Analysis of stochastically disturbed industrial plants within short measuring time

Automatica, Vol. 10, pp. 545-550. Pergamon Press, 1974. Printed in Great Britain.

Brief Paper Analysis of Stochastically Disturbed Industrial Plants Within Short Measuring Time* Analyse d'Installations Industrielles Stochastiquement Perturb6es dans un Temps Court de Mesure Analyse stoehastisch gest6rter industrieller Regelstrecken mit kurzer MeBzeit AHaJIH3bI l-lpOMb]m.qeHamXyCTaHOBOKCO cny,.iarm~aMH BO3MylI.~CHH~MH B TeqeHHe

r p a r r o r o nHTepaa.ria H3MCpeHH,q

HANS-HELMUT WILFERTt and HORST MAIER + Corresponding with these requirements this paper presents analysis strategies with short measuring time for identifying stochastically disturbed linearizable plants by means of a combination of aperiodic and periodic test signals. For plants with an unknown structure of the transfer function, N suitable logarithmically equidistant distributed frequency response points* are to evaluate in the interesting frequency interval (o~in~
Snmnmry--For the analysis of heavily stochastically disturbed, linearizable control systems by means of predetermined test signals, measuring strategies are derived which ensure a short measuring time for the analysis with a prescribed accuracy and given technical limitations. The time favourable analysis strategy is obtained by resolving an isoperimetric variation problem with inequality constraints and a super-ordinated optimizing problem. A digital computer is utilized to calculate the optimum analysis strategy. An example is given which demonstrates the application of the programmed ALGOL algorithms.

---The dynamical behavior of the control loop configuration to be considered, shown in Fig. 1, can be described in the frequency domain.

1. Introduction ACCeFrAnLEsynthesis or optimization of industrial control systems requires the knowledge of a mathematical process model of satisfactory accuracy. A theoretical analysis often lacks the desired accuracy and the mathematical process model must be found by experiment. Because stochastic disturbances will practically always appear in the plant under analysis, expenditure for a process analysis depends largely on required model accuracy. As experimental analysis, in most cases, interferes with normal production, even leading to loss of productivity, the question of economy becomes critical, forcing analysis to be handled with a minimum outlay of time and an accuracy level only as high as absolutely necessary. This requires:

1 ~sturbonce Z(t) +o

)

FoCj~) = F,(j~! 2.F~(I~) =

z ( t ) ' W ( t +v)= 0

i

control deviotion Xw ( t )

(1) fixing the accuracy of the mathematical process model of the plant to the minimumnecessary for a satisfactory solution of the synthesis problem and (2) the derivation of time optimal analysis strategies for the experimental identification of the plant with the desired accuracy requirements.

1~0. 1. Structure of the control loop. - - A rough, approximate frequency response of the plant is known from a rough time-function analysis or from a priori information. --The power distribution spectra of the disturbance and the reference input S~,(o~) and S,ow(co) can be measured

* Received 7 December 1972; revised 10 September 1973; revised 20 February 1974. The OrJ~nal version of this paper was not presented at any IFAC meeting. It wasrecommended for publication in revised form by Associate Editor K. J. Astrom. ? Academy of Science of the GDR, Institute for Cybernetics and Information Processes, Department of Technical Cybernetics, Dresden, German Democratic Republic, HaeckeistraBe 20. :[ College of Transport and Communications, "Friedrich List", Dresden, German Democratic Republic.

* The logarithmically equidistant distribution of the measuring points is advantageous because for the frequency response analysis a relative frequency range of more than 1½ decades must be analysed. The knowledge of the lower frequency range is important for the valuation of the influence of the stochastical input signals, the knowledge of the higher frequency range for the guarantee of the control loop stability.

545

546

Brief paper

or estimated; in the case of industrial plants simple assumptions for S,,~(co) are possible. --The synthesis of the controller will be based on the minimum means square control deviation.

jm {Fstj~)}

Re {F.(it 1~)}

The calculation of analysis strategies is part of an iterative algorithm for the computer-aided automatic analysis and synthesis of heavily stochastically disturbed control systems in industry [1]. The principle of this algorithm is shown in Fig. 2.

I

Evoluation of "the o-pr/oH-informoflon: I [Determlnotlon of on opproximote frequency response [ [for F (J~); Estlmotlon or essump"tlon of Szz(~u) and choice lot o suitoble model of Sww(uJ),

I i

!

2.

i~ore

3.

Determination of 'the necessory occurocy of 'F, (J~) by o given vo?ue / X ~ Determination of the optimol measurement strategy ~

Provisional synthesis of the controller minimum control deviotion Xw-Xw--~{-t-~{t) for mean

t"

4,

J

---T-'I

error.

Y

Actual measuring errors then are situated, with a certain level of confidience, within the tolerance circles having the radius re(co); [Fm(fio)[. Thus, a tolerance band is established around the measured frequency response within which the true frequency response curve is located, also with a certain level of confidence. Basis of the controller synthesis shall then be to obtain that frequency response curve which is determined by the smallest mean square deviation from the measured frequency response points. According to WIUP~T [I, 2] the maximum control deviation due to erroneous frequency response of the controlled system is determined with

f I

Synthesis of the controller (Porometer optlrnizoton of [ 5, I ~ J l ~ with O given structure for the minimum of Xw (~r) or struc"ture optlrnizo"tlon occordlr@ ~o WIENER.)

1

Y

i

Delerrnlnoflon of 1~e possible deviotfon 6, from the compu%,~l mlnlmal square control deviation

eq~o*lon

',os'I }

IFs (J%)l 2

FIG. 3. Measure of the equivalent frequency response

Experlmentol fdentificotion of the plorrP occording to the deter~ned meosurement slTategy

[ A~

mt~,a-v

(z)

l

I

ho

! ~ ) ;J o

/IX'~

f(m, o~) do~

(2)

where *This Is "l'he problem of this poper

FIG. 2. Principle of the analysis and synthesisalgorithm.

f(m, o,)=(s=(o,) + s_(o~))/([1 +F, Oco)l -- m(ta)[F,(fio)[)

The paper is organized as follows: In Section 2 the relation between the analysis errors of the plant and the possible synthesis accuracy of the control loop will be given. In the next section the analysis errors, expected when using determined test signals, arc discussed. Using these relations in Section 4 the time favourable measuring strategies when using combined periodic and aperiodic test signals are developed and an example for the application end the paper. 2. Relation between the analysis errors and the synthesis accuracy For the development of the analysis strategies, the influence of plant analysis errors of the means square control deflation must be known. Analysis errors which occur when using aperiodic and periodic test signals can be compared by calculating the equivalent frequency response error. This error is defined as the relative mean square of the deviations at the various frequency response points, indicated in Fig. 3, and described

in [1-3, 9]. m2(o

=

with Fe(]¢o), Fa(]co)---frequency response of the controlled system and of the controller. According to the Nyquist criterion, the relation [I, 2]

ms*(og) =

[Fo(jco) + ll/IFo(Y~)I

Such a comparative error specification is specially advantageous when comparing the analysis errors from the time mad frequency domain,

(3)

describes the value of the equivalent frequency response error, that which is the maximal permissible for reasons of stability. Hereby it is presumed that the open control is stable and the closed control loop also is stable for m(oJ)-- 0 In the practical application, only a value tin(to)< m**(o~) is allowed. Thus, the desired relation is given between accuracy of analysis-characterized by the equivalent frequency response error re(co)--and the accuracy of synthesis---descfibe,d by the maximum deviation of the mean square control deviation, as e x p ~ by

(l)

F~jco)~frequency response of the controlled system E{ }--expectation value

2

Fo(jCo) ffi F~(o,) . F ROco)

2

max

--xw(t) I~,)=o.

(4)

In this connection it must be mentioned that S~PAN [10, 11] has shown another way for the valuation of measuring based on the analysis of the variances of the

Brief paper coefficients of the transfer function. Thus, the obtained results cannot be directly compared. However, a comparative investigation of the results of the different methods could be fruitful.

3. Analysis errors when using determinated test signals For the calculation of the frequency response by evaluating stochastically disturbed response functions to aperiodic test signals, SrRon~ [4] set up the following approximate frequency response error equation based on a condition of a measuring time of sufficient duration or a very wide band disturbance signal:

m.(o)~ 4T.~' S.,(o)

/(4~. IEUo)I"IFMo)I)

(5)

where E(jo~)--the Fourier transforms of the aperiodic test signal M--the number of evaluated response functions Ta~,---the duration of an aperiodic test signal. With a prescribed shape of the input signal the error behaviour is therefore fixed to a scale factor of M - Z By using periodic test signals, the frequency response points ¢o, will be determined by correlation of the output signal of the disturbed plant with the periodic test signal over n periods. For the frequency response error mf(¢o) is valid [3, 9]

547

where Ts--the settling time of the system n(mj)---the number of averaged cycles at the frequency response point m3 me(m)---the analysis error on averaging one cycle of the periodic test signals. Then the total measuring time is

Te= MT,,p + Tp.

(9)

To be able to formulate the optimization problem in its entirety, the limitations on the size of the equivalent frequency response error have to be taken into account. Hence, when m, is introduced as a lower measurement device conditioned accuracy limitation, the analysis error with aperiodic test signals ma(m) can be

m._< m,(o) < m,(o).

(10)

When employing periodic test signals, averaging is done for no less than one cycle--corresponding to the error mo(m)--.and of most for nm~, cycles of the te~t frequency--corresponding to m~a0. Consequently when using periodic test signals, the possible error interval is expressed by m a x [ m , , mob(O)] = m l ( o ) < rap(o) < r a i n [ m e ( o ) , m , ( o ) ] ----m 2 ( o ).

(11)

Os

m2(°~) = n2. n " ]FMo,)I2A 2

~

+,0 sin2x [ e

[ XO,

+ S , , (~2n~ + o , ) } dx,

(6)

with x-----re, n. n/o~, Ao test signal amplitude. By chen~ng the number of the output signal periods

nfn(m,) to be averaged, a prescribed error behavior m~(m) can be maintained.

4. Deriving time optimal measuring strategies when using combined periodic and aperiodic test signals As is well known, it is advantageous to determine the frequency response of the controlled system in the lower frequency range up to mT by way of the response functions from aperiodic test signals while periodic signals have to be used in the upper frequency range greater than the frequency coT. Des~bing the frequency m~, in compliance with the logarithmic equidistant distribution of measuring points with

or(i)- Omtn" 10 °'tg¢~m~x/"

)m) with 0 ~ i < N

(7)

The shape of the applied aperiodic and periodic test signals and the signal amplitude are fixed according to given technical conditions. Favourable test signals are e.g. for the periodic case, sine or rectangular waves, and for the aperiodic case, step functions or rectangular impulses. The independent variables of the problem then are the division of the analysing range mT with respect to i, the number of aperiodic test signals M to be used, and the behaviour of the error m#(m) for the interval cor
(12)

For this problem there exists an analytical solution [11, which was found with the penalty function technique. The function of the analysis error m~(m) securing minimal time for analysis is

mop,(O, 4) =

f n~(o)= m,(o). ,O(A~+ ~.~)for mr(o) < n~(o)< m2(o) ml(o) for r~(o)
the analysis measuring time with periodic test s/gnal~ then becomes

m 2 ( o ) for ~ ( o ) > m 2 ( o ) where

N

T,-- ~ n(oj). 2. ~/oj + (N- ~+ 1). r~

A(03) = [Szz((D)2."~-Sww(fD)] " O2/g(omax/Omin)

1=1

2n

+(N-i+

l ) T E ~ T t , [ m , ( o ), i]

(8)

N " lgelFo (-- .,

u. .

(14)

and 2 a Lagrange parameter which has to be selected so that by inserting the optimum error behaviour, according to equation (13), into equation (12), the prescribed value

548

Brief p a p e r

A2x2dt) ' is obtained from the superordinated optimizing problem. The respective A-values are found with an iterative method. From the behaviour moat(co), defined by 2, the calculation of the required averaging times n(co) is made according to a relation given by WILFERT[3, 9]. Utilizing the optimum error behaviour for the frequency range where the frequency response is determined by means of periodic test signals, the following superordinated optimizing problem has yet to be resolved:

02 :"

T-

" " "..".,,,.....,....., ......

02 ,,~ " / / # /

-~- .

IV2

01 T g = Tg[2, M , i ] ~ m i n

(15)

with the inequality constraints equations (7) and (10) and prescribed accuracy Ax2w(t)=const. The most suitable approach to solve this problem turned out to be the direct search of the optimal parameters 2, M, i, for cot with the aid of a digital computer. The result is an algorithm in the form of an ALGOL programme. 5. Example for the application of the measurement strategy In the following the determination and application of the proposal measurement strategy shall be demonstrated by an example. The experimental data was obtained by simulation on a computer. Corresponding to Fig. 2, the following steps resulted: (1) From a transient function of the disturbed plant, shown in Fig. 4, an approximate frequency response was obtained and plotted in Fig. 5. The power spectrum of the disturbance was approximated by

t"/S

Fie. 4. Disturbed transient function, •.. stochastic disturbed transient function, undisturbed transient function for comparison.

- -

(2) The given controller has the frequency response F . ( j c o ) = 11,1 + 0,011/(jco) + 161jco/(1 + 1,61jco) (3) The allowed enlargement of the mean square control deviation resulting from the analysis errors of the plant is given by AxZw(t)/x2w(t)<0,25. For the

1 --2,2" 1 0 7 0 2 + 1,78" lOlaco4 + 9,3" 10 t 7co6 S=(CO) = Soz 1 + 9,26" 106co2 + 6,83" lOtZco*+ 1,13" lOt sco6 + 5,83" 1022(,os and that of the command variable

analysis, square wave pulses for aperiodic, and sine waves for periodic, test signals were used with a magnitude Ao=8. Figure 6 shows the determined optimal distribution of the equivalent frequency response errors. Measurement strategy consists of the evaluation of response functions of 23 aperiodic test signals and the measurement of 11 frequency response points according to Table 1.

1

Sww(co) = 2,7 • Sos

1+10

7

"co

2

with Soz=247.

Jm {Fs (Jw)}.I / _

o.o5~6.-~_

-2 / 39XI 0 ~ - - ' ~

5 33Xl0-2

/

/ /~3×to / ~ -0,-:2~ 1.7xt0-2~ 1.7xlo- ~ ,k

\"°'

~

;-

~,4xlo-3 ADproximotton by 6 . ~ biPl ~

x o, ¢

-

.....

1.4XlO

X~10-3

.=~.o.~o_ . . . .

i-~xlo-~"~: ~ ~

~7'3XI0-4

I=°;:,

d

; Y 2x'°3 j 7 - 3 O6×10-3 ~/2-8XI0

,r

f

¢'3.~,o-'

/=,,o.,Sj;::t

_o~×,o-~~_ - ~Trr'THxlo~"~.~_~ ' 5 ×KT3 ixlO-2 8.exlo-3

,o . T. bl # Fs (p)= i~o

FIG. 5. Approximate frequency response computed from the disturbed transient function in Fig. 4 and frequency response from the optimal nmasumtmmt strategy ( . . . ) and the undisturbed frequency response of the plant ( ) for comparison.

Brief paper

549

6. Concluaion

-

I

IO II

0

~.

00

o o o o oo oo o o

o

4

o

2 I- 00 ,oo

tO~3

I

I

I l]

1

I

Ill

2

4

6 810-`4

2

4

6 8 I0-~

ll,[ ../'27r 4

6 8 10"~'

W / 2Tr/s - I Ident-ifico, ion w l ~ ol~rlodic "l'es'ksignols (Evolul"lon of 23 rectongle impulses)

Calculation of measuring strategi~ using a combination of aperiodic and periodic test signals with prescribed synthesis accuracy of the closed control loop presents a complicated optimizing problem. This problem can be divided into an isoperimetric variation complex with inequality constraints and a superordinated simpler optim/zing component. A relatively simple analytical solution exists for the variation problem. The whole problem solution gives the measuring strategy which prescribes the type and parameters of the test signals to be employed, so that the total measuring time becomes a minimum and the prescribed value of Ax2w(t)~is maintained. For the whole analysis and synthesis it is not necessary to know the true structure of the transfer function if the frequency response has that accuracy given by the allowed equivalent frequency response error re(co).

rdentlficatlon wHh i~flodk: 1"estslgnab MeosurmentB'~ II f r e quency responsepoints)

Fie. 6. Optimal equivalent frequency response error inept(co) and the maximal allowed error m,(co) and the real expecting values of the error m(o~), determined with the measured frequency response of the plant (o o). TASTE 1. N U M a ~ OF THE PERIODS TO BE AVERAGED W H E N MAKING FREQUENCY R.~PONSE MEASUREMENTS FOR THE EXAMPLE

Frequency of measuring points f/Hz

Error behaviour mo~(o~) /per cent

Number of the periods to be averaged

1.95 10-3 2.30 10 -3 2.72 10-3 3-21,10-3 3.79 10 -3 4.47 10-3 5.27 10 -3 6.22 10 -3 7.35 10-3 8.67 10 -3 1.02 10-2

14.6 12.6 10.6 11"1 45.8 47"1 24.8 14.8 8.8 15.1 23.5

2 3 9 27 20 10 6 10 41 8 6

References [1] H. H. WILn~T: Uber die Forderung an die Analysegenauigkeit stochastisch gestSrter Regelnngssysteme. Dr.sc.techn.-Diss. TU Dresden (1971). [2] H. H. Wmn~T: The Determination of Optimal Tolerance Areas for the Stochastic Measurement of Errors in Identifying Disturbed Plants. Preprint of the 2nd Prague IFAC-Symposium (1970). [3] H. H. WILn~T: Beitrag zur expedmentellen Systemanalyse bei Anwesenbeit stochastischer St6rungen. Dissertation TU Dresden (1966). [4] H. STROa~L: Zur Analyse stochastisch gestt~rter Systeme dutch kombimerte Answertung yon Zeit- und Frequenzfunktionen. Messen-steuern-regeln (msr) 11, 216-219 (1968). [5] R. ISeRMANN: Zur Messung des dynamiscben Vorhaltens verfahrenstechnischer Regelstrecken mit determinierten Testsi.~alen. Reeelungstechnik(rt) 15, 249-257 (1967). [6] R. I s ~ , ~ , r s : Required Accuracy of Linear Time Invariant Mathematical Models of Controlled Elements. Preprint of the 2nd Prague IFAC-Symposium (1970). [7] H. H. WtLl~tT: Zur Wechselbeziehung zwischen der Regelstreckenanalyse und der Regelkreissynthese stochastisch gest6rter Regelungssysteme. Messensteuern-regeln (msr) 14, 430-437 (1971). [8] H. STROeEL: Systemanalyse mit determinierten Testsignalen. VEB Verlag Technik, Berlin (1968). [9] H. H. WRX~T: Signal- und Frequenzganganalyse an stark gest'drten Systemen. VEB Verlag Technik, Berlin (1969). [10] J. STEPAN: MeBgenauigkeit und Regelungstheurie. Regelungstechnik (rt) 20, 203-207 (1972). [11] J. S ~ P ~ : Limit Identifiability of Control Systems. Preprints of the 3rd Hague IFAC-Symposinm (1973).

(4) The result of the experimental analysis using this strategy is given in Fig. 5. (5) For the optimal controller the following fnnotion is then valid

F(jco) = 4,9 + 0,019/(jco) +

72,6jco/(I+ 0,72jco).

(6) The new expectation values re(co) of the frequency

response errors, resulting from the better knowledge of the transfer behaviour of the plant are entered in Fig. 6. A verification of the possible unaccuracy of

the synthesis gives the value Ax2~(t)/xZw(t)----O,23. With that the desired accuracy is obtained and an iteration of the strategy determination is not necessary. Experience has shown that for practical applications, relative rough a.priori information about the frequency of the plant and of the power distributions of the d mmh will suffice for the determination of a sati~factory measurement strategy.

R(~sum~--Pour l'analyse de syst~nes de contr61e lin~isables lourdement stochastiquement perturb~ par des signaux de tests pr6determin~s, des strat6gies de mesures sent d6rivbys qui assurent un temps de mesure court pour l'analyse avec une pr~lsion donn~e et des limitations techniques donn6es. La strat6gie d'analyse favorable au temps est obtenue en r~olvant un probl6me de variation isop6rim6trique.avec contraintes d'in6galit6, et un probl~me d'optimisation superordinatoire. Un ordinateur digital est utilis6 pour calcuier la strat~gie d'analyse optimale. On donne un exemple qui d6montre l'application des algorithmes ALGOL programm~s.

Z j m m ~ Z u r Analyse stark stochastlsch gest~rter, linearisierbarer Regelstrecken mittels determinierter Testsignale werden Mel~trategien aufgestellt, die unter Berflcksichtigung gegebener technischer Bedingungen tilt eine geforderte

550

Brief paper

Synthesegenauigkeit eine kurze Analysezeit sichem. Diese Analysestrategien sind L6sungen eines komplizierten Optimierungsproblems, dasein isoperimetrisches Variationsproblem mit Nebmbedingun~n in Ungleichtmgsform enth~ilt. Die Berechaung der optimalen Analysestrategien erfolgt auf dem Digitalrechner. Die Anwendung der in A L G O L programmierten Algorithmen wird an einem Beispiel demonstriert. Pe3mMe---~.TLqaHSJIH30BmmcapmoBaHm~x c~creM y n p a ~ e m ~ pa3pa6a~maorcx CTpaTersx ~3Mepemml~ c noMom~o

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