Digital Transposition and Extension of Classical Analogical Control Algorithms

DIGITAL TRANSPOSITION AND EXTENSION OF CLASSICAL ANALOGICAL CONTROL ALGORITHMS

Pierre VANOENBUSSCHE Ingenieur-Chercheur du FRFC Automatic Co~trol Department University of Brussels 50 , av . F . O. Roosevelt B- 1050 Brussels(BELGIUM)

For many industrial applications , s i mple control loops wit h a P . I . D. regulator meet al l imposed criteria . Neverthe l ess , a more adequate control structure is being more frequently conceived . It uses several regulating variables to ensure very simple static optimization O F it takes into account const r aints concerning as well regulating variables as other physicaL usually not controlled variables to which security limits are imposed . These more sophisticated stru c tures are generally obtained with a compute r wh ich allows to introduce easily all functions needed by the chosen control s t r u c t u re • We have studied a few possible numerical structures including sampled P . I . O. control algorithms and which meet industrial requirements . Nevertheless integral action demands great caution in defining algorithms placed after the integrator in the numerical control loop to prevent the latter to drift during a more or less long time and to disturb the behaviour of the controlled system . Notations

- -- - - -- - -

u

s s CO

SI TPD U U

s

contin u e regulating variable a p plied to the process continue output variable of the process numerical set point deviation integral action of the regulator proportional and derivative action o f t here g u I a tor regulating variable from the regu l ator regulating variable applied to the statio n sampling period gain of the regu l ator rese t time rate t i me ga i n of the station

- pT B (p) =_1_- _e __ o p

p z

REME

complex variable (Laplace transform) complex variable ( z transform) measure of the regulating variable u

REMET

theoretical measure of the variable u REINI reinitialization variable SL slope limitation AL absolute limitation X X value at time NT

N

6X ;0(

X

SIR : ADC : OAC:

increment of X (6X =X - X _ ) N N N 1 limited value of X reinitialized value of SI selector) analog - digital converter digital-analog converter

(extremum

1. A rudimentary numerical control loop is represented in figure 1 . The interface station is fundamental as it transforms discontinuous signals coming from the computer into a continuous step signal applied to the process . Two station patterns are to be considered: the absolute regulating variable stations and the incremental regulating variable stations . As to the first ones , the computer sends at each sampling instant the absolute regulating variable to the digital-analog converter . On the other hand , the second ones include an integrator and the computer only sends the increment of the regulating variable at each sampling instant . In a similar way, we will distinguish an incremental algorithm which computes d irectly an incre ment of the regulating variable from a abso lute algorithm which computes the real ~alue of the regulating variable at each sampling instant . For a sampled P.I . D. algorithm we find 6U r,

" incremental algorithm "

transmittance of an holding circuit of order 0

41

SI N=S I N_ 1 + UN=

G{ ~

~N

+T

0

~~

EN

(~l ~ )}+ ~N-~N-1

SI

N

=TPO +SI N N

{ "absolute algorithm " For as simple a structure as the one in figure 1 we generally associate incremental algorithm to incremental regulating variable station or absolute algorithm to absolute regulating variable station . We want to i mprove the structure of f i gure 1 and to introduce 1°) A positioning loop of the station to annul the static error between the regulating variable U created by the computer and the regulating variab l e u really sent to the process , even if the gain of the station is not exactly equal to one . 2°) Absolute limitations of the regulating variable due to the maxima possibilities of the corresponding physical quantity. 3°) S l ope limitations of the regulating variable imposed by the physical system in order to avoid too fast a disturbance of all other physico-chemical phenomena influenced by the regulating variable . If the co ntr ol algori th m has a reset action, special precautions must be taken to prevent this integrator to drift as the regulating variable is blocked by the li rnitors , which would involv,e impor tant overshoots {1} . ~oreover, it is advisable to put the abso lute limitation to an absolute regulating variable station and the slope limitation to an incremental regulating variable station j ust above the digital-analog converter, and so be sure not to overflow the capacity of the part of the register reserved to the value of the regulating variable in the code word used by the converter . Four sol u tions are being proposed in figures 2 , 3,4 and 5 allowing to combine fr eely an absolute algorithm or an incre mental one with an absolute or incremental regulating variable station. We assume i n the four structures that the station has a theoretical gain equal to 1 . (G s 11 to take the imperfections of the station into account) . The test of the station will be easily done by comparing the measure of the regulating variable (REME)~ to its theorotically calculated val ue lREMET)N ' Th e memor izations required for the algo rithms clearly appear in the diagrams ("box z-1) . In figures 2 and 3 , we see that the me morizations have been done with limited values to avoid the drift of the reset action . Formula (1 . 1 }clearly shows that the reset act i on of the absolute algorithm is sub -

jected to the limited regulating variable . that is to say , it has been cut off f rom all limitations which have happened . Under these conditions , the incrementa l and absolute algorithms of figures 2 and 3 lead to an identical " overall " behavio u r of E and u provided they are correctly initialized . In figures 4 et 5 concerning absolute stations , the same principle of reinitialization is applied . N8vert h ~less , we had to define a new variable REINI for the reinitialization because of the static error which may exist between U~ and REMET (G I 1) .

s

In fact, if the stations have the same gain , the va l ue of (REINI)N inside the algorithms with "absolute station " is e;actly the same as the val~e of (REME T )~1 in the algorithms with " incremental station" (if same'E and correct initiali zation) . Those last structures seem to be compli cated , but , in reality , they are the only ones which meet requirements wi t h only three memorisations . On condition that you correct l y initialize the algorithms and you have perfect station (G s =1) , the "overall" behaviour between E and u will be the same for the four considered control structures . You will choose by taking other reasons into account . We will make some remarks which may influence that choice . The structure with "i ncremental station " are safer in case of failure of the converters. The system may indeed be conceived so that a zero increment will be sent in that case , which would only block the regulating variable sent to the process . On the 0 the r h and, wit h an " ab sol u t e s t a t ion" there is a great risk of sending a zero regulating variable and considerably dis turbing the process in case of failure of the system . Additional functions (split range , extremum selectors) are more easily imple mented into an " absolute algorithm (see § 2 . for extremum selectors) . Coefficients of the regulator must often be modified on l ine to insure correct behaviour of the controlled system in the whole desired working range. m the incremental algorithm, parametriza tion is ma de on the i ncrement of the regulating variab le (llU=llSI+ll TPO ) , whereas in the absolute algorithm , it is made on the absolute value of the proprotional and de riv a ti ve term and on the increment of the integral term (TPO+llSI) . If one of the parameters is the regUlating

42

variable itself , the parametrization of the absolute algorithm may give divergent oscillat i ons in some cases (For a regulator PI pattern with a very slow process ( 6 ~cst ) and a parametrization of the gain G function of the regulating variab l e U, the re g u 1 a tor is un s tab 1 e if

dG

I Edu l

>

)

.

2. To solve practical problems where A limitation doesn't operate on the regulating variable itself but on another varia ble controlled by that regulating variabl~ we introduce extremum selectors . We may correctly formulate the problem like this Let si and s2 be two physical quant i ties controlled by the regulating variable u. The regulating system ~ust.regulate si on condition that s7 remalns In an area limited by the v~lue (s2)o(for instance s2«s2)0)' If s2 has a tendency to go out bf the permltted range , the system must switch on control of s2 with the set point (s2)0 and let si "drift" . A solution with PlO cont r ol algorithm is proposed in figure 6 . Some

remarks are to be made

- Selection is made on the reg u lating variables and not on the deviatian s, which allows to switch in terms of respective dynamics of the two processes without waiting for the limited variable to enter the forbidded area with unco ntrolled initial speed and so remain there during a more or less long time .

-If the control algorithm, of which the regulating variab l e is blocked by the selector , has another control device on which it operates with split range, its integral term has of course not to be reinitialized in order to allow it to reach the control area by that other regulating variable . - Abso lute limitations and slope l imitations may be introduced in each control algorithm provided tho se algorithms are reinitia liz ed from the regulating var iable chose n by the selector . CONCLUSION With the help of the above mentioned algorithms, of split range and of parametrization of control algorithms , we may solve a lot of industrial control problems . We may choose , among others , control structures minimizing power and material losses in the system . We have applied thos~ algorithms (absolute P l O, incremental or absolute stations acccording to the disponibilities) to a lot of practical cases the most complicated one includes no less than three cont rolled variables , one limited variable , six control devices and has needed four control a lgorith ms and three extremun selectors . The results are satisfactory . We may , of course, generalize to algorithms that are more Sophisticated than simp l e PlO ones . You only need to include all the other actions in the TPO term except the integral term and to use again the same structures . BIBLIOGRAPHY {1}

Transposition numerique et extension d'algorithmes classiques de regula tion analogique

- A reinitialization of the integral term appears i n the algorithm where the regulating variable is refused by the selector each time a " drift " appears . It consists in substituting the integral term of the "drifting" algorit',m by the one of the algorithm chosen by the selector . In that case , comparison is in fact made on the su m TPO+6S1 of each algorithm. The presence of a lead compensation in the TPO term brings to a switch before reaching the forbidden area .

P . VANOENBUSSCHE - Laboratoire d ' Automatique , Universite Libre de Bruxelles , septembre, 1974.

- The structure defined above insures the stabilization of the system in a point with respect all limitations; neverthe less important transient overshoots or too long settling time may occur if the coef ficients of the algorithms are not very well chosen . - We have omitted in figure 6 the inter face station and its control a l gorithm which don ' t give any special diffic ul ty .

43

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~

R EH E

----!:----i

D

\.-/-------

~

f i

IT~----­ EJ Rud i mentar y

nu~erical

r.:untrol

11

r: '~-----'-----'

loop

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sent

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to

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44

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