. 5th IFAC Symposium on Identification and System Parameter Est imat i on, Pergamon Press, Oxford.
Cop~
IDENTIFICATION OF A LARGE WATERHEATED CROSSFLOW HEAT EXCHANGER WITH BINARY MULTIFREQUENCY SIGNALS G. Franck and H. Rake lnstitut .fiir Rl'gl'iulIgstl'rllllik. I?h l'illi.lrh-IVI'.lt/fi/isr!II' T n h11 i ll /II' IJo r!tsr!1II11' ,40 rhl'll . 1"1'1 /1'1'0/ RI'/iII"'ir of (;I'I'I/III1l)'
Abstract: The dynamic behaviour of a large water-heated crossflow heat e xchanger, the heatlng power of which can be influenced by varying the water flow, is investigated by experimental identification with binary multifrequency test signals. It will be shown that it is possible to describe this non-linear distributed parameter system using linear non-parametric and parametric models. Both correlation analysis and direct leastsquares parameter estimation are employed in order to do this, and the identification results are compared with each other. Furthermore, the necessary signal processing uSing antialiasing filters and digital highpass filters is described, as well as the evaluation of the qual ity of estimation results with the aid of the coherence function. It is shown that the identification errors of the non-linear heat exchanger decrease when the appropriate choice for characteristics is made for the binary multifrequency signal. Keywords. Identification; crossflow heat exchanger; non-linear distributed parameter system; binary multifrequency test signal; antial iasing filter; highpass filter; correlation analysis ; coherence function; direct least-squares parameter estimation.
INTRODUCTION Crossflow heat exchangers in air-conditioning systems are subject to frequently changing operation points and therefore it is necessary to adapt the heating power to the level requi red in each case. In general, heat exchangers can be described as non-linear systems with distributed parameters. Knowledge of their dynamic behaviour is therefore important e.g. for the design of effective automation concepts, for the determination of (sub-) optimal controller parameters, or for val idation of theoretically calculated models (Franck, 19B5). In thi s paper the dynamic behaviour of a 1arge water-heated crossfl ow heat exchanger during normal operation in an air-conditioning system for an office bui 1ding is invest igated. A few important technical data of the heat exchanger are summarized in Table 1. TABLE
1 iJata of the investigated crossflow heat exchanger
by correlation analysis for the non-parametric model and by direct least-squares estimation for a parametric model of the heat exchanger. The nonparametric model provides a particularly impressive picture of the process in graphic form, whereas the parametric model can give a very cl ear overall description of the heat exchanger at different operation pOints by means of a few parameters of the 1 inear discrete time process mOdel. It is shown that the coherence function presents a suitable criterion fol' eval uation of the qual ity of the estimated results. Thi s function is further used as an aid in order to optimize the characteristics of the test signal s used in prel iminary experiments with the purpose to achieve to good identification results. EXPERIMENTAL SET-UP For identification of the dynamic behaviour between the water flow and the average ai rout 1et temperature a measuring system was installed at the heat exchanger. This system is schematicall y illustrated in Fig. 1.
Width 1. 75 m 1.45 m Height 0,16 m Air flow path 19 . 00 m Water flow path Hea t i ng power 600,000 kJ/ h Volume flow rate of the air 27 , 000 m3/ h Nominal temperatures: -12 °C Air inlet 5°C Ai r outlet 100 °C Water inlet 70 °C Water outlet For identification purposes only binary multifrequency test signals were used, and the response of the local mean value of the air outlet temperature to changing water flow is investigated. At first the experimental set up and important c harac teristics of the test signal used are described. Finally, the data processing using antialiasing filters and digital highpass filters, whi c h is necessary because the measured signals are disturbed, is described. The evaluation of the data has been done
Fi g.
Ex perimental set-u p
IH60
(; . Franck a nd H . Rake
The test signal Y which activates the system is given out by a process computer as analogue voltage and transferred to the pneumatic valve via a voltage to pressure converter. In thi s way the water flow through the heat exchanger is changed. A built-in inductive magnetic flow meter produces an output voltage which is proportional to the velocity of the water Vw. The input signal referring to the maximum velocity VWN is chosen for the identification U
=~ VWN
(1)
: "' urnunmu I I Umi n i t . .
2 ••
n-
Fig. 2 One period of binary multi frequency test si gnal Table 2 seruences of Signs of the used binary mu tlfreguency test slgnal
Thus the resul ts become independent from the val ve characteristics and from the dynamic behaviour of the heating water supply. However the measured signal still contains the dynamics of the flow meter. The response of the distributed air outlet temperature to the test signal is approximated by the mean val ue of the temperatures measurea with the eight electronic temperature sensors
8 ALA = 1. • L ALAi • 8 i=1
(2 )
Therefore the sensors were placed in the air current at equidistant points along the flow path of the water. Further sensors for the control of boundary conditions were installed to measure the air inl et temperature ALE' and the water temperatures A and AWA • A process minicomputer MODC(»\P MODACS 11Wf was used for the test signal output, the data acquisition, and the monitoring of the experiment. For data storage and off-l ine analYsi s a master process computer MOOC(»\P CLASSIC 7870 was used, which is linked to the minicomputer. BINARY MULTIFREQUENCY TEST SIGNALS Due to some advantages over other test signal s it was decided to stimulate the investigated heat exchanger by using binary multifrequency test signals. It is particularly advantageous that the signal amplitude is limited to two extremal values i.e. the signal is binary, and that the number, pOSition, and power of the corresponding sinusoidal signal components can be chosen with few 1 imitations. While the bounded signal amplitude is particularly important for the identification of non-linear systems using linear methods, high test signal power has an influence on the available accuracy in the case of disturbed systems and therefore also on the identificaton time. Opposite to the application of mono frequency signals, the heat exchanger investigated is stimulated at several frequencies, however it is only necessary to wait for one transient. After preliminary experiments, the signal, one period of which is plotted in Fig. 2 as an example, was taken from a large signal 1 ibrary of about 40 binary multifrequency signals (Paehlike, 1980). The chronological course of the signal is included in Tab 1e 2 as sequences of si gns, whereby the number of elementary intervals (equal to the sampling time) between the change from max ima 1 to mi n ima 1 value is specified. Figure 3 shows the power density spectrum of the si gnal. The synthesi s and further c haracteristics of binary multi frequency signals were described by Paehlike and Rake, (1979).
"'''1 Su~JIIIIIJ •
I, .. 25
"""I,. '" "1,,,, 5B
Fig. 3 Power density spectrum of the test signal
lul
The signal amplitude was calculated for each operation point investigated with the relation empirically developed in preliminary experiments IUI = 0,27 • U~ + 0.022
(3 )
Finally, the analogue output voltage to the pneumatic val ve was determined by means of the experimentally measured static valve characteristic Yo = f(U o )· DATA PROCESSING In order to st imul ate the heat exchanger in an interesting frequency range an elementary interval of the binary multifrequency signal is required, and therefore a sampl ing time of T = 10s was used. To avoid aliasing errors by sampl ing the analogue measuring signal an antialiasing filter with very low cut-off frequency and hi gh dampi ng in the stop band is therefore necessary. Here the antial iasing filter was realised by an analogue and a digital part. At first the analogue measuring signals were sampled for the identification, admittedly with higher sampl ing rate than necessary, but nevertheless in such a way that Signal s components above the respective f'iyqui st frequency were damped by at 1 east 40 dB due to the lowpass behaviour of the measuring sensors and converters. Finally the data were filtered digitally. Because of its high signal dampi ng in the stop band and it s zero pha se shift, a finite impul se response digital filter with a non causal symmetric weight function was designed by use of a Blackmann window (Oppenheim, Schafer, 1975 ) g(n)=0.42+O.5 cos(2!.fl.)+O.08 cos(l!ill.) -M
M
M
for this purpose and realized with the window 1 ength N = 2M + 1 = 31 according to the convol ution
WattT-Heatccl ( :rossllo\\' I kat M
g(O)a(k)+ S(k)
2 g(n) (a(k+n)+a(k-n))
n=l
(5 )
The frequency response of thi s fi 1 ter is shown in Fig. 4. At the convolution length M = 15 and the short sampl ing time T = Is the first zero of the fi 1 ter frequency response was chosen to coinc ide with the ~qui st frequency of the enl arged sampl ing time on the digital computer at T = 10s. Thus the digital lowpass filter covers the frequency range betvleen the ~qui st frequency of short sampl ing time and the Nyqui st frequency necessary for analysis.
~
.,
..., 0.'"
~
...., • 1!In1
.. Ill
. . .1
1.1
,-'
I H()I
the transient response time to the highpass filter, which is the slowest part of the system, until it comes to a steady state. The steady state is thereby important for the development of reliable estimations. In order to el iminate the influences o f the digital filters in the evaluation both the input and output signals were filtered. The data thus produced contain therefore only those signal components, which have significant power in the frequency range stimulated by the periodic binary multifrequency test signal. IOENTIFICATIUN In order to evaluate the filtered test signals and average air outlet temperature signals a non-parametric and a parametric procedure were introduced. Thi s was done because we were not sure whether a distributed parameter system could be described precisely enough by a rational transfer function. Thus for small deviations from an operating point the 1 inear non-parametric model should yield the best possible description of the non-linear heat exchanger with distributed system parameters.
I
Correlation Analysis
w-
Fig. 4 Magnitude of the digital lowpass filter Low-frequency disturbance, steady-state values and drift ing were el iminated by means of a digital highpass filter. An infinite impul se response digita 1 fi 1ter of second order wa s introduced for thi s (Oppenheim, Schafer, 1975). The cut-off frequency was chosen to be equal to the basic frequency w1 = "/1000 s-l of the binary signal. The filter works according to the relation Y( k)=PY(k-l )+QY(k-2 )+R (S (k) -2 S(k-l)+ S (k-2))
(6)
with P Q R
E~thallgn
0.9780 -1. 9556 0.9566
The orthogonal correlation, identical to the Fourier analysis of the signals, was used to estimate the non-parametric model. For thi s purpose the Fourier transformation was executed with data segments of the signals. The segments were not shifted aga in st each other and had a 1ength equa 1 to one period of the test signal. The computational effort is thus reduced due to the fact that Fourier coefficients of every period only have to be calculated at n frequencies '"k which are activated by the test signal. The expected val ues of the Fourier coefficients are approximated by using the arithmetic mean va 1 ue of the coeffi c i ents of a 11 per i od s m measured so that the frequency response of the investigated heat exchanger can be defined as
(7)
The frequency response of this highpass filter with the sampl ing time T = 10s is plotted in Fig. 5. 11
Coherence Function
,
., L.,
I1
Since the measuring time is finite, errors arise from the disturbance Signal and affect the accuracy of the results just as the non-l inearity of the heat exchanger does.
/
...., [7
..
"'0 ,
..
-110
-,
The following coherence function was chosen as a measure for evaluating these errors :
'""
(9 )
w-
Fig. 5 Frequency response of the digital fi lter
highpass
In general a transient response time is necessary when working with filters, particularly when highpass fi lters with very low cut-off frequency are involved. As a rule this leads to a loss of data in the off-line evaluation. It is however possible to operate these digital filters on-line and to adapt
Thi s can be cal cuI ated from the expected val ues E of the estimated power spectral S( w). The coherence function for the frequencies wk considered here WhlCh are contained in the test si gnal is approx imately detenni ned by
m 1;11 (a0i+ jb Oi) (aUi -jb Ui ) 12
m
m
( 10)
i l l 1aUi +j b Ui 12 i 111 a0 i +j b0i 12 the deriviation of this is given in Paelike, Ra ke (1979). It is thus available as a c riterion whi c h
(;. Frallck alld I-I. Rake
IH(i~
not only allows to frequency response the characteristics large the measuring
determine the quality of the estimation but al so to change of the test signal, or to entime.
and z-transformation correspondences. This transfer function can then be plotted in the Bode diagram as frequency response F(jw).
In this way the test signal ampl itude, for example, can be enlarged if the coherence function has too small values for all frequencies. In doing this a comprom i se mus t be found between the mea sur i ng time and the test signal ampl itude because the process being investigated has non-linear characteristics. If the coherence function has a small value only for some specific frequencies, an improvement can be reached by choosing another binary multifrequency signal with high signal power of the correspondi ng frequency. Fig. 6 Model spl itted into parall el branches RESULTS Least-squares parameter estimation A parametric description is chosen for the determination of the dynamic behaviour of the investigated heat exchanger, which depends on the operating point. The few characteristic data of the 1 inear discrete-time model
(11 )
n 1+
L a. ·z-l
i=l
1
thereby give an excellent survey of the operating point dependance on the process. tt is guaranteed by the activating test signal that the linear range of the non-l inear heat exchanger is not or only insigni ficantly exceeded.
First the behaviour of the heating water supply was investigated. The steady state relation of the pneumatic valve and the pipes between the test signal Yo and the water flow Uo is depicted in Fig. 7 as a measured characteristic curve. An example of the dynamic behaviour of the water flow is shown for an operating pOint Uo = 0.54 as a frequency response in the Bode diagram, Fig. 8. It is recognised that the water flow responds to the test signal approximately proportionally with a dead time. Thus the dynamics of the water flow have a negligible influence on the overall behaviour of the heat exchanger, in the same way as the dynamics of the air temperature meas1lri"0 elements investigated in preliminary experiments.
U 0.8
The parameter estimation was excecuted using the comput i ng-t ime-sav i ng-procedure with the di rect I east-squares method desc ribed by Breddermann (19Bl) for a set of model s of 3rder n and dead time The systematical bias is small, because the disturbance part in the measured output signal is low due to the high test signal power and because of estimation with a model of enlarged order.
0,6 0,4
a·I.
The evaluation of the models with 1<;;<4 and time delay 02a~6 is done off-line. The bestmodel order and tllne delay were determined from these 28 model combinations by finding out the model with minimal equation error loss function. The thus resulting model order was = 3 and = 4 in all cases. A reduction of the order could be achieved by a decomposition of the model. The model is thereby spl it into parallel branches, Fig. b, by determining the partial fractions of the rational component of the z-transfer function (Eq. 11). Branches with a comparatively small steady state value
n
0,2
6
8 V 10
Y --
Fig. 7 Static relation between test signal Yo and water fl ow Uo
n
" 1' 1
0. 1
Ri
R·
=_1_ l-z Pi
(12 )
could be neglected 1n th1S way. The same appl1es for conjugated complex poles ZP1 and zPi+l' Thus a satisfactory description of the heat exchanger could normally be achieved by the model order = 1 with a dead time about 10s.
n
...
1. 21 \
o·
--.......
- 27111 -
111. 0 1
In order to be able to compare the estimated parameters of the discrete-time model with the results of the Fourier-analysi s in the Bode diagram, a transformation to a time-continuous model was executed. Starting from the description of a continuous process using a first order holding element, the Laplace-transfer function of the heat exchanger G(s), and a sampl ing element, the continuous transfer function G(s) can be determined by the Laplace-
111. 1
. -'
I
Fig. 8 Frequency response of the water flow In Fig. 9 one period of the test signal, the measured lowpass filtered air outlet temperature over 10 periods, and the high pass filtered air outlet temperature are plotted. The drift, which is due to a slowly rising air inlet temperature, is thereby el iminated.
lHfi3
Water-Heated ( :rossllol\' Heat Exch;llIger
~~~:gll Ila
I
2ae
n --
the coherence-function Y~O(~) was greater than U.Y for the majority of the actlvating frequency components. Generally thi s could be achieved by increasing the test signal ampl itude and keeping the identification time constant. Figure 12 shows the corresponding frequency responses of the di screte time models of 1st order to 4th order with the dead time between zero and 10s in comparison to the non-parametric model of the Fourier-analysis. Fourier-analysis
lowpass filtered 17.1
15. 1 '.I!II I---t-----f----'~..-I
21ea
1511
n -a. eel
~=~~=:::::==~
f _~:~--j---,----1 ~ -. u·~-~~~~~~~ -36.- L-_-.-J~.!.!.::.=_.J.~_>_-.J
Isaa
IDaa
•. 001
2Dli!la
n-
Fig. 9 Test signal and air outlet temperature at Uo = 0.29 The result of the Fourier-analysis is shown in the Bode diagram in Fig. 10 using four experiments as exampl es. In order to be able to eval uate the quality of the frequency response ~~surement, the corresponding coherence function Yeu(~) is depicted in Fig. 11. .1
I
IFI
UOOO,o"", 0,35 -
0" , -
0,111
L.
~
...
-
.........
I..
e. eD)
I
Fig. 12 Compari son of Fourier-analysis and parameter estimation of the models 1st order to 4th order The model of first order describes the investigated heat exchanger insufficiently in general, considerably better results could be achieved by choosing the order of the model s = 3 and 4 according to the procedure described above. Thus the process characteristics in dependance on the setpoint can be described by the dominatinq parameters Rand T1 of the respective s-transfer function (Fig. 13, 14). The dead time varies between 0 and 10s and is not prese!'teq separately. In non-linear systems the results K, T1 cannot be appointed to one value of h! water flow Uo due to the test signal ampl itude U and so the confidence area of the parameters In h diagram is valid.
n
n ..
4,0
K
~~ u.~
9LAN-9LEN 3 , 0
2,0 ' . 81
LI
"
I
,
.. Y0U
,' 1
w-
I
Fig. 10 Frequency response of air outlet temperature at several set points
1 '2 •
11. 1
I
"\
0.1 •
I·
e.ll l
1,0
O,O +-~~-~~-r-~~~~-'---'
0,0
I I a,al
"'" e, •
C,I
Fig. 13 Gain K in dependance on Uo for the crossflow heat exchanger
.-'
•
Fig. 11 Corresponding coherence-function Y~ (~) at several set points The parameter estimation showed that convenient and interpretable results, which could be compared with the Fourier-analysis, were only achieved, when
(;. FralHk and 1-1. Rake
IH(i4
O +-~~~~--.-~~-r~-,
0,0
0,5
1,0
uo
Fig. 14 Time constant T1 in dependance on Uo for the crossflow heat exchangr SUMMAR Y For the identification using binary multi frequency test signal s of the heat exchanger investigated a measured data processing with an antialiasing filter and a highpass filter was necessary. The limited signal ampl itude and process excitation in a wi de frequency rang e with hi g h si gna 1 power of the test signal made a satisfactory identification possible. The evaluation with the aid of the Fourier-analys is and the di rect 1east-squares parameter estimation yielded corresponding results in the stimul ated frequency range. The coherence function proved to be hel pful and expedient as criterion for the qual ity of the resul ts and choice of the test signal. The identification of a non-linear system using linear models is generally possible when important boundary conditions are taken into consideration.
REF ERE NC ES Breddermann, R. (1981). Ei nsparurlS von Rechenaufwand bei der off-l ine Parameterschiitzung mit Modellstrukturbestimmung. Regelungstechnik, 29, pp. 380-386. Franci<:" G. (1985). Zum dynamischen Verhalten wasserstrom estellter Kreustromwarmeubertra er 1 n K lmaan afen. Fortsc rltt-Berlc t Rel e 8, to be pub ished, VDI-Verlag, Diisseldorf. Oppenheim, A. and R. Schafer. (1975). Digital Signal Processing. Prentice-Hall Inc., Englewood Cliffs. Paehl ike, K.-D. (1980). Regelstreckenidentifikation mit binaren Mehrfreguenzslgnalen. Dlssertatlon, RWTH Aachen. Paehlike, K.-D. and H. Rake. (1979). Binary M.Jltifrequency Signals-Synthesis and Application. 5th IFAC Symposium on Identification and System Parameter Est imat i on, Pergamon Press, Oxford.