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Practical Aspects of Process Identification
0005-1098 (80(0901-05 75 $02.00(0
AII/rmwriw . Vol. 16. pp . 575- 587 Pergamon Press Ltd . 1980. Printed in Great Britain {) International Federation of Automatic Control
Practical Aspects of Process Identification * R. ISERMANNt Key Words- Identification: input signals; sampling time: comparison of identification methods: model verification: program packages.
Abstract-After the presentation of various identification and parameter estimation methods in the previous papers, some selected practical aspects of process identification are discussed. This includes, for a given identification method, the steps from the design of experiments to the verification of the final model. Therefore a general procedure of process identification, the selection of input signals, the selection of the sampling time, off-line and on-line identification, comparison of parameter estimation methods, model order testing and model verification is presented. A short discussion on program packages for process identification follows.
TABLE THE
1.
SOME EXAMPLES FOR THE RELATIONSHIPS BETWEEN
FINAL
GOAL
OF
THE
MODEL
APPLICATION
AND
SPECIFICATIONS OF THE PROC ESS IDENTIFI CATION
Type of process model
Final goal of model application
I. INTRODUCTION THE PRECEDING tutorial papers have been maInly concerned with identification and parameter estimation methods. that means roughly the various procedures to extract mathematical process models from measured noisy input/ output data. However, to apply process identification to a real technical or nontechnical process requires many additional (more practical) preparations, which becomes obvious if the general procedure of process identification is regarded, Fig. l. It is important to regard first the final goal for the application of the process model, since this determines the type of the model and its accuracy requirements and the identification method . Table 1 shows some examples for the relationships between different final goals and some specifications of process identification . The (/ priori knowledge of the process is for example based on the general process understanding, on physical laws of the process and on premeasurements and gives hints on linear or nonlinear, time-invariant or time-variant, proportional or integral behaviour, settling time (dominant time constants), time delays and noise characteristics. By these features and the process operating conditions, the design of experiments and the selec-
Required accuracy of model
Identification method off-line, step response, frequency response, parameterestimation
verification of theoretical models
medium/high linear/ continuoustime nonparametric/ parametric
controller parameter tuning
linear, nonparametric, continuoustime
low for input/output behaviour
off-line step response
computer aided design of digital control algorithms
linear, parametric (nonparametric) discretetime
medium for input/output behaviour
on-line/ off-line parameter estimation
self-adaptive digital control
linear, parametric, discretetime
medium for input/output behaviour
on-line parameter estimati on in closed loop
process parameter monitoring and failure detection
linear/ nonlinear, parametric, continuoustime
high for process parameters
on-line parameter estimation
tion
of the
identification
method
IS
directly
influenced. The design of experiments includes selection and determination of : input signals (amplitudes, spectra); sampling time; measuring time period (identification time); open or closed loop identification; off-line or on-line identification; equipment for signal generation, data storage and computation; signal filtering [high and low frequencies (drift )]. taking into account constraints given by the operating conditions of the process, the properties of the actuators (proportional or constant
'Received March 31, 1980. The original version of this paper- was presented at the 5th If AC Symposium on Identification and Parameter Estimation which was held in Darmstadt. federal Republic of Germany during September. 1979. The published Proceedings of this IfAC Meeting may be ordered from: Pergamon Press Limited_ Headington Hill Hall. Oxford, OX3 OBW, England. This paper was recommended for publication in revised form by the Automatica Editorial Board. tlll$titut flir Regelungstechnik , Techni sche Hochschule Darmstadt. Darmstadt. F.R. Germany. 575
R . IsERMANN
576
PROCESS PHYSICAL LAWS PREMEASUREMENTS { OPERATING CONDITIONS
TASK } ECONOMY
NO
1'1(;.
I. General procedure of proee" ilkntificatiol1
speed action; nonlinearities) and the sensors. Some of these topics will be regarded later in more detail. DiftCrent ways of illput sigllol generatioll, 0111pUI siglllll l1!ellsurl'l1!l'lll and datil storage are shown in Fig. 2. Table 2 shows the main classes of idelll if/catioll lIli'thods which are mostly applied on the measured data. The identified process model finally has to be railled, e.g. by comparing the measured output with the output calculated by the model. Depending on the result of this verification the model structure (order, time delay etc.) has to be changed (model structure search) or the ex periments have to be repeated.
TABLE
2.
MAIN CLASSES OF ID EN TIFICATION METHODS
Model Identification method s step response techniques
non parametric
parametric
x
x
x
x
frequency respo nse techniques Fourier- and spectralanalysis
x
correlation techniques
x
parameter est imation
x
The scheme in Fig. I indicates that process identification in general is an iterative procedure. In the following sections some of these practical aspects of process identification will be discussed.
577
Practical aspects of process identification
J
-4
PROCESS
H cm I RECOR DER
(a )
(b )
IGl iwll 't'lw)
MAGNETIC TAPE
signal has to be sufficiently rich to excite all process modes of interest during the experiments. This leads to the requirement of persistently exciting input Signals. A maximum requirement would be that the input signal be constructed to minimize certain model errors with respect to noise, input and output signal constraints and measuring time. This leads to optimal input signals. First some often used identification methods are considered with respect to the minimal requirements on the input signals. If the F ourier or spectral analysis is applied to identify the frequency response by
' 'G(iW)=F{y(tj} = Y(iw) F{ u(t)} U(iw)
(1)
or
(c) PAPER TAPE
(2 )
FIG. 2. Different ways of input signal generation, output signal measurement and data storage (a) Simple process identification by manually changing the input and data storage by a recorder. (b) Frequency response analyser (using orthogonal correlation for harmonic inputs) (cl Off-line identification using a signal generator and a magnetic tape or punched paper tape as storage (d) On-line identification using a process computer.
2. SELECTION OF INPUT SIGNALS
To identify the controllable and observable (identifiable) part of a dynamic process the input signal has to satisfy certain conditions. A minimum requirement is that the dynamics of the process have to be excited persistently by the input signal over the measurement period. Roughly speaking this means that the input
where
I
g(v)
(3)
\1= 0
has to be used as a basic input/output relationship which leads to the system of equations
<1>uu( - P + I)
g(O) g( 1)
<1>U)'( - 1)
<1>u..! - 1)
<1>uu( - 1 -I)
<1>u)'(O)
<1>u.(O)
<1>uu( -I)
<1>uy( 1 )
<1>uu(i )
<1>uu(i - I)
(4)
g(l ) I A UTO- f
R.IsERMANN
578
For M=l and P=l the (/+1) unknown g(v) can directly be calculated from (/ + 1) equations by
and one imaginary part Im(wJ of the frequency response and therefore two parameters. Persistent excitation therefore means that the input has to have at least j ~ (n + m + 1 )/2 harmonics, van den Bos (1970). Now the parameter estimation for a linear difference equation or for the z-transfer function
The requirement for persistently excltmg input signals in this case is that the inverted matrix 4'::;.1 exists, hence 4' •• may not be singular
(9 )
G(z)
det4' •• oFO.
(6) is regarded. For the nonrecursive method of least squares the estimation equation is
4' •• is symmetric if all
tP •• (r)=
1
lim N- co
N
(10)
N- I
L
u(k)u(k+r)
(7)
with the parameter vector
k=O
do exist. A further discussion for persistently exciting input signals for correlation analysis is given later. If the sampling time To is sufficiently small, equation (3) and the results given above can directly be transferred to the case of continuous signals.
<1>~y (0 )
and 'I' and y being a matrix and a vector containing the measured inputs and outputs, see e.g. Strejc (1980). After dividing ['I'T'I'] by (N + 1) the elements of this matrix are estimates for correlation functions.
<1>~" (1 )
<1>;'v(m - 1)
<1>~y(O)
<1>~y(m-2)
<1>~. (0)
I
- <1>:y(d)
- <1>~. (d + m - I )
-<1>:y(d-l)
-:y (d +m-2)
, - <1>:y(d - m + 1 )
---- - - - --------- --- - -- - - - ----: --
---$~)O)--
---- ---- --
- <1>:y(d) -:J~--=-l-)
----
<1>:.
If the parameters of the continuous-time lrans~ fer function G(s)
b o + bls + .. . + bms m 1 +als+ .. . +ans n
(8 )
have to be estimated by parameter estimation methods by applying periodic (multifrequency) input signals each harmonic with frequency Wj can be used to determine one real part Re(wj)
HII i-·! H12j = [----------:- = H H 21 ;H 22 '
(12)
This matrix is quadratic and symmetric for large N, if the input u(k) is a stochastic or periodic signal with constant meanvalue. To solve equation (10) it must be required that ['I' T 'JI] is not singular, hence detH+O.
(13)
579
Practical aspects of process identification Two cases are possible : det H > 0 or' det H < O. Regarding the successive principal minors of this determinant, it follows, starting with the lower right corner
•u..(r)=a 2
r=O, NJe, 2NJe, .. .
for a2
(r)= - u" N
for
Je(1 +vN)
(14)
1_
H = 1
v=O, 1,2, . .. (PRBS with amplitude a, clock time Je = To and period N . Persistently exciting of order m, if N=m+ 1).
if N is large [ 0 and H 2 > 0, the matrix H cannot be negative definite according to the criterion of Sylvester. Therefore the matrix H has to be positive definite, if equation (3) is to be satisfied. This means also that det H > 0 and all successive principal minor determinants Hi =det Hi > 0. Therefore the condition for persistent excitation is, that
From Fourier analysis it is known that a necessary condition for the existence of the power spectral density
(16)
is, that for all O-:5. w -:5. njTo, 0 for all w, persistently excitation is satisfied for any order, Astrom and Eykhoff (1971). Persistent excitation of finite order implies that Suu(w) may vanish for certain frequencies (like for the Fourier transform of pulses). These examples show that the conditions for persistently exciting input signals are similar for different identification methods and can be transformed to each other. Theoretical investigations on optimal input signals are mostly performed by assuming an asymptotically efficient pa rameter estimation (e.g. maximum likelihood) so that the parameter covariance matrix can be approximated by the inverse of the Fisher information matrix, i.e. the Cramer- Rao lowerbound. Optimal inputs then result from minimizing a scalar function of the inverse of the information matrix, e.g. the trace (sum of diagonal elements) or the determinant, see e.g. Aoki and Staley (1970), Mehra (1974). However, the results hold only for special cases, as efficient estimators and long measurement times. In practice optimal inputs can be required if the experimentation time is very limited because of operating conditions or costs (e.g. aircraft flight tests).
or the matrix
has to be positive definite. In addition
_ 1 U= lim N ~ oc,
N
N- I
I
(18 )
U(k)
k=O
and N- I
I
[U(k)-O][U(k+r)-O]
k= O
(19) must exist, as it was already noted. The con'ditions (17)-(19) have been given in Astrom and Bohlin (1966) for maximum likelihood estimation and are called " persistently exciting of order m". The conditions for persistently excitation for correlation analysis, equation (5) and (6), are the same with exception of the different order I and m. Therefore input signals do exist which are persistently exciting for both identification methods. Special cases of persistently exciting inputs of order m are signals which satisfy
• . .. > 1>"u(m) (coloured noise, if
•
+0;
111 ->
XJ )
uu(2) = ... = 1>uu(111) = 0
(white noise, if m-> x )
a:>
I
Suu(w)= 1: =
-
a:>
=
I
t=
(20)
3. SELECTION OF SAMPLING TIME
F or the identification of discrete-time models the sampling time has to be selected properly before the experiment starts, because it cannot be decreased afterwards, if the sampled signals are
580
R . IsERMANN
stored. Of course, the sampling time can be doubled, tripled etc. by deleting corresponding data . The sampling time To depends primarily on: (a) Sampling time for the final application (e.g. digital control) (b) Accuracy of the resulting model (c) Numerical problems, if To is too small. These points will be discussed as follows : (a) If the final application of the identification is the design of a digital control system, the sampling time for digital control, as for example the control performance, the type of system. Many characteristics influence the sampling time for digital control, as for example the control performance, the type of control algorithm, the spectrum of disturbances, the valve and its actuator, the sensors and the computational burden on the digital processor, Isermann (1977, 1980). As a rule of thumb 6 to 15 samples per 95 % settling time T95 are often a good choice for PIOcontrol algorithms, i.e. T95 / To = 6 ... 15. (b) To discuss the effect of the sampling time on the absolute values of the parameters an example is shown in Table 3. The gain is
K = Lb;/(l +LaJ
TABLE
3.
PARAM ETER S
OF
THE
(21 )
z-TRANSFER
WITH ZERO-ORDER HOLD FOR DIFFERENT SAMPLING TIMES To .
4
8
16
az a3
-2.48824 2.05387 -0.56203
- 1.49863 0.70409 -0.09978
- 0.83771 0.19667 -0.00995
-0.30842 0.02200 -0.00010
bo bt bz b3
0 0.00462 0.00169 -0.00273
0 0.06525 0.04793 -0.00750
0.06525 0.25598 -0.02850 - 0.00074
0.37590 0.32992 0.00767 -0.00001
1
1
0.10568 0.10568
0.34899 0.34899
0.71348 0.71348
d
4
Lb i 1 +r.a i
0.00358 0.00358
A proper choice of the sampling interval in most cases is not critical, because the possible range between too small and too large values is relatively wide. Good experiences have been made with the rule T95 / To~5
... 15
(23 )
where T95 is the 95 % settling time of a transient function, Isermann (1974). 4. OFF-LINE AND ON-LINE IDENTIFICA nON
K = I ; Tt = IOsec; T2=7sec; T3 = 3sec; T4=2sec; T,=4sec
at
(c) If the sampling time is chosen too small, ill conditioned equation systems (matrices) result, because the single difference equations become approximately linear dependent (numerical problems).
FU NCTION,
C ORRESPONDING TO
To [sec]
Small errors in the parameters can then have a significant influence on the input/output behaviour of the model, since the value of e.g. Lb i depends highly on the 4th or 5th place of b i after the decimal point. If the sampling time, on the other hand, is chosen too large, the dynamic behaviour is described too unexact. For To = 8 sec the model reduces practically to 2nd order, because of hl ~ 11+Lail and Ib31 ~ ILbil and for To = 16 sec even to 1st order. This example shows that the sampling time should not be selected too small nor too large, because of accuracy problems of the resulting model.
With decreasing sampling time To the magnitudes of the ai-parameters increase and those of bi decrease. Therefore for a small sampling time, e.g. To = 1 sec, it is
(22)
With regard to the application of general purpose digital computers and process computers (or other equipments) a process identification can be performed off-line or on-line, Fig. 3. For off-line identification the process data is first stored in a data storage (e.g. recorder, mag- netic tape, punched tape). Later on the data is transferred to a computer and evaluated. For this type of identification mostly batch processing of the data is applied, that means, the whole data set is evaluated at once. Then direct identification algorithms (identification in one shot or one step, for example least squares estimation) and iterative identification algorithms (stepwise, for example maximum likelihood estimation) can be distinguished. In the case of on-line identification the process identification is performed in on-line operation with the process. Then, two ways of processing the data can be distinguished: Real-time processing, i.e. the data is evaluated immediately after each sample or batch processing, i.e. the data set is evaluated at once after all the measurements have been made.
Practical aspects of process identification
mated recursively and parameter estimation is done non-recursively. This is referred to as rea/-
$)
time/ batch processing.
Again the final goal of identification and the type of process and the available computing power determines the type of identification. If a not (too much) time varying process has to be identified for general process analysis or for the design of control algorithms, off-line identification often is preferred. For time varying processes and especially for adaptive control on-line identification with real-time processing, also called real-time identification, has to be used .
MODEL
~
+
MODEL
OFF-LINE IDENTIFICATION
ON- LINE IDENTIFICATION
1 ------------
BATCH
581
PROCESSING
REAL - TIME
1----------- --
NONRECURSIVE ALGORITHMS
RECURSIVE
PROCESSING
5. COMPARISON OF PARAMETER-ESTIMATION METHODS ALGORITHMS
/~
ITERATIVE
DIRECT
IDENTIFICATION
IDENTIFICATION
lONE SHOT, ONE STEP)
ISTEPWISE)
11( ,. 3. Various wa ys of processing the measured data for process identification.
Real-time processing in general needs no storage of data. Recursive identification algorithms of the form O(k+l)=O(k)+y(k)e(k+ 1)
(24)
where (J means the parameter vector, y a correcting (weighting, gain) vector and e an error signal, normally are used . The model develops as the data comes in. Time varying processes can be identified by these type of algorithms. For batch processing the data is stored (core memory, disc or magnetic tape) and evaluated at the end of the measurements. Mostly non-recursive identification algorithms are used in this case. The parameter estimation methods least squares, instrumental variables and maximum likelihood and also correlation techniques, Fourier and spectral analysis, are generally first derived as nonrecursive algorithms. They can then be transferred to recursive algorithms by subtracting two directly following estimates for times k + 1 and k. Recursive algorithms are well-known for the following parameter estimation methods : Least squares, instrumental variables, (approximate) maximum likelihood and also for correlation techniques. They have many connections to stochastic approximation and to state variable filtering and digital filtering . Also combinations between pure real-time processing and pure batch processing are possible. For example correlation functions can be esti-
Once the external conditions for process identification have been classified a proper identification method has to be selected. Since in most cases several methods can be applied to obtain a certain type of process model, a comparison of these methods can be of help, indicating the different advantages and disadvantages. Therefore some remarks will be given concerning the comparison of identification methods. Only parameter-estimation methods for linear discretetime parametric models will be regarded. What can be compared?
With respect to parameter estimation methods and given model order following items are important for the selection of a method : the performance with regard to the accuracy of the identified model (model errors) dependent on time (convergence properties); the computational effort (computer storage, computation time) ; the a priori assumptions and a priori factors to select. H (iit' c{ln comparisons he made? It is generally no problem to directly compare the computational effort and the a priori assump-
tions and factors. For example Table 4 shows the assumptions made for the noise models of different parameter estimation methods, which are required for unbiased estimation. The least squares methods presumes a very special noise model, l /A, which does not exist for real processes. The maximum likelihood method assumes a special nominator in the noise model (D/ A). The most general noise models (D/ C) are allowed for the instrumental variables method and for correlation combined with least squares. The comparison of the performance is much more involved . If more than just two methods have been compared, it has been mainly done by
582
R.IsERMANN
TABLE 4. PROCESS AND NOISE MODELS FOR UNBIASED PARAMETER ESTIMATION AND SPECIAL ASS UMPTIONS FOR DIFFERENT PARAMETER ESTIMATION METHODS
.\'11
Parameter estimation method
Process model
Noise model
....
- _\"
Special a-priori assumption
Autoregr . process Moving average process
I/ C Least squares (LS)
D B/A
Stochastic approximation (STA) General least squares (GLS) Maximum likelihood (ML) Extend. least squares (ELS) Instrum. variable (IVA) Correl. and least squares (COR-LS)
Remarks
I/ A Only recursIve
B/A
8/ A
8/A
I/A
I/AF
D/A
B/A
D/A
BIA
DIC
8/A
D/C
e resp. 11 normal distribut.
simulations, e.g. Saridis (1974), Isermann et ul. (1974). This means that a certain process model was selected and simulated together with a certain realization of a stochastic noise process. Test cases have been proposed for this purpose and several authors presented their results, Isermann and colleagues (1973). Of course, questions come up how many runs have to be taken and if this way is realistic enough. For some nonrecursive methods like the least squares, weighted least squares or maximum likelihood method, an analytical comparison can be made by directly comparing the variances and covariances of the parameter estimation methods. However, the covariance matrices can only be given for special assumptions on the error signal and for a relatively large identification time. An analytical comparison of different recursive parameter estimation methods has been made by Soderstrom and colleagues (1978). Based on an unified recursive estimation algorithm the asymp-
totic behavior is described, i.e. the results are only valid for rather long identification times. The third way for comparisons is to use the measured data of a real process. However, also this way has some drawbacks. Mostly the exact model is not known and the process behaviour and the noise characteristics often change with time, so that the results are rather special. This discussion shows that there seems to exist no unique way which results in a clear comparison of the most important performance measures of parameter estimation methods . Therefore all three ways, simulations, analytical studies and applications on real processes have to be undertaken to come to more general results. A further difficulty arises if the error between the process P and its model M has to be selected. Generally following errors are regarded: Parameter error:
!18; = 8iO - (];
Output error:
!1y(k)= yp(k)- YM(k);
Equation error:
elk);
i = 1,2, ... ;
Input/ Output behaviour error: e.g. impulse response error !1g(r) = go(r) - g(r). Furthermore these errors may be represented as: absolute values ; relative values; averaged values (mean squared over i or k or r). So a great variety of error measures does exist. The selection of an error measure can influence the final results of a comparison. For example small input/output behaviour errors may be accompanied by large parameter errors, especially for higher order models . The final goal of the application of a model should decide which model error has to be used . It is, of course, important that the comparisons are made for the same process, the same noise and the same input and the same measuring time. In the following an attempt will be made to summarize results which have been obtained by simulations, Isermann and colleagues (1973), by analytical investigations, Soderstrom and colleagues (1978) and practical experience, Baur (1976). Some results of comparisons (a) Overall comparison of the performance. For long identification time the parameter estimation methods with unbiased estimates for noise filte'rs D/e and D/ A (IVA, ELS, ML, eOR-LS) lead to approximately the same accuracy of the input/ output model. For short identification all methods, including LS, show little difference. However, in this case the nonrecursive method s
Practical aspects of process identification are better than the mentioned recursive methods. the estimates of the process parameters generally converge quicker than the estimates for the noise parameters (since in the first case the input is known). (b) Properties of important methods. (For abbreviations of the methods see Table 4, R stands for Recursive). Least squares (LS) Biased estimates for real noise applicable for short identification time if noise acts on the process sensitive to unknown Yoo relative small computational expense good starting method for IV A or ML a priori factor: 1 matrix PtO) RLS: reliable convergence RELS : similar as RML. Generali zed least squares (G LS) Biased estimates possible relative large computational expense noise model identified a priori factors: 2 matrices, filter order.
I nstrumental variables (J V A) Good performance for wide range of nOIse models medium/small computational expense insensitive to unknown Yoo if u(k) = 0 (/ priori factors : 1 matrix, filter factors RIV : no reliable convergence. Therefore start with RLS is recommended . Maximum likelihood (ML) Good performance for special nOIse models D/A
large computational expense noise model identified problems if local minima of loss function theoretically well funded a priori factors : dep. on optimization procedure RML: Slow convergence in the starting phase. Convergence more reliable than RELS . Parameters of the polynomial D( Z- I) are also estimated, however, slower convergence than for A(Z - I) and 8(Z - I). Correlation and least squares (COR-LS) Good performance for wide class of nOIse models relati ve small computational expense access to intermediate results
insensitive to unknown Yoo if u(k) = 0 search of model order : small computational expense easy verification
583
a priori factors: only number of correlation functions RCOR-LS: reliable convergence. 6. DATA FILTERING
The process input signal u(k)=U(k)-U oo
(25)
and the output signal y(k) = Y(k) - Yoo
(26)
generally have unknown d.c. values, U 00 and Yoo , and are disturbed by high and low frequent noise. Therefore the following steps have to be taken to eliminate those parts of the disturbances which cannot be eliminated by the identification method. I dentification of d.c. values Only the variations of the output signal with respect to variations in the input signals must be used for the estimation algorithms. If the input signal is generated by a process computer, U 00 is known . Yoo , however, is not known. Some parameter estimation methods need the exact knowledge of Yoo , some not. For example, LS needs exact Yoo , IVA only if E{u(k)} f O and COR-LS just a rough estimate (to avoid an ill-conditioned matrix), see Isermann (1974) and Baur and colleagues (1977). Yoo can be estimated just by averaging _ 1 Yoo = -
NG
L Y(k)
(27)
NGk = 1
before the identification of the dynamics starts. N G depends on the noise. Elimination of high and low fr equency noise Very high frequency noise is properly eliminated by passive or active analogue filters of first or second order. Various methods for digital filtering of low frequency disturbances, as e.g. drift, have been considered, including parameter estimation of drift polynomials, Baur (1976). It then turned out that simple high pass filtering is best suited. In order to avoid an influence on the identified process model, not only the output but also the input has to be filtered. First order high pass filters (HP) seem to be sufficient in many cases.
ii (k) = cxii (k - 1 ) + u (k ) - u (k - 1 ) J:(k) =cxf(k - l) + y (k) - y (k - l) where cx=e - T oi T :
To = sampling time T = time constant.
(28 )
584
R.IsERMANN
u( k) and y( k) are filtered signals used for para-
meter estimation. T has to be chosen so that the high pass filter, if possible, does not eliminate the lowest frequency of the input. For PRBS signals with clock interval A and period N the time constant should be
process with true order mo and true dead time m= mo and = do should result. The dead time can be simply estimated by assuming d = 0 and increasing the order m to at least mo + do. Then
a
do. In the ideal case
m
T> N PRBSA/ 2n.
IEII, 1621, .. .,16p l~ I 6; i= 1
7. MODEL ORDER TESTING AND MODEL VERIFICATION
and
As already discussed in section 1, model order testing and model verification form the last stages in the procedure of process identification. After obtaining an identified model, all a priori assumptions and simplifications and the performance of the model should be tested. In the case of parametric models in general the model structure has to be 'identified' also. For lumped single-input, single-output processes this reduces to the determination of the model order. Model order testing and model verification are very much related to one another, because some methods of verification are used as a feedback in the order search procedure. Before some methods of order testing and verification are discussed some general remarks are given which may underline the difficulties in the judgement of process models. Process models are mostly an approximate description of a part of a process. Therefore an exact agreement between a model and a real process cannot be expected. The evaluation of a model depends very much on the final goal of its application. Therefore the objective of the model has to be taken into account. A priori assumptions of identification methods often do not coincide with the situation for which an .identification has to be carried out, taking into account practical limitations. The experimental conditions for the identification (input signals, time variation of the process, noise characteristics, drift) may differ from the (unknown) conditions under which the model is applied (to control system design, optimization, monitoring, etc.). Therefore model evaluations on an objective basis are not the only way, but also heuristic and subjective components have to be included. 7.1. Model order testing The determination of the order of a parametric
single-input, single-output model of type
for f3 = do can be used as a criterion, Isermann (1974). The search of the dead time can, however, also be included in the general search procedure for the model order. Several methods for order testing have been proposed in recent years. The commonly used order testing methods can be divided into the groups. loss function tests pole-zero cancellation tests residual tests. These methods are summarized for example in van den Boom and van den Enden (1974), Unbehauen and Gohring (1974) and Soderstrom (1977). Only a brief description of some of these methods is given here. Loss function tests
The most simple method is just to regard the loss function V(m)=eT(m)e(m)
(30)
in dependence on the order m, where e is the equation error or residual of the used parameter estimation method, using the same input-output data. For m = 1,2, . .. the loss function for higher order processes first decreases until V (m) and then remains constant or changes slightly, Fig. 4.
0 .001 0 0,-
1 1
1 I I I 1
v
I
I 1 1 1 I
d=3
3
3
3
3
2..........
0
......... 12
.........
2 I
"
1 2
2 0
oooooL----------2.===£ ==-=--J5 1 2 3 4 m
_y(z)_ bIZ-I+ ... +bmz - m z z G p () u(z) 1 +alz I. .. +amz - m
means to estimate
mand
the dead time
d
(29)
aof the
FIG. 4. Loss function V(m , d) in dependence on the order m and dead time d for an identified heat exchanger (a distributed parameter system). Result: m=3 and .1 =0. (Baur, 1977).
Practical aspects of process identification Estimates m are found if V (m + 1) and V (m) do not differ significantly. This can be checked based on plots, van den Boom (1974), Isermann (1974). A statistical method to check if the loss function reduces significantly is the F -test. This test is based on the statistical independency of V(m 2 ) and V(md- V(m 2 ), which have X2 distributions for normal residuals. To test if the reduction of the loss function is significant when the order is increased from m l to m2 (number of parameters increased from 2ml to 2m 2 ) the test quantity t=V(m l )-V(m2) . N-2m2 V(m 2 ) 2(m 2 -m l )
(31)
is used . For large samples N the random variable t is asymptotically F[2(m2 - m l )], [(N - 2m 2 )] distributed. A risk level is defined and from the tabulated F-distribution the corresponding t* can be taken . If t < t* the probability of V(m 2 ) is smaller than V(m l ) and therefore m l is the estimated order, see Astrom (1967). Other order tests based on loss functions have been proposed by Wilks, Akaike and Parzen, see Soderstrom (1977). Pole-zero cancellation tests If for the parameter estimation a higher order m than for the real process with order mo is assumed (m - mo) additional pole-zero pairs result which at least approximately cancel each other. This effect can be used for order testing by calculating the roots of the polynomials B(Z-I) and A (Z-I) for different orders m, van den Boom and van den Enden (1974). Residual tests Some parameter estimation methods as LS, ELS, GLS, ML require a white noise of the residuals for the case of unbiased estimates. A whiteness test of the residuals, for example by calculating the autocorrelation function, indicates if this requirement is fulfilled or not and is therefore a general verification procedure for the parameter estimation. However, it can also be used to test for the right order, van den Boom and van den Enden (1974). Several methods for order testing have been compared in van den Boom and van den Enden (1974) and Unbehauen and Gohring (1974). However, only second order models have been used, for which the order search in most cases does not lead to decision problems. Soderstrom (1977) has shown analytically that some loss function and pole-zero cancellation methods are asymptotically equivalent. The practice shows that the order determination for single-input, single-output processes in general is not critical.
585
In many cases there is not one definite 'best' order, because of lumped higher order processes with many small time lags or distributed parameter processes which are approximated by finite order lumped models. However, the test for the structure of multi-input, multi-output processes is a highly difficult task. Blessing (1979) gives a survey and has proposed a method for rank determination of state-space system matrices for the case of noisy processes. For all nonrecursive or recursive parameter estimation methods it is required that the whole data set be stored and passed through several times until the right order is found. If, however, correlation functions are first calculated, leading to a nonparametric model, and the parameter estimation method is based on these correlation functions (method COR-LS) considerable storage and computing time can be saved, Isermann and Baur (1974). 7.2. Model verification Model verification means to check if the identified model agrees with the real process behaviour, taking into account the limitations of any identification method and the final goal of model application. This includes a check to determine if the a priori assumptions of the identification method used are true and to compare the input/output behaviour of the model and the processes, Isermann (1974). The verification of the a priori assumptions can be performed as follows : (a) N onparametric identification methods. Linearity: Comparison of models obtained
for different input amplitudes. Comparison of models with measured transient functions in both directions. Time variance: Comparison of models for different data sets. Noise : Is the noise n statistical independent from the input u? Checking the signals: Are there outliers? Drift : Averaging of output. Comparison of models with and without drift elimination. Input signal: Can it be measured without noise? Is it persistently exciting? D.c. values: Are U 00 and Yoo exactly known? (b) Parametric identification methods.
In addition to the already listed items, the following a priori assumptions should be tested . These, of course, depend on the estimation method. Residuals: Are the residuals statistically independent; CPee(')=O for +O? Are they independent from the input, CPue(')=O for all ,? E{e(k)} =O? Normal distribution?
1,1
586
R . IsERMANN
Covariance matrix of the parameter estimates: Do the variances and covariances decrease with increasing measuring time? Are they small enough ? A final overall judgement of the identified model is obtained by the verification of the input/output behaviour, i.e. by comparing the measured and the model predicted input/output behaviour. This can be made by (a) Comparison of measured yp(k) and calculated y(k) output signal: for the input u(k) used for the identification ; for other input signals like steps or pulses. (b) Comparison of the crosscorrelation function cPu/r) based on measured signals and on the model. For processes with no noise (or small noise only) (a) is straightforward. However, if the process is contaminated by noise the comparison of the outputs leads to ~ y (Z)=
yp(z) - y(z) Bp(Z) B(Z)] = [ -- - -~- u(z)+n(z) Ap( z ) A(z )
(32)
and the difference signal ~y( z ) depends on the model errors and the noise. If the noise model IS also identified, ~y( z ) reduces to ~y(z) =
Bp(Z) B(Z)] u(z) [ ----~ Ap(z) A(z) +[
D(z) v(z ) C( z)
n( z ) - ~-
A
]
(33)
Then the one step ahead prediction ~y(k) can be used, taking into account estimates of u(k - 1). A proper criterion, e.g. (34 )
can be used to compare different models. Another way is to compare directly the crosscorrelation function , case (b), since the effect of the noise is already removed, at least partially, for the crosscorrelation function from measured signals. Of course, this has to be accompanied by checking the a priori assumption for the crosscorrelation function . An additional way is cross-checking, that means to verify the identified model for another set of measurements. As the results of the identification depend on the input signals used, compare section 2, also this effect has to be taken into account for the verification, Ljung and Overbeek (1978).
Finally, the most important evaluation can only be given if the model is applied for the final purpose, i.e. for control, optimization, monitoring or just verification of a theoretical obtained model. For all these cases different requirements on the model have to be satisfied, see Table 1, and therefore different measures of model performance have to be applied.
8. PROGRAM PACKAGES FOR PROCESS IDENTIFICATION
Program packages for process identification have been developed which are complete sets of programs according to the various steps of an identification procedure. One has to distinguish: (A) Program packages for general digital computer applications III off-line (batch) operation (B) Program packages for process computer applications in on-line (real-time) operation. Packages of type (A) consist of, for example, data filtering, one or several methods for identification and parameter estimation, methods for search of model order, model verification and proper graphic presentation of the results. Packages of type (B) may consist of the same items, but in addition have to take into account real-time execution, process dependent interrupts, foreground and background organization, use of internal and external storages, signal generation, etc. A big advantage can be an interactive dialog with the operator. Fig. 5 gives an example for the procedure of a program package for on-line identification, Baur (1976) and Mann (1978). The identification methods need only 20 % of the length if the total package. During the 4th IF AC -Symposium on Identification and System Parameter Estimation in Tbilisi, 21 - 27 September, 1976, a round table was held on these program packages. Questionnaires have been used for the preparation of this discussion session. As a result 21 program packages have been described, and the main results can be summarized as follows . For all 21 packages a high level programming language was used , mostly FORTRAN (17), but also ALGOL (3) and PL/ l (1). Of these, 12 packages are written for process computers and nine for general purpose digital computers. In all cases linear processes can be identified, nonlinear processes in nine cases (mostly linear in parameters) and time-variant processes in eight cases (mostly slowly time varying). As process and noise models differential equations or s-transfer function s (4) or vector differential equations (1) (continuous signals), difference equations or z-
Practical aspects of process identification ON LINE
IDENTlFCATION 'OllD-SISO'
all the available methods. Good identification of real pro£.~sses is an art.
Method A
o
_______ F:l
r.=~~~~----L-,
P
~
---[[Ji. ~ "om" = U.,mtn
•
E R
Display
A
,
T
-
o
ir"_- ~"J_ ........... . -.. ...., .. ..." u
ll
.......
R ~--I--6'f-Tl 1 2 3 4
5 m
estimated
parameters
18 2 , ······m b1tJ, ·····-bm
8
PRINT
587
FIG. 5. Simplified procedure of the program package OLIDSISO (Parameter estimation of linear discrete-time models of orders III = I to 5 and dead time d = 0 ... 3).
transfer functions (15) or impulse responses (1) (sampled signals) are used. Hence, most of the authors regarded discrete-time models. As well single-input (7) as multi-input-single-output (9) as muIti-input-multi-output processes (7) are taken into account. The packages furthermore distinguish in the applied identification and parameter estimation methods, recursive or nonrecursive algorithms, ranges of model order and time delay, methods for search of model order, drift elimination and model verification and final results. The required computer storage ranges from 20 to 1.000 K on the disk, however with smaller modules 2 and 256 K required in the core memory. The packages have been developed mostly at universities and research institutes. In the meantime probably many more program packages do exist. 9. CONCLU DING REMARKS
An attempt has been made to discuss some topics on practical aspects of process identification. In addition to an identification method many other tasks have to be performed to identify a real process. This indicates also that process identification is not just a mathematical routine on the one side and not just a measuring technique on the other side. It consists of many procedures which require a good understanding of the theoretical background, a good understanding of the process and good knowledge of
REFERENCES Aoki, M. and R. M . Staley (1970). On input signal synthesis in parameter estimation. Automatica 6, 431--440. Astrom, K . J. (1967). On the achievable accuracy in identification methods. Preprints of the 1st IFAC-Symposium on Identification, Prague. Academia, Prague. Astrom, K. J. (1980). Maximum likelihood and prediction error methods. Automatica 16: this issue. Astrom, K. 1. and T. Bohlin (1966). Numerical identification of linear dynamic systems from normal operating records. IFAC-Sy mposiu m on Theory of self-adaptive control systems, Teddingt on. Plenum Press, New York. Astrom, K. J. and P. EykhofT (1971). System identification- a survey. Preprints of the 2nd IFAC-Symposium on Identification, Prague. Also: Automatica 7, 123- 162. Baur, U. (1976). On-line Parameterschatzverfahren zur Identifikation linearer dynami scher Prozesse mit Proze I3rechnern-Entwicklung, Vergleich, Erprobung. PDVBericht KfK-PDV-65 . Baur, U. and R. Isermann (1977). On-line identification of a heat exchanger with a process computer- a case study. Automatica 13, 487- 496. Blessing, P. (1979). Ein Verfahren zur Identifikation von Iinearen , stochastisch gestorten MehrgroJ3ensystemen. PDVBericht KFK-PDV 181 , Kernforsch ungszentrum Karlsruhe. Van den Boom, A. J. W. and A. W. M. van den Enden (1974). The determination of the orders of process and noise dynamics . A utomatica 10, 245- 246. Van den Bos. A. (1970). Estimation of linear system coefficients from noisy responses to binary multifrequency test signal s. 2nd IFAC-Symposiu m on Idel1lijicacion , Pra gue. Fasol, K.-H. (1980). Modelling and identification. Automatica 16: this issue. Godfrey, K. R. (1980). Correlation methods. Automatica 16: this issue. Isermann , R. (1973). Test cases for evaluation of difTerent identification method s using simulated processes (test case A). Proc. of the 3rd IFAC-Symposiwn on Identification, The H ague, North-Holland, Amsterdam 1, pp. 101 - 1. 106. Isermann , R. (1974). Prozej3 ident ijikat ion. Springer, Berlin. Isermann, R. (1977). Dig irale Regelsysteme. Springer, Berlin. Isermann, R. (1980/81). Digital C onrrol System s. Springer, Berlin. Isermann , R. and U. Baur (1974). Two-step process identification with correlation analysis and least squares pa ra meter estimation. J. D,l'Iwm. Melts. Control ASME 96G (4), 426432. Isermann , R., U. Baur, W. Bamberger, P. Kneppo and H. Siebert (1974). Comparison of six on-line identification a nd parameter estimation methods. Automacica 10,81 - 103. Ljung, L. and A. J . M ,. va n O verbeek (1978). Validation of a pproximate models o btained from prediction error identification. Proc. of the 7th IF AC-Collgress, Helsinki. M a nn , W. (1978). OLID-SISO. Ein Program zur On-lineIdentifikation dynami sc her Prozesse mit Prozel3rechnernBcnut zeranleitung. Ellt\\'icklulIgslloti: PDV-E-II4 , CfK Kar/sruhe. Mehra. R. K . (1974). Optimal input signals for parameter estimation in dynamic systems- a survey and new results . IEEE Trail S. AC-19 (6). 753- 768. Rake, H . (1980). Step response and freque ncy respo nse meth ods. Autol1latica 16: this issue. Saridis, G. N. (1974).Comparison of six on-line identification algorithms. Automatica 10,69 79. Soderstrom, T. (1977). On model structure testing in system identification. lilt. J. COlllrol 26 , 1 18. Soderstrom, T. , L. Ljung and I. Gustavsson (197 8). A theoretical analysis of recursive identification methods. Autol1llttica 14,231 - 244. Strejc, V. (1980). Least sq uares parameter estimation. Autol1latica 16: thi s issue. Unbehauen, H . and B. Gohring (1974). Tests for determining the model order in parameter estimation. Automarica 10, 233- 244.
AII/rmwriw . Vol. 16. pp . 575- 587 Pergamon Press Ltd . 1980. Printed in Great Britain {) International Federation of Automatic Control
Practical Aspects of Process Identification * R. ISERMANNt Key Words- Identification: input signals; sampling time: comparison of identification methods: model verification: program packages.
Abstract-After the presentation of various identification and parameter estimation methods in the previous papers, some selected practical aspects of process identification are discussed. This includes, for a given identification method, the steps from the design of experiments to the verification of the final model. Therefore a general procedure of process identification, the selection of input signals, the selection of the sampling time, off-line and on-line identification, comparison of parameter estimation methods, model order testing and model verification is presented. A short discussion on program packages for process identification follows.
TABLE THE
1.
SOME EXAMPLES FOR THE RELATIONSHIPS BETWEEN
FINAL
GOAL
OF
THE
MODEL
APPLICATION
AND
SPECIFICATIONS OF THE PROC ESS IDENTIFI CATION
Type of process model
Final goal of model application
I. INTRODUCTION THE PRECEDING tutorial papers have been maInly concerned with identification and parameter estimation methods. that means roughly the various procedures to extract mathematical process models from measured noisy input/ output data. However, to apply process identification to a real technical or nontechnical process requires many additional (more practical) preparations, which becomes obvious if the general procedure of process identification is regarded, Fig. l. It is important to regard first the final goal for the application of the process model, since this determines the type of the model and its accuracy requirements and the identification method . Table 1 shows some examples for the relationships between different final goals and some specifications of process identification . The (/ priori knowledge of the process is for example based on the general process understanding, on physical laws of the process and on premeasurements and gives hints on linear or nonlinear, time-invariant or time-variant, proportional or integral behaviour, settling time (dominant time constants), time delays and noise characteristics. By these features and the process operating conditions, the design of experiments and the selec-
Required accuracy of model
Identification method off-line, step response, frequency response, parameterestimation
verification of theoretical models
medium/high linear/ continuoustime nonparametric/ parametric
controller parameter tuning
linear, nonparametric, continuoustime
low for input/output behaviour
off-line step response
computer aided design of digital control algorithms
linear, parametric (nonparametric) discretetime
medium for input/output behaviour
on-line/ off-line parameter estimation
self-adaptive digital control
linear, parametric, discretetime
medium for input/output behaviour
on-line parameter estimati on in closed loop
process parameter monitoring and failure detection
linear/ nonlinear, parametric, continuoustime
high for process parameters
on-line parameter estimation
tion
of the
identification
method
IS
directly
influenced. The design of experiments includes selection and determination of : input signals (amplitudes, spectra); sampling time; measuring time period (identification time); open or closed loop identification; off-line or on-line identification; equipment for signal generation, data storage and computation; signal filtering [high and low frequencies (drift )]. taking into account constraints given by the operating conditions of the process, the properties of the actuators (proportional or constant
'Received March 31, 1980. The original version of this paper- was presented at the 5th If AC Symposium on Identification and Parameter Estimation which was held in Darmstadt. federal Republic of Germany during September. 1979. The published Proceedings of this IfAC Meeting may be ordered from: Pergamon Press Limited_ Headington Hill Hall. Oxford, OX3 OBW, England. This paper was recommended for publication in revised form by the Automatica Editorial Board. tlll$titut flir Regelungstechnik , Techni sche Hochschule Darmstadt. Darmstadt. F.R. Germany. 575
R . IsERMANN
576
PROCESS PHYSICAL LAWS PREMEASUREMENTS { OPERATING CONDITIONS
TASK } ECONOMY
NO
1'1(;.
I. General procedure of proee" ilkntificatiol1
speed action; nonlinearities) and the sensors. Some of these topics will be regarded later in more detail. DiftCrent ways of illput sigllol generatioll, 0111pUI siglllll l1!ellsurl'l1!l'lll and datil storage are shown in Fig. 2. Table 2 shows the main classes of idelll if/catioll lIli'thods which are mostly applied on the measured data. The identified process model finally has to be railled, e.g. by comparing the measured output with the output calculated by the model. Depending on the result of this verification the model structure (order, time delay etc.) has to be changed (model structure search) or the ex periments have to be repeated.
TABLE
2.
MAIN CLASSES OF ID EN TIFICATION METHODS
Model Identification method s step response techniques
non parametric
parametric
x
x
x
x
frequency respo nse techniques Fourier- and spectralanalysis
x
correlation techniques
x
parameter est imation
x
The scheme in Fig. I indicates that process identification in general is an iterative procedure. In the following sections some of these practical aspects of process identification will be discussed.
577
Practical aspects of process identification
J
-4
PROCESS
H cm I RECOR DER
(a )
(b )
IGl iwll 't'lw)
MAGNETIC TAPE
signal has to be sufficiently rich to excite all process modes of interest during the experiments. This leads to the requirement of persistently exciting input Signals. A maximum requirement would be that the input signal be constructed to minimize certain model errors with respect to noise, input and output signal constraints and measuring time. This leads to optimal input signals. First some often used identification methods are considered with respect to the minimal requirements on the input signals. If the F ourier or spectral analysis is applied to identify the frequency response by
' 'G(iW)=F{y(tj} = Y(iw) F{ u(t)} U(iw)
(1)
or
(c) PAPER TAPE
(2 )
FIG. 2. Different ways of input signal generation, output signal measurement and data storage (a) Simple process identification by manually changing the input and data storage by a recorder. (b) Frequency response analyser (using orthogonal correlation for harmonic inputs) (cl Off-line identification using a signal generator and a magnetic tape or punched paper tape as storage (d) On-line identification using a process computer.
2. SELECTION OF INPUT SIGNALS
To identify the controllable and observable (identifiable) part of a dynamic process the input signal has to satisfy certain conditions. A minimum requirement is that the dynamics of the process have to be excited persistently by the input signal over the measurement period. Roughly speaking this means that the input
where
I
g(v)
(3)
\1= 0
has to be used as a basic input/output relationship which leads to the system of equations
<1>uu( - P + I)
g(O) g( 1)
<1>U)'( - 1)
<1>u..! - 1)
<1>uu( - 1 -I)
<1>u)'(O)
<1>u.(O)
<1>uu( -I)
<1>uy( 1 )
<1>uu(i )
<1>uu(i - I)
(4)
g(l ) I A UTO- f
R.IsERMANN
578
For M=l and P=l the (/+1) unknown g(v) can directly be calculated from (/ + 1) equations by
and one imaginary part Im(wJ of the frequency response and therefore two parameters. Persistent excitation therefore means that the input has to have at least j ~ (n + m + 1 )/2 harmonics, van den Bos (1970). Now the parameter estimation for a linear difference equation or for the z-transfer function
The requirement for persistently excltmg input signals in this case is that the inverted matrix 4'::;.1 exists, hence 4' •• may not be singular
(9 )
G(z)
det4' •• oFO.
(6) is regarded. For the nonrecursive method of least squares the estimation equation is
4' •• is symmetric if all
tP •• (r)=
1
lim N- co
N
(10)
N- I
L
u(k)u(k+r)
(7)
with the parameter vector
k=O
do exist. A further discussion for persistently exciting input signals for correlation analysis is given later. If the sampling time To is sufficiently small, equation (3) and the results given above can directly be transferred to the case of continuous signals.
<1>~y (0 )
and 'I' and y being a matrix and a vector containing the measured inputs and outputs, see e.g. Strejc (1980). After dividing ['I'T'I'] by (N + 1) the elements of this matrix are estimates for correlation functions.
<1>~" (1 )
<1>;'v(m - 1)
<1>~y(O)
<1>~y(m-2)
<1>~. (0)
I
- <1>:y(d)
- <1>~. (d + m - I )
-<1>:y(d-l)
-:y (d +m-2)
, - <1>:y(d - m + 1 )
---- - - - --------- --- - -- - - - ----: --
---$~)O)--
---- ---- --
- <1>:y(d) -:J~--=-l-)
----
<1>:.
If the parameters of the continuous-time lrans~ fer function G(s)
b o + bls + .. . + bms m 1 +als+ .. . +ans n
(8 )
have to be estimated by parameter estimation methods by applying periodic (multifrequency) input signals each harmonic with frequency Wj can be used to determine one real part Re(wj)
HII i-·! H12j = [----------:- = H H 21 ;H 22 '
(12)
This matrix is quadratic and symmetric for large N, if the input u(k) is a stochastic or periodic signal with constant meanvalue. To solve equation (10) it must be required that ['I' T 'JI] is not singular, hence detH+O.
(13)
579
Practical aspects of process identification Two cases are possible : det H > 0 or' det H < O. Regarding the successive principal minors of this determinant, it follows, starting with the lower right corner
•
r=O, NJe, 2NJe, .. .
for a2
(r)= - u" N
for
Je(1 +vN)
(14)
1_
H = 1
v=O, 1,2, . .. (PRBS with amplitude a, clock time Je = To and period N . Persistently exciting of order m, if N=m+ 1).
if N is large [
From Fourier analysis it is known that a necessary condition for the existence of the power spectral density
(16)
is, that for all O-:5. w -:5. njTo,
or the matrix
has to be positive definite. In addition
_ 1 U= lim N ~ oc,
N
N- I
I
(18 )
U(k)
k=O
and N- I
I
[U(k)-O][U(k+r)-O]
k= O
(19) must exist, as it was already noted. The con'ditions (17)-(19) have been given in Astrom and Bohlin (1966) for maximum likelihood estimation and are called " persistently exciting of order m". The conditions for persistently excitation for correlation analysis, equation (5) and (6), are the same with exception of the different order I and m. Therefore input signals do exist which are persistently exciting for both identification methods. Special cases of persistently exciting inputs of order m are signals which satisfy
•
•
+0;
111 ->
XJ )
(white noise, if m-> x )
a:>
I
Suu(w)= 1: =
-
a:>
=
I
t=
(20)
3. SELECTION OF SAMPLING TIME
F or the identification of discrete-time models the sampling time has to be selected properly before the experiment starts, because it cannot be decreased afterwards, if the sampled signals are
580
R . IsERMANN
stored. Of course, the sampling time can be doubled, tripled etc. by deleting corresponding data . The sampling time To depends primarily on: (a) Sampling time for the final application (e.g. digital control) (b) Accuracy of the resulting model (c) Numerical problems, if To is too small. These points will be discussed as follows : (a) If the final application of the identification is the design of a digital control system, the sampling time for digital control, as for example the control performance, the type of system. Many characteristics influence the sampling time for digital control, as for example the control performance, the type of control algorithm, the spectrum of disturbances, the valve and its actuator, the sensors and the computational burden on the digital processor, Isermann (1977, 1980). As a rule of thumb 6 to 15 samples per 95 % settling time T95 are often a good choice for PIOcontrol algorithms, i.e. T95 / To = 6 ... 15. (b) To discuss the effect of the sampling time on the absolute values of the parameters an example is shown in Table 3. The gain is
K = Lb;/(l +LaJ
TABLE
3.
PARAM ETER S
OF
THE
(21 )
z-TRANSFER
WITH ZERO-ORDER HOLD FOR DIFFERENT SAMPLING TIMES To .
4
8
16
az a3
-2.48824 2.05387 -0.56203
- 1.49863 0.70409 -0.09978
- 0.83771 0.19667 -0.00995
-0.30842 0.02200 -0.00010
bo bt bz b3
0 0.00462 0.00169 -0.00273
0 0.06525 0.04793 -0.00750
0.06525 0.25598 -0.02850 - 0.00074
0.37590 0.32992 0.00767 -0.00001
1
1
0.10568 0.10568
0.34899 0.34899
0.71348 0.71348
d
4
Lb i 1 +r.a i
0.00358 0.00358
A proper choice of the sampling interval in most cases is not critical, because the possible range between too small and too large values is relatively wide. Good experiences have been made with the rule T95 / To~5
... 15
(23 )
where T95 is the 95 % settling time of a transient function, Isermann (1974). 4. OFF-LINE AND ON-LINE IDENTIFICA nON
K = I ; Tt = IOsec; T2=7sec; T3 = 3sec; T4=2sec; T,=4sec
at
(c) If the sampling time is chosen too small, ill conditioned equation systems (matrices) result, because the single difference equations become approximately linear dependent (numerical problems).
FU NCTION,
C ORRESPONDING TO
To [sec]
Small errors in the parameters can then have a significant influence on the input/output behaviour of the model, since the value of e.g. Lb i depends highly on the 4th or 5th place of b i after the decimal point. If the sampling time, on the other hand, is chosen too large, the dynamic behaviour is described too unexact. For To = 8 sec the model reduces practically to 2nd order, because of hl ~ 11+Lail and Ib31 ~ ILbil and for To = 16 sec even to 1st order. This example shows that the sampling time should not be selected too small nor too large, because of accuracy problems of the resulting model.
With decreasing sampling time To the magnitudes of the ai-parameters increase and those of bi decrease. Therefore for a small sampling time, e.g. To = 1 sec, it is
(22)
With regard to the application of general purpose digital computers and process computers (or other equipments) a process identification can be performed off-line or on-line, Fig. 3. For off-line identification the process data is first stored in a data storage (e.g. recorder, mag- netic tape, punched tape). Later on the data is transferred to a computer and evaluated. For this type of identification mostly batch processing of the data is applied, that means, the whole data set is evaluated at once. Then direct identification algorithms (identification in one shot or one step, for example least squares estimation) and iterative identification algorithms (stepwise, for example maximum likelihood estimation) can be distinguished. In the case of on-line identification the process identification is performed in on-line operation with the process. Then, two ways of processing the data can be distinguished: Real-time processing, i.e. the data is evaluated immediately after each sample or batch processing, i.e. the data set is evaluated at once after all the measurements have been made.
Practical aspects of process identification
mated recursively and parameter estimation is done non-recursively. This is referred to as rea/-
$)
time/ batch processing.
Again the final goal of identification and the type of process and the available computing power determines the type of identification. If a not (too much) time varying process has to be identified for general process analysis or for the design of control algorithms, off-line identification often is preferred. For time varying processes and especially for adaptive control on-line identification with real-time processing, also called real-time identification, has to be used .
MODEL
~
+
MODEL
OFF-LINE IDENTIFICATION
ON- LINE IDENTIFICATION
1 ------------
BATCH
581
PROCESSING
REAL - TIME
1----------- --
NONRECURSIVE ALGORITHMS
RECURSIVE
PROCESSING
5. COMPARISON OF PARAMETER-ESTIMATION METHODS ALGORITHMS
/~
ITERATIVE
DIRECT
IDENTIFICATION
IDENTIFICATION
lONE SHOT, ONE STEP)
ISTEPWISE)
11( ,. 3. Various wa ys of processing the measured data for process identification.
Real-time processing in general needs no storage of data. Recursive identification algorithms of the form O(k+l)=O(k)+y(k)e(k+ 1)
(24)
where (J means the parameter vector, y a correcting (weighting, gain) vector and e an error signal, normally are used . The model develops as the data comes in. Time varying processes can be identified by these type of algorithms. For batch processing the data is stored (core memory, disc or magnetic tape) and evaluated at the end of the measurements. Mostly non-recursive identification algorithms are used in this case. The parameter estimation methods least squares, instrumental variables and maximum likelihood and also correlation techniques, Fourier and spectral analysis, are generally first derived as nonrecursive algorithms. They can then be transferred to recursive algorithms by subtracting two directly following estimates for times k + 1 and k. Recursive algorithms are well-known for the following parameter estimation methods : Least squares, instrumental variables, (approximate) maximum likelihood and also for correlation techniques. They have many connections to stochastic approximation and to state variable filtering and digital filtering . Also combinations between pure real-time processing and pure batch processing are possible. For example correlation functions can be esti-
Once the external conditions for process identification have been classified a proper identification method has to be selected. Since in most cases several methods can be applied to obtain a certain type of process model, a comparison of these methods can be of help, indicating the different advantages and disadvantages. Therefore some remarks will be given concerning the comparison of identification methods. Only parameter-estimation methods for linear discretetime parametric models will be regarded. What can be compared?
With respect to parameter estimation methods and given model order following items are important for the selection of a method : the performance with regard to the accuracy of the identified model (model errors) dependent on time (convergence properties); the computational effort (computer storage, computation time) ; the a priori assumptions and a priori factors to select. H (iit' c{ln comparisons he made? It is generally no problem to directly compare the computational effort and the a priori assump-
tions and factors. For example Table 4 shows the assumptions made for the noise models of different parameter estimation methods, which are required for unbiased estimation. The least squares methods presumes a very special noise model, l /A, which does not exist for real processes. The maximum likelihood method assumes a special nominator in the noise model (D/ A). The most general noise models (D/ C) are allowed for the instrumental variables method and for correlation combined with least squares. The comparison of the performance is much more involved . If more than just two methods have been compared, it has been mainly done by
582
R.IsERMANN
TABLE 4. PROCESS AND NOISE MODELS FOR UNBIASED PARAMETER ESTIMATION AND SPECIAL ASS UMPTIONS FOR DIFFERENT PARAMETER ESTIMATION METHODS
.\'11
Parameter estimation method
Process model
Noise model
....
- _\"
Special a-priori assumption
Autoregr . process Moving average process
I/ C Least squares (LS)
D B/A
Stochastic approximation (STA) General least squares (GLS) Maximum likelihood (ML) Extend. least squares (ELS) Instrum. variable (IVA) Correl. and least squares (COR-LS)
Remarks
I/ A Only recursIve
B/A
8/ A
8/A
I/A
I/AF
D/A
B/A
D/A
BIA
DIC
8/A
D/C
e resp. 11 normal distribut.
simulations, e.g. Saridis (1974), Isermann et ul. (1974). This means that a certain process model was selected and simulated together with a certain realization of a stochastic noise process. Test cases have been proposed for this purpose and several authors presented their results, Isermann and colleagues (1973). Of course, questions come up how many runs have to be taken and if this way is realistic enough. For some nonrecursive methods like the least squares, weighted least squares or maximum likelihood method, an analytical comparison can be made by directly comparing the variances and covariances of the parameter estimation methods. However, the covariance matrices can only be given for special assumptions on the error signal and for a relatively large identification time. An analytical comparison of different recursive parameter estimation methods has been made by Soderstrom and colleagues (1978). Based on an unified recursive estimation algorithm the asymp-
totic behavior is described, i.e. the results are only valid for rather long identification times. The third way for comparisons is to use the measured data of a real process. However, also this way has some drawbacks. Mostly the exact model is not known and the process behaviour and the noise characteristics often change with time, so that the results are rather special. This discussion shows that there seems to exist no unique way which results in a clear comparison of the most important performance measures of parameter estimation methods . Therefore all three ways, simulations, analytical studies and applications on real processes have to be undertaken to come to more general results. A further difficulty arises if the error between the process P and its model M has to be selected. Generally following errors are regarded: Parameter error:
!18; = 8iO - (];
Output error:
!1y(k)= yp(k)- YM(k);
Equation error:
elk);
i = 1,2, ... ;
Input/ Output behaviour error: e.g. impulse response error !1g(r) = go(r) - g(r). Furthermore these errors may be represented as: absolute values ; relative values; averaged values (mean squared over i or k or r). So a great variety of error measures does exist. The selection of an error measure can influence the final results of a comparison. For example small input/output behaviour errors may be accompanied by large parameter errors, especially for higher order models . The final goal of the application of a model should decide which model error has to be used . It is, of course, important that the comparisons are made for the same process, the same noise and the same input and the same measuring time. In the following an attempt will be made to summarize results which have been obtained by simulations, Isermann and colleagues (1973), by analytical investigations, Soderstrom and colleagues (1978) and practical experience, Baur (1976). Some results of comparisons (a) Overall comparison of the performance. For long identification time the parameter estimation methods with unbiased estimates for noise filte'rs D/e and D/ A (IVA, ELS, ML, eOR-LS) lead to approximately the same accuracy of the input/ output model. For short identification all methods, including LS, show little difference. However, in this case the nonrecursive method s
Practical aspects of process identification are better than the mentioned recursive methods. the estimates of the process parameters generally converge quicker than the estimates for the noise parameters (since in the first case the input is known). (b) Properties of important methods. (For abbreviations of the methods see Table 4, R stands for Recursive). Least squares (LS) Biased estimates for real noise applicable for short identification time if noise acts on the process sensitive to unknown Yoo relative small computational expense good starting method for IV A or ML a priori factor: 1 matrix PtO) RLS: reliable convergence RELS : similar as RML. Generali zed least squares (G LS) Biased estimates possible relative large computational expense noise model identified a priori factors: 2 matrices, filter order.
I nstrumental variables (J V A) Good performance for wide range of nOIse models medium/small computational expense insensitive to unknown Yoo if u(k) = 0 (/ priori factors : 1 matrix, filter factors RIV : no reliable convergence. Therefore start with RLS is recommended . Maximum likelihood (ML) Good performance for special nOIse models D/A
large computational expense noise model identified problems if local minima of loss function theoretically well funded a priori factors : dep. on optimization procedure RML: Slow convergence in the starting phase. Convergence more reliable than RELS . Parameters of the polynomial D( Z- I) are also estimated, however, slower convergence than for A(Z - I) and 8(Z - I). Correlation and least squares (COR-LS) Good performance for wide class of nOIse models relati ve small computational expense access to intermediate results
insensitive to unknown Yoo if u(k) = 0 search of model order : small computational expense easy verification
583
a priori factors: only number of correlation functions RCOR-LS: reliable convergence. 6. DATA FILTERING
The process input signal u(k)=U(k)-U oo
(25)
and the output signal y(k) = Y(k) - Yoo
(26)
generally have unknown d.c. values, U 00 and Yoo , and are disturbed by high and low frequent noise. Therefore the following steps have to be taken to eliminate those parts of the disturbances which cannot be eliminated by the identification method. I dentification of d.c. values Only the variations of the output signal with respect to variations in the input signals must be used for the estimation algorithms. If the input signal is generated by a process computer, U 00 is known . Yoo , however, is not known. Some parameter estimation methods need the exact knowledge of Yoo , some not. For example, LS needs exact Yoo , IVA only if E{u(k)} f O and COR-LS just a rough estimate (to avoid an ill-conditioned matrix), see Isermann (1974) and Baur and colleagues (1977). Yoo can be estimated just by averaging _ 1 Yoo = -
NG
L Y(k)
(27)
NGk = 1
before the identification of the dynamics starts. N G depends on the noise. Elimination of high and low fr equency noise Very high frequency noise is properly eliminated by passive or active analogue filters of first or second order. Various methods for digital filtering of low frequency disturbances, as e.g. drift, have been considered, including parameter estimation of drift polynomials, Baur (1976). It then turned out that simple high pass filtering is best suited. In order to avoid an influence on the identified process model, not only the output but also the input has to be filtered. First order high pass filters (HP) seem to be sufficient in many cases.
ii (k) = cxii (k - 1 ) + u (k ) - u (k - 1 ) J:(k) =cxf(k - l) + y (k) - y (k - l) where cx=e - T oi T :
To = sampling time T = time constant.
(28 )
584
R.IsERMANN
u( k) and y( k) are filtered signals used for para-
meter estimation. T has to be chosen so that the high pass filter, if possible, does not eliminate the lowest frequency of the input. For PRBS signals with clock interval A and period N the time constant should be
process with true order mo and true dead time m= mo and = do should result. The dead time can be simply estimated by assuming d = 0 and increasing the order m to at least mo + do. Then
a
do. In the ideal case
m
T> N PRBSA/ 2n.
IEII, 1621, .. .,16p l~ I 6; i= 1
7. MODEL ORDER TESTING AND MODEL VERIFICATION
and
As already discussed in section 1, model order testing and model verification form the last stages in the procedure of process identification. After obtaining an identified model, all a priori assumptions and simplifications and the performance of the model should be tested. In the case of parametric models in general the model structure has to be 'identified' also. For lumped single-input, single-output processes this reduces to the determination of the model order. Model order testing and model verification are very much related to one another, because some methods of verification are used as a feedback in the order search procedure. Before some methods of order testing and verification are discussed some general remarks are given which may underline the difficulties in the judgement of process models. Process models are mostly an approximate description of a part of a process. Therefore an exact agreement between a model and a real process cannot be expected. The evaluation of a model depends very much on the final goal of its application. Therefore the objective of the model has to be taken into account. A priori assumptions of identification methods often do not coincide with the situation for which an .identification has to be carried out, taking into account practical limitations. The experimental conditions for the identification (input signals, time variation of the process, noise characteristics, drift) may differ from the (unknown) conditions under which the model is applied (to control system design, optimization, monitoring, etc.). Therefore model evaluations on an objective basis are not the only way, but also heuristic and subjective components have to be included. 7.1. Model order testing The determination of the order of a parametric
single-input, single-output model of type
for f3 = do can be used as a criterion, Isermann (1974). The search of the dead time can, however, also be included in the general search procedure for the model order. Several methods for order testing have been proposed in recent years. The commonly used order testing methods can be divided into the groups. loss function tests pole-zero cancellation tests residual tests. These methods are summarized for example in van den Boom and van den Enden (1974), Unbehauen and Gohring (1974) and Soderstrom (1977). Only a brief description of some of these methods is given here. Loss function tests
The most simple method is just to regard the loss function V(m)=eT(m)e(m)
(30)
in dependence on the order m, where e is the equation error or residual of the used parameter estimation method, using the same input-output data. For m = 1,2, . .. the loss function for higher order processes first decreases until V (m) and then remains constant or changes slightly, Fig. 4.
0 .001 0 0,-
1 1
1 I I I 1
v
I
I 1 1 1 I
d=3
3
3
3
3
2..........
0
......... 12
.........
2 I
"
1 2
2 0
oooooL----------2.===£ ==-=--J5 1 2 3 4 m
_y(z)_ bIZ-I+ ... +bmz - m z z G p () u(z) 1 +alz I. .. +amz - m
means to estimate
mand
the dead time
d
(29)
aof the
FIG. 4. Loss function V(m , d) in dependence on the order m and dead time d for an identified heat exchanger (a distributed parameter system). Result: m=3 and .1 =0. (Baur, 1977).
Practical aspects of process identification Estimates m are found if V (m + 1) and V (m) do not differ significantly. This can be checked based on plots, van den Boom (1974), Isermann (1974). A statistical method to check if the loss function reduces significantly is the F -test. This test is based on the statistical independency of V(m 2 ) and V(md- V(m 2 ), which have X2 distributions for normal residuals. To test if the reduction of the loss function is significant when the order is increased from m l to m2 (number of parameters increased from 2ml to 2m 2 ) the test quantity t=V(m l )-V(m2) . N-2m2 V(m 2 ) 2(m 2 -m l )
(31)
is used . For large samples N the random variable t is asymptotically F[2(m2 - m l )], [(N - 2m 2 )] distributed. A risk level is defined and from the tabulated F-distribution the corresponding t* can be taken . If t < t* the probability of V(m 2 ) is smaller than V(m l ) and therefore m l is the estimated order, see Astrom (1967). Other order tests based on loss functions have been proposed by Wilks, Akaike and Parzen, see Soderstrom (1977). Pole-zero cancellation tests If for the parameter estimation a higher order m than for the real process with order mo is assumed (m - mo) additional pole-zero pairs result which at least approximately cancel each other. This effect can be used for order testing by calculating the roots of the polynomials B(Z-I) and A (Z-I) for different orders m, van den Boom and van den Enden (1974). Residual tests Some parameter estimation methods as LS, ELS, GLS, ML require a white noise of the residuals for the case of unbiased estimates. A whiteness test of the residuals, for example by calculating the autocorrelation function, indicates if this requirement is fulfilled or not and is therefore a general verification procedure for the parameter estimation. However, it can also be used to test for the right order, van den Boom and van den Enden (1974). Several methods for order testing have been compared in van den Boom and van den Enden (1974) and Unbehauen and Gohring (1974). However, only second order models have been used, for which the order search in most cases does not lead to decision problems. Soderstrom (1977) has shown analytically that some loss function and pole-zero cancellation methods are asymptotically equivalent. The practice shows that the order determination for single-input, single-output processes in general is not critical.
585
In many cases there is not one definite 'best' order, because of lumped higher order processes with many small time lags or distributed parameter processes which are approximated by finite order lumped models. However, the test for the structure of multi-input, multi-output processes is a highly difficult task. Blessing (1979) gives a survey and has proposed a method for rank determination of state-space system matrices for the case of noisy processes. For all nonrecursive or recursive parameter estimation methods it is required that the whole data set be stored and passed through several times until the right order is found. If, however, correlation functions are first calculated, leading to a nonparametric model, and the parameter estimation method is based on these correlation functions (method COR-LS) considerable storage and computing time can be saved, Isermann and Baur (1974). 7.2. Model verification Model verification means to check if the identified model agrees with the real process behaviour, taking into account the limitations of any identification method and the final goal of model application. This includes a check to determine if the a priori assumptions of the identification method used are true and to compare the input/output behaviour of the model and the processes, Isermann (1974). The verification of the a priori assumptions can be performed as follows : (a) N onparametric identification methods. Linearity: Comparison of models obtained
for different input amplitudes. Comparison of models with measured transient functions in both directions. Time variance: Comparison of models for different data sets. Noise : Is the noise n statistical independent from the input u? Checking the signals: Are there outliers? Drift : Averaging of output. Comparison of models with and without drift elimination. Input signal: Can it be measured without noise? Is it persistently exciting? D.c. values: Are U 00 and Yoo exactly known? (b) Parametric identification methods.
In addition to the already listed items, the following a priori assumptions should be tested . These, of course, depend on the estimation method. Residuals: Are the residuals statistically independent; CPee(')=O for +O? Are they independent from the input, CPue(')=O for all ,? E{e(k)} =O? Normal distribution?
1,1
586
R . IsERMANN
Covariance matrix of the parameter estimates: Do the variances and covariances decrease with increasing measuring time? Are they small enough ? A final overall judgement of the identified model is obtained by the verification of the input/output behaviour, i.e. by comparing the measured and the model predicted input/output behaviour. This can be made by (a) Comparison of measured yp(k) and calculated y(k) output signal: for the input u(k) used for the identification ; for other input signals like steps or pulses. (b) Comparison of the crosscorrelation function cPu/r) based on measured signals and on the model. For processes with no noise (or small noise only) (a) is straightforward. However, if the process is contaminated by noise the comparison of the outputs leads to ~ y (Z)=
yp(z) - y(z) Bp(Z) B(Z)] = [ -- - -~- u(z)+n(z) Ap( z ) A(z )
(32)
and the difference signal ~y( z ) depends on the model errors and the noise. If the noise model IS also identified, ~y( z ) reduces to ~y(z) =
Bp(Z) B(Z)] u(z) [ ----~ Ap(z) A(z) +[
D(z) v(z ) C( z)
n( z ) - ~-
A
]
(33)
Then the one step ahead prediction ~y(k) can be used, taking into account estimates of u(k - 1). A proper criterion, e.g. (34 )
can be used to compare different models. Another way is to compare directly the crosscorrelation function , case (b), since the effect of the noise is already removed, at least partially, for the crosscorrelation function from measured signals. Of course, this has to be accompanied by checking the a priori assumption for the crosscorrelation function . An additional way is cross-checking, that means to verify the identified model for another set of measurements. As the results of the identification depend on the input signals used, compare section 2, also this effect has to be taken into account for the verification, Ljung and Overbeek (1978).
Finally, the most important evaluation can only be given if the model is applied for the final purpose, i.e. for control, optimization, monitoring or just verification of a theoretical obtained model. For all these cases different requirements on the model have to be satisfied, see Table 1, and therefore different measures of model performance have to be applied.
8. PROGRAM PACKAGES FOR PROCESS IDENTIFICATION
Program packages for process identification have been developed which are complete sets of programs according to the various steps of an identification procedure. One has to distinguish: (A) Program packages for general digital computer applications III off-line (batch) operation (B) Program packages for process computer applications in on-line (real-time) operation. Packages of type (A) consist of, for example, data filtering, one or several methods for identification and parameter estimation, methods for search of model order, model verification and proper graphic presentation of the results. Packages of type (B) may consist of the same items, but in addition have to take into account real-time execution, process dependent interrupts, foreground and background organization, use of internal and external storages, signal generation, etc. A big advantage can be an interactive dialog with the operator. Fig. 5 gives an example for the procedure of a program package for on-line identification, Baur (1976) and Mann (1978). The identification methods need only 20 % of the length if the total package. During the 4th IF AC -Symposium on Identification and System Parameter Estimation in Tbilisi, 21 - 27 September, 1976, a round table was held on these program packages. Questionnaires have been used for the preparation of this discussion session. As a result 21 program packages have been described, and the main results can be summarized as follows . For all 21 packages a high level programming language was used , mostly FORTRAN (17), but also ALGOL (3) and PL/ l (1). Of these, 12 packages are written for process computers and nine for general purpose digital computers. In all cases linear processes can be identified, nonlinear processes in nine cases (mostly linear in parameters) and time-variant processes in eight cases (mostly slowly time varying). As process and noise models differential equations or s-transfer function s (4) or vector differential equations (1) (continuous signals), difference equations or z-
Practical aspects of process identification ON LINE
IDENTlFCATION 'OllD-SISO'
all the available methods. Good identification of real pro£.~sses is an art.
Method A
o
_______ F:l
r.=~~~~----L-,
P
~
---[[Ji. ~ "om" = U.,mtn
•
E R
Display
A
,
T
-
o
ir"_- ~"J_ ........... . -.. ...., .. ..." u
ll
.......
R ~--I--6'f-Tl 1 2 3 4
5 m
estimated
parameters
18 2 , ······m b1tJ, ·····-bm
8
587
FIG. 5. Simplified procedure of the program package OLIDSISO (Parameter estimation of linear discrete-time models of orders III = I to 5 and dead time d = 0 ... 3).
transfer functions (15) or impulse responses (1) (sampled signals) are used. Hence, most of the authors regarded discrete-time models. As well single-input (7) as multi-input-single-output (9) as muIti-input-multi-output processes (7) are taken into account. The packages furthermore distinguish in the applied identification and parameter estimation methods, recursive or nonrecursive algorithms, ranges of model order and time delay, methods for search of model order, drift elimination and model verification and final results. The required computer storage ranges from 20 to 1.000 K on the disk, however with smaller modules 2 and 256 K required in the core memory. The packages have been developed mostly at universities and research institutes. In the meantime probably many more program packages do exist. 9. CONCLU DING REMARKS
An attempt has been made to discuss some topics on practical aspects of process identification. In addition to an identification method many other tasks have to be performed to identify a real process. This indicates also that process identification is not just a mathematical routine on the one side and not just a measuring technique on the other side. It consists of many procedures which require a good understanding of the theoretical background, a good understanding of the process and good knowledge of
REFERENCES Aoki, M. and R. M . Staley (1970). On input signal synthesis in parameter estimation. Automatica 6, 431--440. Astrom, K . J. (1967). On the achievable accuracy in identification methods. Preprints of the 1st IFAC-Symposium on Identification, Prague. Academia, Prague. Astrom, K. J. (1980). Maximum likelihood and prediction error methods. Automatica 16: this issue. Astrom, K. 1. and T. Bohlin (1966). Numerical identification of linear dynamic systems from normal operating records. IFAC-Sy mposiu m on Theory of self-adaptive control systems, Teddingt on. Plenum Press, New York. Astrom, K. J. and P. EykhofT (1971). System identification- a survey. Preprints of the 2nd IFAC-Symposium on Identification, Prague. Also: Automatica 7, 123- 162. Baur, U. (1976). On-line Parameterschatzverfahren zur Identifikation linearer dynami scher Prozesse mit Proze I3rechnern-Entwicklung, Vergleich, Erprobung. PDVBericht KfK-PDV-65 . Baur, U. and R. Isermann (1977). On-line identification of a heat exchanger with a process computer- a case study. Automatica 13, 487- 496. Blessing, P. (1979). Ein Verfahren zur Identifikation von Iinearen , stochastisch gestorten MehrgroJ3ensystemen. PDVBericht KFK-PDV 181 , Kernforsch ungszentrum Karlsruhe. Van den Boom, A. J. W. and A. W. M. van den Enden (1974). The determination of the orders of process and noise dynamics . A utomatica 10, 245- 246. Van den Bos. A. (1970). Estimation of linear system coefficients from noisy responses to binary multifrequency test signal s. 2nd IFAC-Symposiu m on Idel1lijicacion , Pra gue. Fasol, K.-H. (1980). Modelling and identification. Automatica 16: this issue. Godfrey, K. R. (1980). Correlation methods. Automatica 16: this issue. Isermann , R. (1973). Test cases for evaluation of difTerent identification method s using simulated processes (test case A). Proc. of the 3rd IFAC-Symposiwn on Identification, The H ague, North-Holland, Amsterdam 1, pp. 101 - 1. 106. Isermann , R. (1974). Prozej3 ident ijikat ion. Springer, Berlin. Isermann, R. (1977). Dig irale Regelsysteme. Springer, Berlin. Isermann, R. (1980/81). Digital C onrrol System s. Springer, Berlin. Isermann , R. and U. Baur (1974). Two-step process identification with correlation analysis and least squares pa ra meter estimation. J. D,l'Iwm. Melts. Control ASME 96G (4), 426432. Isermann , R., U. Baur, W. Bamberger, P. Kneppo and H. Siebert (1974). Comparison of six on-line identification a nd parameter estimation methods. Automacica 10,81 - 103. Ljung, L. and A. J . M ,. va n O verbeek (1978). Validation of a pproximate models o btained from prediction error identification. Proc. of the 7th IF AC-Collgress, Helsinki. M a nn , W. (1978). OLID-SISO. Ein Program zur On-lineIdentifikation dynami sc her Prozesse mit Prozel3rechnernBcnut zeranleitung. Ellt\\'icklulIgslloti: PDV-E-II4 , CfK Kar/sruhe. Mehra. R. K . (1974). Optimal input signals for parameter estimation in dynamic systems- a survey and new results . IEEE Trail S. AC-19 (6). 753- 768. Rake, H . (1980). Step response and freque ncy respo nse meth ods. Autol1latica 16: this issue. Saridis, G. N. (1974).Comparison of six on-line identification algorithms. Automatica 10,69 79. Soderstrom, T. (1977). On model structure testing in system identification. lilt. J. COlllrol 26 , 1 18. Soderstrom, T. , L. Ljung and I. Gustavsson (197 8). A theoretical analysis of recursive identification methods. Autol1llttica 14,231 - 244. Strejc, V. (1980). Least sq uares parameter estimation. Autol1latica 16: thi s issue. Unbehauen, H . and B. Gohring (1974). Tests for determining the model order in parameter estimation. Automarica 10, 233- 244.