Control Engineering [1].

7/29/2019 Control Engineering [1].

http://slidepdf.com/reader/full/control-engineering-1 1/132





Control Engineering

A guide for beginners

Manfred Schleicher

Frank Blasinger



Preface

This w ork is intend ed to b e o f pra c tic a l a s s is ta nce in control eng ineering te chnology. It w ill helpyou to select and set up a suitable controller for various applications. It describes the differenttypes of controller and the op tions for setting them up. The explana tions a nd d efinitions a re provid-ed without using a dva nced ma thema tics , a nd a re ma inly a pplied to temperature-control loo ps.

In this new a nd revise d ed ition, C hap ters 3 and 5 have be en extensively upda ted.

We w ish to tha nk our collea gues for their valua ble support in w riting this bo ok.

Fulda , J a nuary 2005

Ma nfred S chleicher Fra nk B la s inger

J UMO G mbH &Co . KG, Fulda , G ermany

Copying is permitted with source citation!

5rd Edition

Part number: 00323761Book number: FAS 525Printing date: 02.04

IS B N: 3-935742-01-0



Inhalt

1 Basic concepts ............................................................................ 7

1.1 Introduction .................................................................................................. 7

1.2 Concepts and designations ........................................................................ 7

1.3 Operation and control .................................................................................. 7

1.4 The control action ...................................................................................... 11

1.5 Construction of controllers ....................................................................... 12

1.6 Analog and digital controllers ................................................................... 18

1.6.1 S igna l types .................................................................................................. 181.6.2 Funda menta l differenc es .............................................................................. 20

1.7 Manipulating devices ................................................................................. 23

1.8 Other methods of achieving constant values .......................................... 25

1.8.1 Utilizing phys ica l effects ......... .................. ................ ................. ................ ... 251.8.2 C ons tructiona l mea s ures ............................................................................. 251.8.3 Ma inta ining c ons ta nt va lues b y operation ................................................... 26

1.9 Main areas of control engineering ............................................................ 27

1.10 Tasks of the control engineer .................................................................... 28

2 The process ................................................................................ 29

2.1 Dynamic action of technical systems ...................................................... 29

2.2 Processes with self-limitation ................................................................... 32

2.3 Processes without self-limitation ............................................................. 33

2.4 Processes with dead time ......................................................................... 35

2.5 Processes with delay ................................................................................. 37

2.5.1 P roc es se s w ith one de la y (first-orde r proc es se s) ........................................ 382.5.2 P roc es se s w ith tw o dela ys (se co nd-orde r proc es se s) ................................. 39

2.5.3 P roc es se s w ith several de la ys (hig her-orde r proc es s es ) ............................. 41

2.6 Recording the step response .................................................................... 41

2.7 Characteristic values of processes .......................................................... 43

2.8 Transfer coefficient and working point .................................................... 43



Inhalt

3 Continuous controllers .............................................................. 45

3.1 Introduction ................................................................................................ 45

3.2 P controller ................................................................................................. 45

3.2.1 The propo rtiona l ba nd ........................ ................ ................. ................ ......... 473.2.2 P erma nent devia tion a nd w orking point ...................................................... 493.2.3 C ontrollers w ith d yna mic a ction ................................................................... 52

3.3 I controller ................................................................................................... 53

3.4 PI controller ................................................................................................ 54

3.5 PD controller ............................................................................................... 57

3.5.1 The pra ctica l D co mponent - the DT1 element ............................................ 60

3.6 PID controller .............................................................................................. 61

3.6.1 B loc k dia gram of the P ID c ontroller ............................................................. 62

4 Control loops with continuous controllers .............................. 63

4.1 Operating methods for control loops with continuous controllers ....... 63

4.2 Stable and unstable behavior of the control loop ................................... 64

4.3 Setpoint and disturbance response of the control loop ......................... 65

4.3.1 S etpoint res pons e o f the co ntrol loop .......................................................... 66

4.3.2 Disturba nce respo nse .................................................................................. 674.4 Which controller is best suited for which process? ................................ 68

4.5 Optimization ................................................................................................ 69

4.5.1 The mea s ure of control q ua lity .................... ............... .................. ............... . 704.5.2 Adjustment by the os cilla tion method .......................................................... 714.5.3 Adjustment a cc ording to the tra nsfer function or proc es s s tep res pons e ... 724.5.4 Adjustment a cc ording to the ra te of ris e ...................................................... 754.5.5 Adjustment w ithout know led g e of the proces s ............................................ 764.5.6 C hec king the c ontroller settings .................................................................. 77



Inhalt

5 Switching controllers ................................................................ 79

5.1 Discontinuous and quasi-continuous controllers ................................... 79

5.2 The discontinuous controller .................................................................... 80

5.2.1 The proc es s varia ble in first-orde r proc es se s .............................................. 815.2.2 The proces s varia ble in highe r-orde r proc es se s .......................................... 835.2.3 The proc es s va ria ble in proc es ses w ithout self-limita tion ............... ............ 85

5.3 Quasi-continuous controllers: the proportional controller .................... 86

5.4 Quasi-continuous controllers: the controller with dynamic action ....... 89

5.4.1 S pec ia l fea tures of the sw itching sta g es ..................................................... 905.4.2 Co mments on disc ontinuous a nd q uas i-co ntinuous

controllers w ith one output ................ ................ ................. ................ ......... 90

5.5 Controller with two outputs: the 3-state controller ................................ 915.5.1 Dis co ntinuous co ntroller w ith tw o outputs ................................................... 915.5.2 Quas i-co ntinuous c ontroller with tw o outputs,

as a proportional controller .......................................................................... 935.5.3 Quas i-co ntinuous controller w ith two outputs and dyna mic a ction ............ 945.5.4 C omments on c ontrollers w ith two o utputs ................................................. 94

5.6 The modulating controller ......................................................................... 95

5.7 Continuous controller with integral motor actuator driver .................... 98

6 Improved control quality through special controls .............. 101

6.1 Base load .................................................................................................. 101

6.2 Power switching ....................................................................................... 103

6.3 Switched disturbance correction ........................................................... 104

6.4 Switched auxiliary process variable correction .................................... 107

6.5 Coarse/fine control .................................................................................. 107

6.6 Cascade control ....................................................................................... 108

6.7 Ratio control ............................................................................................. 110

6.8 Multi-component control ......................................................................... 111



Inhalt

7 Special controller functions .................................................... 113

7.1 Control station / manual mode ............................................................... 113

7.2 Ramp function .......................................................................................... 114

7.3 Limiting the manipulating variable ......................................................... 114

7.4 Program controller ................................................................................... 115

7.5 Self-optimization ...................................................................................... 116

7.6 Parameter/structure switching ............................................................... 118

7.7 Fuzzy logic ................................................................................................ 118

8 Standards, symbols, literature references ............................ 121



7J UMO, FAS 525, Edition 02.04

1 Basic concepts

1.1 Introduction

Automa tic co ntrol is bec oming more a nd mo re importa nt in this a ge of a utoma tion. In manufac tur-ing proc es se s it ensures that c ertain parame ters, s uch a s tempe ra ture, pres sure, sp eed o r volta ge ,ta ke up s pec ific co nsta nt values rec og nized a s the optimum, or a re ma intained in a pa rticula r rela -

tionship to other variables. In other words, the duty of control engineering is to bring these param-eters to ce rta in pre-defined values (se tpoints), a nd to ma intain them c ons ta nt a ga inst a ll disturbinginfluences. However, this apparently simple duty involves a large number of problems which arenot ob vious a t first g la nce.

Modern control engineering has links with almost every technical area. Its spectrum of applicationrang es from elec trica l eng ineering, through d rives , mec ha nica l eng ineering, right up to ma nufac tur-ing processes. Any attempt to explain control engineering by referring to specialized rules for eacha rea wo uld mea n that the co ntrol engineer has to ha ve a thorough knowledge of ea ch s pec ia l fieldin which he ha s to p rovide c ontrol. This is s imply not pos s ible w ith the c urrent s ta te o f technolog y.

However, it is obvious that there are certain common concepts behind these specialized tasks. Itsoon becomes clear, for example, that there are similar features in controlling a drive and in pres-

sure and tempe ra ture co ntrol: these fea tures ca n be d es cribed by using a sta nda rd proced ure. Thefundamental laws of control engineering apply to all control circuits, irrespective of the differentforms of equipment and instruments involved.

A practical engineer, trying to gain a better understanding of control engineering, may consult vari-ous b ooks o n the subjec t. These b ooks usua lly sug ge st tha t a mo re de ta iled knowledg e of co ntrolengineering is not pos sible, w ithout extensive ma thema tica l know ledg e. This impression is com-

pletely wrong. It is found a ga in a nd a ga in that, provided sufficient effort is ma de in pres enta tion, aclear understa nding c a n be a chieved, even in the ca se of rela tionships which a ppear to d emand a nextensive mathematical knowledge.

The rea l req uirement in so lving c ontrol ta sks is no t a know led ge of ma ny formula e or ma thema tica l

methods, but a clear grasp of the effective relationships in the control circuit.

1.2 Concepts and designations

Tod a y, thanks to increas ing s ta nda rdization, we ha ve definite concepts a nd designa tions for use in

co ntrol engineering. G erman de signa tions a re la id d ow n in the well-know n DIN S ta nda rd 19 226

(C ontrol Engineering, Definitions a nd Terms). Thes e c onc epts a re no w widely a cc epted in G ermany.

Interna tiona l ha rmoniza tion o f the des igna tions then led to DIN S ta nda rd 19 221 (S ymbo ls in co n-trol engineering), which permits the use of most of the designations laid down in the previous stan-da rd. This b oo k keep s ma inly to the de finitions a nd c onc epts g iven in DIN 19 226.

1.3 Operation and controlIn many processes, a physical variable such as temperature, pressure or voltage has to take up aspecified value, and maintain it as accurately as possible. A simple example is a furnace whosetemperature ha s to be m a intained c ons ta nt. If the energy supply, e.g . elec trica l pow er, ca n be va r-ied , it is possible to us e this fa c ility to ob ta in different furna ce tempera tures (Fig . 1). Ass uming t ha texternal conditions do not c hang e, there will be a definite tempe ra ture c orres ponding to ea ch va lueof the energy supply. Specific furnace temperatures can be obtained by suitable regulation of theelectrical supply.

However, if the external conditions were to change, the temperature will differ from the anticipatedvalue. There a re ma ny different kinds of s uch d isturbanc es or cha nges , w hich ma y be introd uced

into the proc es s a t different po ints. They c a n be d ue to va ria tions in externa l tempe rature o r in the



1 Basic concepts

8 J UMO, FAS 525, Edition 02.04

Fig. 1: Operation and control



9

1 Basic concepts

J UMO, FAS 525, Edition 02.04

hea ting c urrent, or c a used by the furna ce do or opening. This type o f temperature c ontrol takes noa cc ount of the ac tual furnac e tempera ture, a nd a devia tion from the required va lue ma y not be no-tice d b y the opera tor.

S ome form of c ontrol is nece ss a ry if the furnac e temp erature ha s to ma intain its value in spite o fchanges in external conditions, or non-constant disturbances which cannot be predicted. In itssimples t form the c ontrol may just be a thermomete r which mea sures a nd indica tes the a ctua l fur-na ce te mperature. The operato r ca n now rea d the furnac e temperature, a nd ma ke a ppropria te a d-justme nts to t he energy s upply, in the event o f a temp erature de via tion (Fig . 1).The energ y s upply is now no longer pre-de termined , b ut is linked to the furnac e te mpera ture. Thismea sure ha s c onverted furnac e ope ra tion into furna ce co ntrol, with the operato r a cting a s the c on-troller.

Control involves a comparison of the actual value with the desired value or setpoint. Any deviationfrom the s etpo int lea ds to a cha nge to the energy supp ly. The energy input is no long er fixed , a s isthe ca se with simple ope ra tion, b ut depend s o n the ac tual proc es s value a ttained. We refer to thisa s a closed co ntrol loop (Fig. 2)

If the c onnec tion to the tempera ture probe is broken, the c ontrol loop is open-circuited. B ec a usethere is no feedb a ck of the proc es s va lue, a n open c ontrol loop ca n only be us ed for opera tion.

Fig. 2: The closed control loop

The c ontrol loo p ha s the follow ing c ontrol pa rame ters (the a bb revia tions conform to DIN 19 226):

Process variable (process value, PV) x: the process value is the control loop variable which ismea sured fo r the purpos e o f co ntrol and w hich is fed into the c ontroller. The a im is tha t it should a l-wa ys b e ma de e q ual to the de sired va lue through the a ction of the co ntrol (exa mple: a ctua l furna cetemperature).

Desired value (setpoint, SP) w: the predetermined value at which the process variable has to bemaintained through the action of the control (example: desired furnace temperature). It is a param-eter which is not influenced by the control action, and is provided from outside the control loop.

Control difference (deviation) e: difference betw een de sired va lue and proc es s va ria ble e = w - x(example: difference between required and actual furnace temperature).



1 Basic concepts


Disturbance z: a n effec t w hos e va ria tion exerts a n unfa vorable influence on the proc es s value (in-fluenc e o n the c ontrolled va ria ble throug h externa l effect s).

Controller output Y R: it represents the input variable of the manipulating device (the manipulator

or actuator).

Manipulating variable y: a varia ble through which the proc es s value c a n be influenced in the re-q uired wa y (e.g . hea ting po we r of the furnac e). It forms the output of the co ntrol syste m a nd, a t thesa me time, the input of the proc es s.

Manipulation range Y h: the range within w hich the ma nipula ting varia ble ca n be a djusted .

Control loop: co nnection o f the output of the proc es s to the input of the co ntroller, a nd o f the co n-troller output to the process input, thus forming a closed loop.It c ons ists of c ontroller, ma nipula tor a nd p roc es s.

The phy s ica l units involved c a n differ wide ly:process value, setpoint, disturbance and deviation usually have the same physical units such as° C, ba r, volts, r.p.m., d epth in metres etc. The ma nipula ting varia ble ma y b e proportiona l to a hea t-

ing c urrent in a mps or ga s flow in m3/min, o r is often a pres sure expres se d in b a r. The m a nipula tionra nge de pends on the ma ximum a nd minimum values of the ma nipula ting va ria ble a nd is thereforeexpres se d in the sa me units.



11

1 Basic concepts


1.4 The control action

The ba sic ta sk of the c ontroller is to mea sure a nd prepare the proc es s value P V, a nd c ompa re itw ith the setpo int S P ; a s a res ult it produc es the a ppropria te ma nipula ting va ria ble MV. The c ontrol-ler has to perform this action in a way which compensates for the dynamic characteristics of the

co ntrolled proc es s. This mea ns tha t the proc es s value P V sho uld reac h the setpo int SP a s ra pidlya s po ss ible, and then fluctuate a s little a s po ss ible ab out it.

The a c tion of the c ontroller on the co ntrol loo p is c ha ra c terized by the follow ing pa rameters:

- the overshoot: Xo ,

- the a pproa ch time: Ta , the time ta ken for the proces s value P V to reac h thenew se tpoint S P for the first time,

- the s t ab iliz a tion tim e: Ts ,

- and a lso a greed tolerance limits ± ∆x (see Fig. 3)

Fig. 3: Criteria for control action

The c ontroller is s a id to have “ sta bilized ” w hen the proc es s is operating with a c ons ta nt manipula t-ing varia ble MV, a nd the proc es s value P V is moving within the a greed tolerance ba nd ± ∆x.

In the idea l ca se the overshoot is ze ro. In most c a se s this c a nnot be c omb ined w ith a s hort sta bili-za tion time. In certain proc es se s, e.g . s peed co ntrols, ra pid s ta biliza tion is important, a nd a slightovershoot beyond the setpoint ca n be tolera ted. Other proc ess es, such a s pla stics production ma -chinery, a re s ensitive to a temperature o vershoo t, since this ca n q uite ea sily da ma ge the tool or theproduct.



1 Basic concepts


1.5 Construction of controllers

The c hoice o f a s uita ble co ntroller depe nds es sentially on its a pplica tion. This c onc erns b oth itsmec hanica l features a nd its electrica l cha ra cte ristics. There is a wide rang e o f different de signs a nda rra nge ments , so o nly a few w ill be d is cuss ed he re. The disc uss ion is limited to electronic c ontrol-

lers, a nd excludes mec hanica l a nd pneuma tic c ontrol sys tems . The user, who is fac ed w ith choos -ing a controller for his pa rticula r ap plica tion, w ill be s how n initia lly w hich types a re a va ila ble. Thelisting is not intended to be comprehensive.

Mechanical variations:

- Compact controllers (proc es s co ntrollers) co ntain all the nec es sa ry c ompo nents (e.g . d isplay,keypa d, input for setpo int etc.) a nd a re mo unted in a ca se which includes a pow er supply. Thehousing usually has one of the standa rd ca se s izes, 48mm x 48mm, 48mm x 96mm,96mm x 96mm or 72mm x 144mm.

- Surface-mounting controllers a re usua lly insta lled inside co ntrol ca binets a nd mo unted on a

DIN-ra il or the like. Indica ting d evices such a s proc es s va lue display o r relay s ta tus LEDs a re notusua lly provided , a s the ope ra tor does not normally have a cc es s to thes e c ontrollers.

- Rack-mounting controllers a re intended for use in 19-inch rac ks. They a re o nly fitted w ith afront pa nel a nd do not have a co mplete housing.

- Card-mounted controllers co nsist of a microproc es so r with suita ble periphera ls, a nd a re use din va rious hous ing formats . They a re freq uently found in la rge-sc a le insta lla tions in conjunctionwith centra l proc es s co ntrol system s a nd P LCs . These co ntrollers a ga in have no operating or in-dica ting d evice s, since they rec eive their proc es s da ta via a n interfac e from the c entra l co ntrolroom through software programs.

Functional distinctions

The terms tha t a re us ed here a re c overed a nd expla ined in more de ta il in la ter cha pters (se e Fig . 4).

- Continuous controllers

(usua lly referred to a s propo rtiona l or a na log controllers)Controllers which receive a continuous (analog) input signal, and produce a controller outputsigna l tha t is a lso continuous (a na log ). The ma nipula ting s igna l ca n take on a ny va lue w ithin thema nipula tion ra nge. They usually prod uce output signa ls in the ra nge 0 — 20mA, 4 — 20mA or0 — 10V. They a re used to c ontrol va lve drives or thyris tor units .

- Discontinuous controllers

2-state controllers ( s ingle-setpo int controllers) w ith one switching output a re c ontrollers tha t pro-duc e a disc ontinuous output for a c ontinuous input signa l. They c a n only sw itch the ma nipula tingvariable on and off, and are used, for instance, in temperature-control systems, where it is onlynece ss a ry to s witch the heating o r co oling o n or off.

3-state controllers (do uble-se tpoint controllers) ha ve two sw itching c ontrol outputs . They a re s im-ila r to 2-sta te c ontrollers b ut have tw o outputs for ma nipula ting va ria bles. Thes e c ontrollers a reuse d for ap plica tions such a s hea ting/coo ling, humidifying/de humidifying etc .



13

1 Basic concepts


- Quasi-continuous controllers

Quasi-continuous controllers with one switching output are controllers that achieve a quasi-continuous a c tion. The a verag e va lue of the controller output over a d efined time interval show sapproximately the same time-dependent variation as a continuous controller. Applications are, forinsta nce , temp erature control (hea ting o r c oo ling), w here improved c ontrol-loo p performanc e is re-q uired . In pra ctice, q uas i-co ntinuous c ontrollers w ith one s witching o utput a re a lso des cribed a s 2-sta te co ntrollers.

Quasi-continuous controllers with two switching outputs ca n steer a process in opposing di-rec tions (for exa mple, he a ting /coo ling or humidifying/dehumidifying). Thes e c ontrollers a lsoa chieve a q uas i-co ntinuous a ction, by pulsing the s witched outputs. In prac tice , a ll co ntrollers tha tuse two outputs to steer a process in opposing directions are referred to as 3-state controllers.Here the outputs need not necessarily be switched, but can be continuous.

- Modulating controllers

Modula ting c ontrollers ha ve two sw itching o utputs a nd a re s pec ia lly des igned for motorized a ctua -tors which are use d, for insta nce, to d rive a va lve to the open a nd close d pos itions .

- Actuating controllers

Actua ting c ontrollers a re a lso used for motorized a c tuators a nd a ga in have tw o s w itching outputs.They d iffer from mod ula ting controllers b y req uiring feed ba ck of the a c tua tor pos ition (s troke re-transmission).



1 Basic concepts


Fig. 4: Difference in controller functions



15

1 Basic concepts


All thes e typ es of c ontroller (a pa rt from the d isc ontinuous controller) c a n be implemented w ith d if-ferent forms o f dyna mic res ponse . This is often referred to a s the “ controller structure”. The te rmsuse d a re P, P I, P D or P ID controllers (see Fig. 5).

Different setpoint arrangements

The s etpoint c a n be se t ma nually on the co ntroller by mea ns of a potentiometer, o r by using keysto input digital values. The s etpo int is indica ted in either a na log form (po inter of a setpo int knob), o rdigitally as a numerical value.

Another pos sibility is the use of a n externa l se tpoint. The s etpo int is then fed in a s a n electric a l sig-nal (e.g . 0 — 20mA) from s ome e xterna l device . As w ell a s thes e a nalog s igna ls, it is a lso pos sibleto us e d ig ita l s igna ls fo r se tting the setpo int. The s igna ls a re fed into the c ontroller throug h a digitalinterface and can be derived from another digital instrument, or from a computer linked to the con-troller. If this external setpoint operates according to a fixed time sequence (program), this is alsoreferred to a s program o r se q uence c ontrol.



1 Basic concepts


Fig. 5: Typical step responses

Evaluation of the process variable

The proc es s va ria ble must b e a vaila ble as a n elec trica l signa l. Its form depe nds on the s enso r usedand on the processing of this signal. One possibility is to connect the transducer signal (sensor,probe) direc tly to the c ontroller input. The c ontroller must the n be ca pa ble of proc es s ing this signa l;

in ma ny temperature probes the output s igna l is not a linea r function o f the tempera ture, a nd thecontroller mus t ha ve a suita ble linea riza tion fac ility.



17

1 Basic concepts


The o ther pos s ibility is the use of a tra nsm itter.The tra nsmitter co nverts the s enso r signa l into a sta nda rd s igna l (0 — 20mA, 0 — 10V) a nd usua llya lso linea rizes the s igna l. In this ca se the c ontroller need o nly ha ve a n input for sta nda rd s igna ls.

The proc es s va lue is norma lly d ispla yed on the controller. This ca n be in the form o f a digita l dis-pla y (numerica l indica tion), w hich has the a dva ntag e o f being read a ble from a longe r dista nce. Theadvantage of the analog display (pointer movement) is that trends such as rising or falling of theprocess variable are clearly visible, as well as the position within the control range.

Fig. 6: Example for external connections to a controller

In ma ny ca se s the proc es s value requires further proc es sing, e.g . for a rec order or for remote indi-ca tion. Most co ntrollers provide a proc es s value o utput where the proce ss varia ble is g iven out a sa s tanda rd signal.In order to signal movements of the proc es s varia ble ab ove o r below ce rta in values , the c ontrollersa re provide d w ith so -ca lled limit co mpa rato rs (limit va lue or a la rm conta ct s ), w hich provide a s igna lif the proc es s va lue infringe s set limits . This signa l c a n then be used to trigger alarms or simila r

equipment.



1 Basic concepts


1.6 Analog and digital controllers

1.6.1 Signal types

Tec hnica l sys tems ca n be c la ss ified a cc ording to the type of s igna ls a t their inputs a nd outputs.

The s igna ls differ in their tec hnica l na ture. In co ntrol sys tems w e o ften find tempera ture, p res sure,current or volta ge a s signal ca rriers w hich, a t the s a me time, determine the units o f mea surement.The s igna ls ca n be divided into d ifferent types , d epending o n their ra nge of va lues a nd va ria tionwith time.

Fig. 7: Various signal forms



19

1 Basic concepts


Analog signals

Analog signa ls have the g rea test number of pos sible s igna l levels. The mea suring device co nvertsthe proces s varia ble P V, for example a temperature, into a signa l co rres ponding to this tempera-ture. Ea ch temperature va lue c orres ponds to a value o f the electrica l signa l. If the tempe ra ture no w va ries co ntinuously, the s igna l w ill a lso va ry co ntinuous ly. We c a ll this a va lue-continuous s igna l.

The es se ntia l element in defining a nalog signa ls is that suc h s igna ls pa ss co ntinuously through afull rang e o f va lues .The time c ourse is a lso co ntinuous; a t every insta nt the signa l value c orres ponds to the te mpera-ture a t this insta nt. It is t herefore a lso a time-co ntinuous s igna l (see Fig . 7a ). In a n a pplica tion w herethe mea suring d evice operate s through a cha nnel se lec tion s witch in which the c onta c t a rm is ro-ta ting c ontinuous ly, the mea sured s igna l is o nly sa mpled a t ce rta in dis c rete times . The s igna l isthen no longer time-continuous, but time-discrete (see Fig. 7b). On the other hand, the measure-ment rema ins value-continuous, s ince the mea sured signal is fully reprod uced a t ea ch sa mpling in-stant.

Digital signals

Digital signals belong to the group of discrete signals. Here the individual signal levels are repre-

sented by numera ls (digitally). This m ea ns tha t disc rete s igna ls c a n only ta ke up a limited numberof values . The varia tion of such d isc rete s igna ls w ith time a lwa ys a ppea rs a s a se ries o f steps .A simple e xample of a sys tem w ith disc rete s igna ls is the c ontrol sys tem of a pa ss enge r lift or ele-va tor, w hich ca n only ta ke up disc rete va lues fo r the height. This typ e of s igna l a ppea rs in co ntrolsys tems using co mputers, o r digital controllers. The important fea ture here is tha t the ana log sig-na ls c a n only be converted into d ig ita l signa ls by d isc retizat ion o f the signa l level. There a re nolong er any intermediate va lues. How ever, a ss uming tha t the co nversion takes p la ce a t a n effec tive-ly unlimited sp eed , it is still pos s ible to ha ve a time-co ntinuous s igna l (see Fig . 7c). In pra ct ice, thetec hnica l methods a vaila ble limit the c onversion to a time-disc rete form. In other wo rds , the a na-log /digita l co nverter, use d in digita l control, only c a rries out the c onversion proce ss a t disc rete t imeintervals (sampling time). From the analog signal we obtain a result which is both value-discrete

a nd time-discrete (see Fig . 7d).It is q uite evide nt that c onversion of ana log to d igital signa ls in this wa y lea ds to a los s of informa -tion a bo ut the mea sured s igna l.

Binary signals

In their simples t form the s igna ls ca n only ha ve two sta tes, a nd a re therefore ca lled bina ry s igna ls.The c ontrol eng ineer is a lrea dy fa milia r with this t ype o f signa l. The tw o s ta tes a re normally d e-sc ribed a s “ 0” a nd “1”. Every sw itch used to turn a voltag e on a nd off produces a binary signa l a sits output variable. Binary signals are also referred to as logic values and are assigned the values“true” and “false”. Virtually all digital circuits in electrical engineering work with this type of logicsignals. Microprocessors and computers are built up from such elements, which only recognizethese two signa l sta tes (se e Fig. 7e).

3-state signalsSignals with the next higher information content after binary signals are 3-state signals (sometimesca lled tri-sta te s igna ls). They a re often used in connec tion w ith motors. Ess entially, a m oto r ca nha ve three o perating s ta tes . The motor ca n be sta tiona ry, or it ca n rota te c loc kwise o r anticlock-wise. Corresponding elements with a 3-state action are frequently found in control engineering,a nd a re of g rea t interes t. Eac h of the three s igna l levels c a n have a ny des ired value; in ce rta in ca s-es ea ch s igna l level ca n be a pos itive signa l, or the mag nitude of the pos itive a nd neg a tive signalscan be different (see Fig. 7f).



1 Basic concepts


1.6.2 Fundamental differences

A co ntroller prod uces a rela tionship betwe en the proc es s varia ble P V a nd the s etpoint S P, a nd d e-rives from it the ma nipula ting va ria ble MV. There a re a number of wa ys to c a rry out this ta sk: me-cha nica l, pneuma tic , elec tric a l, ma thema tica l. The mec ha nica l controller, for exa mple, alters a sig-

na l throug h a lever s ys tem, the e lec tronic c ontroller throug h op erationa l amplifiers. With the intro-duc tion o f more pow erful a nd low-co st microproc es so rs, a nother type of e lec trica l co ntroller hascornered the ma rket in rec ent ye a rs, the mic roproc es sor c ontroller (digital co ntroller). The me a -surement signal is no longer proc es se d in a n ope ra tiona l amplifier, b ut is now ca lcula ted using amicroproc es sor. The d ifferent structures found in thes e d ig ita l controllers c a n be de sc ribe d d irec tlyin ma thema tica l terms.

The term “ digital” mea ns tha t the input va ria ble, the proce ss va lue, mus t initia lly be digitized , i.e.converted into a numerical value, as described in Chapter 1.6.1, before the signal can be pro-ces sed b y the microproc es sor. The ca lcula ted o utput signa l (the ma nipula ting va ria ble) then has tobe converted back to an analog signal, by a digital to analog converter, to control the process, ora lterna tively, fed direc tly to a digital a ctua tor. There is very little functiona l difference be tw een digi-

ta l a nd a nalog co ntrollers, s o this is not c overed in-dep th in the c ontext of this bo ok.Use of a digital display is, in itself, not an adequate criterion for calling an instrument a digital con-troller. There a re instruments w hich w ork on a na log principles , b ut w hich ha ve a digital displa y.They d o not ha ve a n interna l mic roproc es sor to ca lcula te the s igna ls , a nd a re therefore s till referredto as analog controllers.

Fig. 8: Principle of analog and digital controllers



21

1 Basic concepts


Fig. 9: Arrangement of analog and digital controllers



1 Basic concepts


Advantages and disadvantages of digital controllers

Ana log controllers a re b uilt up from ope rationa l a mplifiers. The c ontrol pa rameters a re s et b ymeans of potentiometers, trimmers or solder links. Controller structure and characteristics arela rgely predete rmined b y the design a nd c ons truc tion. They a re use d w here there is no req uirementfor very high accuracy, and where the required features of the controller, such as its dynamic ac-

tion, a re a lrea dy know n at the pla nning s ta ge . Be ca use of its s peed of rea ction, the ana log co ntrol-ler ha s c lea r a dva ntag es in extremely fa st c ontrol loo ps.

In digital controllers a microprocessor converts all analog inputs into numerical values, and usesthem to c a lcula te the ma nipula ting va ria ble. This ha s c erta in ad vanta ge s c ompa red with a nalogprocessing:

- increa sed a cc ura cy of control, depending on the meas urement signa l a nd the technology used(e.g . A/D co nverter). Unlike co mponents w hich a re a ffec ted by to leranc es a nd d rift, the ma the-matica l relationships used have a co nstant a cc ura cy a nd a re unaffected by a geing, varia tions incomponents a nd temperature e ffects.

- high flexibility in the structure a nd c hara cteristics o f the co ntroller. Instea d o f having to a djust

pa ra meters or unsolder compo nents, a s in ana log co ntrollers, a digita l co ntroller ca n be mod i-fied by s imply prog ramming a new linea riza tion, c ontroller structure e tc . by inputting numerica lvalues

- fac ility for data tra nsfer. There is often a nee d to mod ify or sto re informa tion a bo ut proc es s s ta -tus va ria bles, o r pas s it on for different uses , a nd this is very s imple to a chieve using digita ltechnolog y. Remote setting of parameters through da ta s ystems, s uch as proc ess mana gementsys tems via a digital interfac e, is a lso q uite s imple.

- control pa rameters c a n be o ptimized a utoma tica lly, under certain conditions.

Digital controllers also have disadvantages compared with controllers operating on analog princi-ples . The d ig ita l disp la y, normally s ta nda rd w ith d ig ita l controllers, ma kes it mo re d ifficult to ide ntify

trends in process values. Digital instruments are more sensitive to electromagnetic interference.The proc es so r needs a c erta in time to c a lcula te pa ra meters a nd to c a rry out other ta sks, s o tha tproc es s values ca n only be rea d in at c ertain time interva ls. The time interva l betw een tw o s ucc es -sive readings of the process variable is referred to as the sampling time, and the term “samplingcontroller” is often used. Typica l va lues of the s a mpling time in comp a c t controllers a re in the ra nge50 — 500ms ec . There a re no tec hnica l rea so ns w hy co ntrollers with sa mpling times les s tha n1 mse c co uld no t be built. If the proc es s is rela tively s low co mpa red with the s a mpling time, thebehavior of a digital controller is similar to that of an analog controller, since the sampling action isno longer notice a ble.



23

1 Basic concepts


1.7 Manipulating devices

The purpose of the ma nipula ting d evice is to influenc e the proc es s va ria ble. Its ma in ta sk is to reg -ulate a mass or energy flow. Mass flows may have either gaseous or liquid state, e.g. natural gas,steam, fuel oil etc.

Fig. 10: Overview of different manipulators



1 Basic concepts


Fig. 11: Overview of different actuators

Energy flows often ta ke the form of elec tric a l energy. The energy supp ly ca n be va ried disc ontinu-

ously through contacts, relays or contactors, or continuously by means of variable transformers,va ria ble resistors or thyristo r units .



25

1 Basic concepts


The ma nipula ting d evice is freq uently o perated by a n a ctua tor where the c ontroller ca nnot ope ra teit d irec tly, for insta nce , if it c a nnot p rovide sufficient po w er, o r where the output o f the c ontroller isin the w rong energy fo rm for driving the ma nipula tor. The c ontroller then op erate s either a mec ha n-ical-pneumatic or electrically powered driver. For example, the relays built into switching control-lers c a n normally only handle currents up to 5 A; external conta cto rs or so lid-sta te rela ys a re then

used to c ontrol the higher pow er req uired by the proce ss .

Ta ble 1 g ives a brief o verview of the va rious ma nipula tors/drivers a nd their op eration from suita blecontrollers.

Table 1: Controller types and manipulators/drivers

1.8 Other methods of achieving constant values

Automa tic co ntrol, i.e. mea surement o f the proces s varia ble P V, c ompa riso n w ith the s etpoint S P,a nd p rod uction o f the ma nipula ting varia ble MV, is not the only pos sible w a y o f ensuring that a pa -

rame ter is kept c ons ta nt. There are several other metho ds of a chieving this, w hich o ften offer amore co st-effec tive s olution, a s a n a lternative to a utoma tic co ntrol.

1.8.1 Utilizing physical effects

There a re a number of physica l values which remain cons ta nt over a w ide ra nge even when s ub-jec ted to va rying external influences . They include, for example, the melting po int of a s ubs ta nce .While ice is melting, the temperature remains co nsta nt a t 0 ° C. P hysica l effec ts like this a re s uc-ce ss fully use d in ma ny mea surements , pa rticula rly in the la bo ra tory. In this wa y, a temperature c a nbe maintained cons tant to a high d egree of a cc ura cy, without the expense of s ophistica ted c ontrolequipment.

1.8.2 Constructional measures

To s ome extent, pa ra meters c a n be held c ons ta nt through s uita ble c onstructiona l features . For ex-ample, a constant liquid level can be maintained in a container or tank, in spite of variations in theinflow rate , just b y providing a n overflow (see Fig .12a). Another exam ple is a sw imming pool, wherethe w a ter level ca n be ma intained co nsta nt by providing a n overflow a ll round the po ol.

Controller type Operated manipulators/drivers

C ontinuous c ontrollers Adjus ta ble res is torThyristor unitVa lves , flaps , s lide sS peed -controlled motors

2-s ta te controllers Conta c tRela y, c onta cto r, s olenoid valveS olid-sta te rela y for heating, co oling e tc.

3-state co ntrollers (switching) Heating, cooling, relays etc.

Mo dula ting c ontro lle rs Ac tua ting mo to rs (AC , DC , 3-pha s e e tc .)



1 Basic concepts


Fig. 12: Methods of achieving constant values

1.8.3 Maintaining constant values by operation

As alrea dy d isc ussed in Cha pter 1.3, “Operation a nd c ontrol”, a parameter ca n be kept c onsta ntby s uita ble operation. An exa mple c ould b e to ma intain a c onsta nt furna ce temperature. Ass uminga cons tant voltag e, i.e. a stea dy pow er supply to a n electrica lly heated furnac e, the setting o f anenergy reg ula tor ca n be varied to provide different furnac e temp eratures . B y noting thes e tempe ra -tures , i.e. by producing a temperature s ca le a nd a tta ching this to the energy regula tor, we ca n thense t any d es ired furnac e temperature. As the a djustment is ma de b y hand, w e refer to this a s ma n-ua l ope ra tion. The input pa rame ter in this form of temp erature control is the s etting of the energyreg ula tor, the o utput va ria ble is the furnac e te mperature, w hich ca n be displa yed on a suita ble indi-ca ting instrument (see Fig. 1).



27

1 Basic concepts


Adjustment of the input parameters need not take place manually, but can be automated: this isthen ca lled a utoma tic operation. As a n exa mple, ta ke the c ontrol of a mixing proce ss . The ta skconsists of producing a flow Q2 which is proportional to an externally determined flow Q1 in order

to a chieve a pa rticular mixture ra tio (se e Fig . 12b). Here the flow Q1 is d etermined a s the input va ri-

a ble, and is a pplied to the o perating e q uipment. The output of the ope ra ting e q uipment ope ra tes ama nipula tor which c hang es the flow Q2.

From this it is c lea r that a proc es s va ria ble ca n a lso be kept co nsta nt by s imple ope ra tion. How ev-er, it should be borne in mind that operation has considerable disadvantages compared with auto-ma tic c ontrol. If the proc es s is sub jec ted to a disturba nce, o r there is a cha nge in the tra nsfer cha r-a cteristic of the ma nipula ting d evice , there c a n be undesira ble cha nges in output, even with a fixedtra nsfer ac tion b etw een input and output varia bles.

1.9 Main areas of control engineering

Tod a y, co ntrol eng ineering ha s a pplica tions in almos t every area o f techno log y. In Cha pter 1.1 w e

have already seen that these different applications have certain common features, which can bedescribed through a standard procedure. A number of main application groups have evolved as aresult of differing process variables, stabilization rates, types of machinery and equipment, andcertain special features of the application field.

Fig. 13: Main areas of control engineering

Industrial process control

This hea ding c overs the c ontrol of tempera ture, p res sure, flow , level etc . in many different industria lapplications. If we look at the criterion “stabilization time”, this can have an order of magnituderanging from milliseconds, e.g. in pressure control, up to several hours in the case of temperaturecontrol of larger installations (industrial furnaces).

Drive control (speed control)This group includes spe ed co ntrol of motors on d ifferent ma chines a nd insta lla tions , s uch a s inplastics manufacture, paper production or textile machinery. Specially designed controllers arenormally used for these applications, since they have to remain stable during fast disturbances inthe ra nge of tenths of sec onds.

Control of electrical variables

This refers to s ta biliza tion of electric a l pa rameters, e .g . voltag e, c urrent, po w er or even freq uency.This typ e of eq uipment is use d in pow er gene ra tion or to s ta bilize c ha rac teristic va lues in supplynetworks. Here again there are very fast disturbances, in the range of tenths of seconds or evenshorter.

! industrial process control

! drive control (speed control)

! control of electrical variables

! pos itiona l control

! course control



1 Basic concepts


Position control

This involves the po s itioning of tools, w orkpiec es or co mplete a ss emb lies , either in tw o o r three d i-mensions. Examples include a milling machine and the positioning of guns on ships and tanks.Once a ga in, sta biliza tion a t the setpoint must be very ra pid a nd very a cc ura te.

Course control

The c ontrol of the co urse of s hips or pla nes. Here the c ontroller has to s a tisfy s pec ia l dema nds ,such as high processing speed and operational safety, combined with low weight.

1.10 Tasks of the control engineer

S o fa r we ha ve disc ussed various c oncepts and des igna tions, the differences betwe en operationa nd c ontrol and the various forms of c ontrollers a nd ma nipula tors. We c a n now summa rize theta sks a co ntrol engineer ha s to fa ce in pra ctice.

The mo st importa nt ta sks for a co ntrol engineer are a s follow s:

Fig. 14: Tasks of a control engineer

By control engineer, we don’t mean specialist engineers and technicians from universities or re-se a rch depa rtments, w ho w ork in the lab oratory de veloping co ntrollers, c ontrol algorithms o r spe-

cial control circuits. Specialists such as these require a much more extensive knowledge. Insteadwe a re a dd res sing people wo rking o n site w ho ma y have to optimize a n unsa tisfa cto ry co ntrol loopor co nvert from ma nual opera tion to a utoma tic co ntrol, or thos e involved in the d es ign o f a co ntrolloop for a new installation. In most cases these operations can be tackled without using advancedmathematics. All that is really needed is a basic understanding, pragmatic rules and knowledgega ined from pa st experience .

As a general principle for planning a control system, it should be borne in mind that when high-performanc e de ma nds a re pla ce d o n a c ontroller, the co sts will increas e c onsiderab ly.

! Determining the proc es s va ria b le

! C hecking w hether a utoma tic c ontroloffers significant advantages

! Determining the me a sureme nt s ite

! Assessing the disturbances

! S elec ting the ma nipula tor

! Selecting a suitable controller! Ins ta lla tion of the c ontroller

in a ccorda nce w ith a pplica ble reg ula tions

! Starting up, adjusting parameters, optimizing




2 The process

2.1 Dynamic action of technical systems

The proc es s is the element of a sys tem w hich ha s to be c ontrolled in ac co rda nce w ith the applica -tion duty. In practice, the process represents either an installation or a manufacturing processwhich requires controlling. Normally, the process covers a number of elements within a system.

The input is the ma nipula ting va ria ble y rec eived from the c ontrol device. The o utput is representedby the proces s va ria ble x. As w ell a s thes e two varia bles there a re the disturba nces z w hich a ffec tthe proc es s to so me extent, through e xterna l influences or proc es s-depe ndent va ria tions .

An exa mple o f a proc es s is a ga s-fired furna ce (se e Fig. 15). At the s ta rt of the proce ss is the valve,w hich has a s its input the ma nipula ting va ria ble of the controller. The va lve c ontrols the ga s flow tothe burner. The b urner prod uces hea t energy by b urning the ga s, which brings the c harge up to ahigher temperature. If the temperature in the charge is measured (process value), this also formspa rt of the proces s. The final comp onent of the proces s here is the s enso r, w hich ha s the job ofconverting the temperature into an electrical signal. Disturbances here are all the variables in theproc es s which, when they c hang e, result in a different temperature for the sa me va lve se tting.

Example: If the manipulating variable is just large enough to give the required temperature in the

cha rge , a nd a disturba nce oc curs due to a fall in outside a mbient temperature, then, if the manipu-lating variable is not changed, the temperature in the charge will also be lower.

Fig. 15: Input and output variables of a process

When de signing a co ntrol loo p, it is important to know how the proc es s res ponds when there is achange in one of the influencing variables mentioned above. On the one hand, it is of interest toknow the new process value reached when stable conditions have been attained, following suchcha nges . On the other ha nd, it is a lso important to find out how the proc es s va lue varied with timeduring the tra nsition to the new stea dy-sta te va lue. A know ledg e o f the cha ra cte ristics dete rmined

by the proce ss is e ss entia l a nd c a n help to a void d ifficulties la ter on, when de signing the proce ss .



2 The process


Although processes have many different technical arrangements, they can be broadly categorizedby the following features:

- with and without self-limitation,

- with and without dead time or timing elements,

- linear or non-linear.

In mos t c a se s, how ever, a co mbina tion o f individua l cha ra cteristics will be pres ent.

An a cc ura te c hara cte riza tion a nd d eta iled knowledg e o f the proc es s is a prereq uisite for the des ignof c ontrols a nd for the optimum so lution o f a co ntrol tas k. It is not po ss ible to s elec t suita ble co n-trollers a nd a djust their paramete rs, without know ing e xac tly how the proc es s beha ves. The de -scription of the dynamic action is important to achieve the objective of control engineering, i.e. tocontrol the dynamic behavior of technical dynamic systems and to impose a specific transient re-spons e on the technica l sys tem.

Static characteristic

The s ta tic beha vior of a tec hnica l sys tem c a n be des cribed by c ons idering the o utput signa l in rela -tion to the input s igna l. In other words , b y d ete rmining the va lue of the output s igna l for different in-put signals. With an electrical or electronic system, for instance, a voltage from a voltage sourceca n be a pplied to the input, and the c orres ponding output voltag e d etermined. When co nsideringthe s tatic b eha vior of c ontrol loo p elements, it is of no importance how a pa rticula r control elementrea ches its fina l s ta te. The only comp a rison ma de is limited t o the va lues o f the input and outputsigna ls a t the e nd o f the s ta biliza tion o r settling time.When mea suring sta tic cha ra cte ristics, it is interes ting to know, a mong st other things , w hether theparticular control loop element exhibits a linear behavior, i.e. whether the output variable of thecontrol element follows the input proportionally. If this is not the case, an attempt is made to deter-mine the exact functional relationship. Many control loop elements used in practice exhibit a linearbe havior over a limited ra nge. With spec ia l reg a rd to the proc es s, this mea ns tha t w hen the ma nip-

ulating variable MV is doubled, the process value PV also doubles; PV increases and decreaseseq ua lly with MV.An exa mple of a transfer eleme nt with a linea r cha rac teris tic is a n RC netw ork. The output voltag eU2 follows the applied voltage U1 with a certain dynamic action, but the individual final values areproportiona l to the a pplied vo lta ges (see Fig . 16). This c a n be expres sed by s ta ting tha t the pro-cess gain of a linear process is constant, as a change in the input value always results in the samecha nge in the output va lue.

How ever, if we now loo k at a n elec trica lly hea ted furna ce , w e find tha t this is in fac t a non-linea rproc es s. From Fig. 16 it is clear that a cha nge in hea ter powe r from 500 to 1000W prod uces a la rg-er tempera ture increa se tha n a c ha nge in pow er from 2000 to 2500W. Unlike the beha vior of an RCnetwork, the furnace temperature does not increase to the same extent as the power supplied, as

the hea t los se s d ue to ra diation be co me more pronounce d a t higher temperatures . The pow ermust therefore be increas ed to c ompens a te for the energy los se s. The transfer co efficient or pro-ce ss ga in of this type o f syste m is not c ons ta nt, but dec rea se s w ith increas ing proces s va lues. Thisis co vered in more de ta il in Cha pter 2.8.



31

2 The process


Fig. 16: Linear and non-linear characteristics

Dynamic characteristic

The dyna mic resp ons e of the proces s is de c is ive for cha rac terizing the c ontrol loo p. The dyna miccharacteristic describes the variation in the output signal of the transfer element (the process)

w hen the input signa l va ries w ith time. In theory, it is pos s ible for the output va ria ble to c ha nge im-media tely a nd to the s a me extent as the input varia ble c hanges . However, in many c as es, the sys-tem res ponds with a ce rta in dela y.

Fig. 17: Step response of a process with self-limitation

Processy

t

y

z

x

t

z

t

x



2 The process


The s implest w a y of es ta blishing the b eha vior of the output signa l is to rec ord the va ria tion o f theproce ss va lue P V w ith time, a fter a s tep c ha nge in the ma nipula ting va ria ble MV. This “s tep re-sponse” is determined by applying a step change to the input of the process, and recording thevaria tion o f P V with time. The s tep c hang e need not nec es sa rily be from 0 to 100%; s tep c hang esover sma ller ra nges ca n be a pplied, e .g. from 30 to 50%. The dyna mic b eha vior of proc es se s c a n

be clearly predicted from this type of step response, which will be discussed in more detail inCha pter 2.6.

2.2 Processes with self-limitation

P roc es se s with self-limita tion respond to a cha nge in the ma nipula ting va ria ble or to a disturba nceby mo ving to a new s ta ble proces s va lue. This type of proc es s c a n diss ipa te the energy s uppliedand achieve a fresh equilibrium.

A clas sic exa mple is a furna ce where, a s the hea ting po we r is increa se d, the tem perature rise s untila new equilibrium temperature is reached, at which the heat lost is equal to the heat supplied.

Howe ver, in a furnac e, it takes so me time to a chieve the new eq uilibrium follow ing a ste p c ha nge inthe ma nipula ting varia ble. In proc es se s without de la ys, the proces s value immedia tely follow s thema nipula ting va ria ble. The s tep respons e o f such a proc es s then has the form show n in Fig. 18.

Fig. 18: Process without delay; P process

In this type of proce ss w ith se lf-limita tion, the proc es s va lue P V is p ropo rtiona l to the ma nipula tingvaria ble MV, i.e. P V increas es to the s a me extent a s MV. S uch proces se s a re o ften ca lled propo r-tiona l proc es se s or P proc es se s. The rela tions hip be twe en proc es s va lue x a nd ma nipula ting va ri-a ble y is given by:

∆x KS ∆y•=



33

2 The process


The fa c tor KS is know n a s the proce ss ga in (trans fer coe ffic ient). The rela tions hip w ill be disc uss edin more d eta il in Cha pter 2.8.

Examples of propo rtiona l proc es se s a re:

- mechanica l gea ring without slip

- mechanica l transmiss ion by lever

- tra nsistor (collector current Ic follow s the ba se current IB with virtually no delay)

2.3 Processes without self-limitation

A process without self-limitation responds to a change in the manipulating variable or to a distur-ba nce b y a permanent co nsta nt cha nge in the proc es s va lue. This type o f proc es s is found in thecourse control of an aircraft, where a change in the manipulating variable (rudder deviation) pro-duc es a n increa se in the proc es s va lue de via tion (course d eviation) w hich is propo rtiona l to time. Inother words, the course deviation continually increases with time (see Fig. 19).

Fig. 19: Process without self-limitation; I processBecause of this integrating effect, such processes are also called integral processes or I process-es . In this type of proces s, the proc es s value increa se s propo rtiona lly w ith time a s a res ult of a stepchange ∆y in the ma nipula ting va ria ble. If the c ha nge in MV is do ubled, the proce ss va lue w ill a lsodo uble a fter a ce rtain time.

If ∆y is constant, the following relationship applies:

KIS is ca lled the trans fer coe ffic ient o f the proc es s w ithout s elf-limita tion. The proces s va lue now

increases proportionally with both the manipulating variable change ∆y, as in a process with self-limita tion, a nd a lso w ith time t.

∆x KIS ∆y t••=



2 The process


Additional examples of processes without self-limitation are:

- an electric motor driving a threa ded s pindle

- the liq uid level in a ta nk (se e Fig. 20)

Fig. 20: Liquid level in a tank; I process

P rob a bly the bes t known exa mple o f a proc es s without s elf-limita tion is a liq uid co ntainer with a ninflow a nd a n outflow. The outlet va lve, w hich here repres ents the disturba nce, is a ss umed to beclosed initially. If the inlet valve is now opened to a fixed position, the liquid level (h) in the containerw ill rise s tea dily a t a uniform ra te w ith time.

The level in the c onta iner rise s fas ter a s the inflow rate increa se s . The w a ter level will continue torise until the c onta iner overflow s. In this c a se, the p roc es s d oes no t se lf-st a bilize. Ta king the effectof outflow into consideration, no new equilibrium is reached after a disturbance (except when in-flow = outflow ), unlike the ca se o f a proc es s w ith se lf-limita tion.

In genera l, proc es ses w ithout s elf-limita tion a re mo re difficult to control tha n thos e w ith s elf-limita -tion, a s they d o not s ta bilize. The rea so n is, tha t follow ing a n overshoot d ue to a n exces sivechange in MV by the controller, the excessive PV cannot be reduced by process self-limitation.Ta ke a ca se where the rudder is mo ved too far when ma king a co urse a djustment, this c a n only beco rrec ted by a pplying a n oppo sing MV. An exce ss ive c hang e in MV co uld ca use the proc es s valueto s wing ba ck be low the des ired se tpoint, which is w hy co ntrol of such a proc es s is mo re d ifficult.



35

2 The process


2.4 Processes with dead time

In proc es se s w ith a pure dea d time the proc es s o nly res ponds a fter a c erta in time ha s elaps ed, thedead time Tt. Similarly, the response of the process value is delayed when the manipulating vari-a ble cha nges ba ck (se e Fig. 21).

Fig. 21: Process with dead time; Tt process

A typica l exa mple here is a be lt co nveyor, whe re there is a certain time d elay b efore a c ha nge in thechute feed ra te is rec orded a t the mea surement loca tion (se e Fig. 22).

Systems like this, which are affected by a dead time, are called Tt proce sse s . The relationship be -twe en proc es s va lue x and ma nipula ting va ria ble y is a s follow s:

but dela yed by the de ad time Tt.

∆x KS ∆y•=



2 The process


Fig. 22: Example of a process with dead time; belt conveyor

Another example is a pressure control system with long gas lines. Because the gas is compress-ible, it takes a certain time for a pressure change to propagate. By contrast, liquid-filled pipelineshave virtually no dea d time, since a ny pres sure cha nge is propa ga ted a t the speed o f sound. Rela ysw itching times a nd a ctua tor stroke times a lso introd uce d ela ys, s o tha t such elements in the con-trol loop frequently give rise to dead times in the process.

Dea d times po se a se rious prob lem in co ntrol engineering, s ince the effec t of a cha nge in ma nipu-lating variable is only reproduced in the process variable after the dead time has elapsed. If thechange in manipulating variable was too large, there is a time interval before this is noticed andacted on by reducing the manipulating variable. However, if this process input is then too small, it

has to be increa sed once more, a ga in after the dead time has elapsed , and s o the seq uence c on-tinues. S ystems a ffected b y dea d time a lwa ys ha ve a tendency to os cilla te. In ad dition, dea d timesca n only rea lly be co mpensa ted for by the use of very co mplex c ontroller des igns . When de signinga nd c onstructing a proc es s, it is very important that d ea d times a re a voide d w herever poss ible. Inmany c a ses this ca n be a chieved by a suitab le a rra ngement of the senso r and the a pplica tion pointof the ma nipula ting va ria ble. Thermal and flow res is ta nces should b e a voided o r kept to a mini-mum. Alwa ys try to mount the se nso r at a suita ble loc a tion in the proc es s where it w ill rea d the a v-erage value of the process conditions, avoiding dead spaces, thermal resistances, friction etc.

Dea d times ca n oc cur in proc es se s with and without s elf-limita tion.



37

2 The process


2.5 Processes with delay

In many processes there is a delay in propagation of a disturbance, even when no dead time ispresent. Unlike the case explained above, the change does not appear to its full extent after thede a d time has ela pse d, but varies co ntinuously, even follow ing a ste p c hang e in the disturbing in-

fluence.Co ntinuing with the e xample of a furna ce , a nd looking close ly a t the interna l temperature propa ga -tion:

If there is a sud den cha nge in hea ting po we r, the e nergy must first of a ll hea t up the hea ting ele-ment, the furnac e ma teria l a nd o ther parts of the furnac e until a prob e inside the furnac e c a n reg is-ter the c ha nge in temp erature. The tempera ture the refore rises slow ly a t first until the temperaturedisturba nce ha s p ropa ga ted a nd there is a co nsta nt flow of energy. The temperature then co ntin-ues to rise . Over a period of time the tempe ra ture of the hea ting e lement a nd the probe c ome clos-er and c los er tog ether; the temperature increas es a t a lowe r ra te a nd a pproa ches a fina l value (se eFig. 23).

Fig. 23: Processes with delay



2 The process


As a n ana log y, co nsider two pres sure vess els w hich a re c onnec ted b y a throttle valve. In this c a se ,the air must flow into the first vessel initially, and build up a pressure there, before it can flow intothe s ec ond vess el. Eventua lly, the press ure in the first vess el rea ches the s upply press ure, a nd nomore air can flow into it. As the pressures in the two vessels slowly come into line with each other,the pressure equalization rate between the two vessels becomes slower and slower, i.e. the pres-

sure in the s ec ond vess el rise s more and more slow ly. Follow ing a ste p c hang e in the ma nipula tingvariable (in this case the supply line pressure) the process value (here the pressure in the secondves sel) w ill ta ke the follow ing course: a very slow rise to be g in w ith until a certain pres sure ha s builtup in the first ves se l, follow ed by a ste a dy rise a nd then finally a n a sympto tic or grad ual approac hto the fina l va lue.

The trans fer function of this typ e of s ys tem is de termined b y the number of energy sto res a va ila blewhich are sepa ra ted from ea ch other by res ista nces . This c onc ept ca n also b e used when referringto the number of delays or time elements present in a proc es s.

Such processes can be represented mathematically by an equation (exponential function) whichhas an exponential term for each energy store. Because of this relationship, these processes are

de signate d a s first-order, s ec ond-order, third-order proc es se s, a nd s o o n.The sys tems ma y be proces se s w ith or without self-limita tion, w hich c a n also b e a ffec ted b y dea dtime.

2.5.1 Processes with one delay (first-order processes)

In a process with one delay, i.e. with one available energy store, a step change in MV causes theP V to c hang e immediately without dela y a nd a t a c erta in initia l ra te of c hang e: P V then approa chesthe fina l va lue mo re a nd more s low ly (se e Fig . 24).

Fig. 24: First-order process; PT1 process



39

2 The process


For a s tep change ∆y the relationship is as follows:

The term in brac kets sho ws that a step cha nge in MV do es not prod uce a co rres ponding immedi-a te c ha nge in P V. Instea d P V slowly approa ches the final value in a cha ra cte ristic ma nner. As thetime t increas es (la rge va lue of t/T), the va lue of the e xpres s ion in the brac kets te nds tow a rds 1, sotha t for the fina l va lue, ∆x = KS • ∆y.

As sho w n in Fig . 24, a fter a time t = T (time c ons ta nt), the P V ha s rea ched 63% of the fina l va lue.After a time t = 5 T, the P V ha s a lmos t rea ched 100% of the fina l va lue.

Such processes are also referred to as T1 processes. If it is a process with self-limitation, it is re-ferred to a s a P T1 process; a process without self-limitation is an IT1 process. Processes with one

delay (first-order) occur very frequently. Examples are:- heat ing and cooling of a hot water tank

- filling a co ntainer with air or ga s via a throttle va lve or a s ma ll bo re pipe

2.5.2 Processes with two delays (second-order processes)

In a proce ss with two d elays there must be two storag e elements, co nnected together by a resis-tanc e. S uch proc ess es ca n be cha ra cterized by s pecifying the transfer co efficient KS a nd the timeconstants T1 and T2. Here, in contrast to a first order process, the step response of the processvalue s ta rts with a ho rizonta l pha se a nd a lso has a point of inflec tion (se e Fig. 25).

The co urse of the step respo nse c a nnot be d ra wn b y simply co mbining T1 and T2. For a stepchange ∆y a nd for T1 not equa l to T2 , the relationship is as follows:

Example:A typical example of a first-orderproc ess is the cha rge or disc harge

of a capacitor through a resistor.The plot of the p roc es s va ria ble(capacitor voltage) follows a typi-cal exponential function.

Uin Uout

R

Uout = Uin (1 - e )

-tRC

∆x KS ∆y 1 e

-tT-----

–⎝ ⎠⎜ ⎟

⎜ ⎟⎛ ⎞

••=

∆x KS ∆y 1T1

T1 T2–-----------------e

t–T1------

+T2

T1 T2–-----------------e

t–T2------

–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

••=



2 The process


Fig. 25: Second-order process; PT2 process

S uch a proc ess is normally ca lled a P T2 proc ess . As a lrea dy discuss ed, s eco nd-order proc ess esa lwa ys ha ve a point of inflec tion, w here the rad ius o f curvature c hang es from a left-hand to a right-hand curve. First-order processes do not have this point of inflection.

Typica l exa mples of this t ype o f ac tion a re:

- filling tw o c onta iners in se ries w ith air or ga s through restricto rs (se e Fig. 23)

- charging up two RC networks in series

- tempera ture rise in a heated hot-wa ter tank, where the thermometer is mounted in a pocket.



41

2 The process


2.5.3 Processes with several delays (higher-order processes)

If there a re mo re tha n two sto ra ge elements , the proces s has a co rres pondingly higher order.Interestingly enough, the transfer function characteristic of a higher-order process shows very littlecha nge from tha t of a s ec ond-orde r proc es s. The rise o f the c urve doe s, how ever, bec ome increas -

ingly stee per and more de la yed , until, w ith an infinite numbe r of time d elay elements , it approa che sa pure d ea d time (se e Fig . 26).

The o rder of a proc es s is a n important c hara cteristic, pa rticula rly w hen de sc ribing it ma thema tica l-ly. In practice, almost every process is made up of a large number of widely differing energy stor-age elements, such as protective fittings, filling materials for temperature probes, dead spaces inma nometers, etc . As a res ult, it is q uite impos sible to g ive a n a cc ura te ma thema tica l des cription ofan actual process.

Fig. 26: Processes with several delays

In practice, the exact order of the process is not as important as might appear at first glance. Of

much g rea ter significa nce a re the longes t delay times, w hich d etermine the na ture of the proce ss .

As the order of the proc es s increas es , it b ec omes more a nd mo re d ifficult to co ntrol, since it a p-proximates more and more to a system with dead time. A combination with a pure dead time isalso possible, when the controllability deteriorates even more. Controllability is improved whenthere are significant differences between the time constants of the individual process elements.The wo rst c a se oc curs when the time co nstants have the sa me value.

2.6 Recording the step response

The ste p res pons e of a proc es s, i.e. the course of the proc es s va lue P V follow ing a ste p cha nge inma nipula ting varia ble MV ca n be cha ra cterized by tw o time values :

- the d elay tim e Tu, a nd the

- response tim e Tg

If thes e times a re known, a q uick estimate o f the c ontrolla bility of a proce ss ca n be ma de, a nd thecontrol pa ra mete rs de termined in a s imple wa y, a s e xpla ined la ter. The order of the proc es s is ig -nored when using this approac h, where it is a ss umed that a ny proc ess is mad e up of a dea d timeTu a nd a first-order proc es s w ith a time co nsta nt Tg .

To d ete rmine such a tra nsfer func tion a nd the resulting d elay a nd respo nse times , a rec order isco nnected to the transd ucer (se nsor) a nd the ma nipula ting varia ble (e.g . hea ting current) cha ngedsud denly. Obviously, the c hang e in MV sho uld be limited to a value s uch tha t the new se tpoint ca n

be rea ched without dama ging the s ystem. The co urse of the proce ss value is reco rded, a tange ntis draw n to the curve a t the po int of inflec tion, a nd Tu and Tg a re d etermined a s sho wn in Fig. 27.

y

t

∆y

t 0

infinite order

Process value

t

x

t 0



2 The process


Fig. 27: Determining the delay time and response timeThe ratio o f dela y time to res pons e time g ives informa tion a bo ut the cha ra cter of the proc es s a ndits con trolla b ility:

Tg /Tu more than 10: process easy to control

Tg /Tu between 10 and 3: process can be controlled

Tg /Tu less than 3: process difficult to control

As the ratio of response time Tg to delay time Tu reduces, there is an increasing delay before thechange in manipulating variable is communicated to the controller, and the controllability is pro-gressively reduced. As explained in Chapter 2.5.3, a low Tg/Tu ra tio c orresponds to a steep g rad i-

ent on the graph, representing a higher-order process which is difficult to control because of itstendency to overshoot.

Fast processes with Tg/Tu less than 3 are comparatively rare in furnaces, for example, since thetemperature d isturbanc e p ropa ga tes rela tively s low ly a nd c ontinuously through the furnac e ma teri-a l. One exc eption is the type o f furna ce where the hea ting a cts direc tly on the c harge. The s ituationis quite different with pressure control: opening an air-lock can lead to a sudden drop in pressure,to which the controller must respond with an equally fast increase in supply pressure. Pressureeq ualiza tion in the s ystem ta kes pla ce just a s q uickly, s o tha t the entire c ontrol proc es s is co mplet-ed within a s hort spa ce of time. In these proce ss es the dea d times a re long in rela tion to the d ela ytimes . C ertain c hemica l proc es se s (rea ctions, neutraliza tion) ca n s ometimes proc eed very q uickly.

As well as the delay and response times, another important characteristic of the process can be

determined, the maximum rate of rise Vma x. It is ob ta ined from the s lope of the ta ngent a t the pointof inflec tion (see Fig . 27):

y

t

Dx

t

x

Dt

inflection point

inflection tangent

Tu

Tg

Dy



43

2 The process


As shown later in the section on optimization, the maximum rate of rise is also referred to whense tting c ontrol pa ra meters.

DIN Spe c ific a tion 19 226 refers to the s ta rt-up va lue A inste a d o f the rate o f rise. This st a rt-up va l-ue is the rec iproc a l of the ma ximum ra te of rise of the proces s value P V for a s udden c hang e in thema nipula ting va ria ble from 0 — 100%:

2.7 Characteristic values of processes

The delay times a nd res pons e times (sta nda rd va lues) of so me typica l proc es se s a re s how n in

Ta ble 2 b elow .

Table 2: Delay times and response times (standard values) for some processes

The values given in the tab le s hould b e ta ken a s a vera ge values a nd s erve only a s a rough g uide.For practical applications, the values of delay time and response time should be determined byca rrying o ut a s tep respons e tes t.

2.8 Transfer coefficient and working point

Previous sections have dealt mainly with the dynamic characteristic of the process (course of theste p res pons e), i.e. its be havior with res pec t to time. Cha pter 2.1 has a lrea dy me ntioned the sta ticcha ra cte ristic, a nd d es cribed the final values for various ma nipula ting va ria bles. No a cc ount is ta k-en of cha nges in the proc es s va lue with res pec t to time.

The trans fer coe ffic ient is g iven by the ratio of o utput to input va lue, in this ca se the ra tio o f thecha nge in proc es s varia ble to the cha nge in ma nipula ting va ria ble.

Process

variable

Type of process Delay time Tu Response time Tg

Tempera ture sma ll electrica lly-heated ovenlarge electrically-heated furnacelarge gas-fired reheating furnaceautoclavehigh-pressure autoclaveinjec tion molding ma chineextruder

pac kag ing ma chine

0.5 — 1 min1 — 5 min0.2 — 5 min0.5 — 0.7 min

12 —15 min0.5 — 3 min1 — 6 min

0.5 — 4 min

5 — 15min10 — 20min

3 — 60min10 — 20min

200 — 300min3 — 300min5 — 60min

3 — 40min

P re s sure d rum b oile r, g a s o r o il-fire ddrum boiler, solid fuel-fired

0sec0 — 2 min

150sec2.5 — 5min

Flow ga s pipelinesliq uid pipelines

0 — 5s ec0sec

0.2 — 10sec0sec

Vm a xchange in process variable

unit time-------------------------------------------------------------

∆x∆t------= =

A1

Vma x-------------=

KSchange in process variable

change in manipulating variable

------------------------------------------------------------------------∆x

∆y

------= =



2 The process


The tra ns fer co effic ient KS is a lso ca lled the proc ess ga in.In many ca ses the proc ess ga in KS is not c onsta nt over the entire rang e of the proc es s va ria ble, asthe following case will explain: in a furnace, a small increase in heating power is sufficient to pro-duc e a la rge increas e in temperature a t the low er end of the temperature rang e; a t the upper end o fthe temperature range, however, a much larger change in energy flow is required to achieve the

sa me e ffec t (se e Fig . 28).

Fig. 28: Process gain and working point

The reas on for this is that the furnac e us ed in the exa mple repres ents a non-linea r proc es s. In a d-dition to temperature processes, such processes also include processes where the friction is pro-portional to speed, relationships between motor power and speed, etc.

The d ela y time a nd respo nse times in non-linea r proc es se s a lso dep end o n the w orking p oint. P ro-cess ga in KS , de la y time, respons e time a nd o ther such va lues must b e referred to a wo rking po int,

i.e. to a pa ir of values o f MV a nd P V, a t w hich they ha ve bee n eva luated . In ge nera l, they a re notvalid for other wo rking points; in non-linea r proc es se s, a new se t of va lues must b e d etermined foreach different working point, since the controller setting depends on the process parameters. Insuc h proc es se s the a ction of the co ntroller is only optimized a t the w orking p oint of the proc es s forwhich the va lues we re e valua ted. If this is cha nged , for example, if a different proc es s temperatureis required, the controller has to be re-tuned to achieve optimum control.

G enera lly, the w orking po int s hould lie in the rang e from middle to upper third of the trans fer func-tion a t full pow er. A w orking po int in the low er third is les s s a tisfa ct ory, be ca use of the la rge exc es spow er. Althoug h the d es ired va lue (setpo int) is rea ched more ra pidly in this ca se, the controlla bilityis ma de wo rse . A wo rking p oint in the upper part of the c hara cte ristic is a lso unsa tisfa cto ry, due tothe la ck of res erve po we r and resulta nt slow sta biliza tion, a nd is a lso unsa tisfa cto ry from the po int

of view of disturba nces .

Temperature ° C

300

200

100

05 10 15

Heating pow er kW

WP1: K

K depends on the working point

AP1

AP2

Dy

Dy

D

x

Dx

WP: working point

ð

=D

D

xy

= = 20

WP2: K =D

D

xy

= = 6

S

S

S30 ° C5 kW

° CkW

100 ° C5 kW

° CkW




3 Continuous controllers

3.1 Introduction

After discus sing p roc es se s in Chapte r 2, we now turn to the se co nd important element of the co n-trol loo p, the c ontroller. The controller has a lrea dy b een des cribe d a s the element w hich ma kes thecomparison between process variable PV and setpoint SP, and which, depending on the control

de via tion, prod uce s the ma nipula ting va ria ble MV. The output of a continuous controller ca rries aco ntinuous or a nalog signa l, either a volta ge or a current, w hich c a n ta ke up a ll intermediate valuesbetw een a sta rt value and a n end value.

The o ther form o f co ntroller is the d iscontinuous or q uas i-continuous co ntroller in w hich the m a nip-ula ting va ria ble c a n only be sw itched on or off.

Continuous controllers offer advantages for certain control systems since their action on the pro-ces s ca n be c ontinuously modified to meet dema nds imposed by proc ess events. C ommon indus-try sta nda rd o utput signa ls for continuous c ontrollers a re: 0 — 10V, 0 — 20mA, 4 — 20mA. On aco ntinuous c ontroller with a 0 — 20 mA output, 10% manipula ting va ria ble c orres ponds to a n out-put of 2mA, 80% co rres ponds to 16 mA, a nd 100% eq uals 20mA.

As d isc uss ed in Cha pter 1, co ntinuous co ntrollers a re used to ope ra te a ctua tors, suc h as thyristo runits, reg ula ting valves etc. which need a co ntinuous s igna l.

3.2 P controller

In a P co ntroller the co ntrol devia tion is produced by forming the difference be twe en the proce ssvariable PV and the selected setpoint SP; this is then amplified to give the manipulating variableMV, w hich ope ra tes a suitab le a c tua tor (se e Fig . 29).

Fig. 29: Operating principle of a P controller

The c ontrol devia tion s igna l ha s to b e a mplified, since it is too sma ll a nd c a nnot be used direc tly a sthe ma nipula ting va ria ble. The g a in (Kp) of a P controller must b e a djus ta ble, so tha t the c ontroller

ca n be matched to the proces s.The c ontinuous output s igna l is d irec tly prop ortiona l to the c ontrol de via tion, a nd follow s t he s a meco urse ; it is merely a mplified by a ce rta in fac tor. A ste p c hang e in the d evia tion e, ca used for exam-ple b y a sudd en c hang e in setpo int, res ults in a ste p c hang e in manipula ting varia ble (se e Fig. 30).

P roc es s value (x)Control

deviatione = (w - x)Amplifier

Manipulating

Setpoint (w)

(Kp)

va ria b le (y)





Fig. 30: Step response of a P controller

The s tep resp ons e o f a P controller is s how n in Fig . 30.

In other words, in a P co ntroller the ma nipula ting va ria ble cha nges to the s a me extent a s the devi-a tion, thoug h a mplified by a fac tor. A P co ntroller ca n be repres ented ma thema tica lly b y the follow -ing c ontroller eq ua tion:

The fa c tor KP is c a lled the proportiona lity fac tor or tra nsfer co effic ient of the P controller and corre-

sponds to the control amplification or gain. It should not be confused with the process gain KS ofthe proc ess .

S o, in an a pplica tion where the use r has se t a KP of 10 %/° C, a P co ntroller will prod uce a ma nipu-la ting va ria ble of 50 % in res ponse to a control difference o f 5 ° C .

Another example would be a P controller for the regulation of a pressure, with a KP se t to 4 %/ba r.In this c a se, a co ntrol difference o f 20 ba r w ill produc e a ma nipula ting va ria ble of 80 %.

e

y

e = (w - x)

t

t

P controller

Step response

t

y = K • (w - x)P

0

y KP w x–( )•=



47



3.2.1 The proportional band

Looking at the controller equation, it follows that, in a P controller, any value of deviation wouldproduce a corresponding value of manipulating variable. However, this is not possible in practice,as the manipulating variable is limited for technical reasons, so that the proportional relationship

be twe en ma nipula ting va ria ble and c ontrol deviation only exists over a c erta in ra nge o f va lues.

Fig. 31: The position of the proportional band

The top ha lf of Fig . 31 show s the c ha ra c teris tic of a P controller, w hich is controlling a n electrica llyheated furnace, with a selected setpoint w = 150° C.

The following relationship could conceivably apply to this furnace

The ma nipula ting va ria ble is o nly proportiona l to the de via tion o ver the ra nge from 100 to 150° C ,i.e. for a deviation of 50°C from the intended setpoint of 150°C. Accordingly, the manipulating vari-able reaches its maximum and minimum values at these values of deviation, and the highest andlow es t hea ter pow er is a pplied res pec tively. No further cha nges a re p os sible, e ven if the d evia tionincreases.

This ra nge is c a lled the proportiona l ba nd XP . Only within this band is the manipulating variablepropo rtiona l to the devia tion. The g a in of the c ontroller ca n be ma tched to the proc es s by a lteringthe XP ba nd. If a na rrow er XP ba nd is c hos en, a sma ll de via tion is suffic ient to tra vel throug h the full

ma nipula ting rang e, i.e. the g a in increas es a s XP is reduced .

The X band

Heater pow erkW

Manipula ting va ria ble MVSetpoint%w

50

25

50

100X

50

100 150 200 T /° C

Different controller ga ins throug h different X ba nds

10080

50

MV%

w

50 100 150 200 250 300 T /° C

X

X

X = 50 ° C

X = 150 ° CX = 250 ° C

P

P

P 1

P 2

P 3P 2

P 1

P





The relationship betw een the propo rtiona l ba nd a nd the g a in or propo rtiona lity fac tor of the con-troller is given by the following formula:

Within the proportional band XP , the controller travels through the full manipulating range y H, sothat KP ca n be d etermined a s follow s:

The unit of the prop ortiona lity fa c tor KP is the unit of the manipulating variable divided by the unitof the proc es s varia ble. In pra ctice, the proportiona l ba nd XP is often more useful than the propor-tiona lity fa c tor KP a nd it is XP ra ther than KP tha t is mos t often s et on the c ontroller. It is sp ec ified inthe sa me unit a s the proc es s varia ble (° C, V, ba r etc.). In the a bo ve exa mple o f furnac e c ontrol, theXP ba nd has a value 50° C. The ad vantag e of using XP is that the va lue of d evia tion, w hich p rod uc-

es 100% ma nipula ting va ria ble, is immediate ly evident. In temp erature c ontrollers, it is of pa rticula rinterest to know the operating temperature corresponding to 100% manipulating variable. Fig. 31sho ws a n example of different XP bands.

An example

An elect ric furna ce is to be co ntrolled by a digital co ntroller. The m a nipula ting va ria ble is to be100% for a devia tion o f 10° C. A propo rtiona l ba nd XP = 10 is therefore s et o n the c ontroller.

Until now, for reasons of clarity, we have only considered the falling characteristic (inverse operat-ing s ense ), in other w ords, a s the proces s va ria ble increas es , the ma nipula ting va ria ble d ec rea se s,until the s etpo int is rea ched . In add ition, the position o f the XP ba nd has b een shown to one side ofa nd be low the setpoint.

However, the XP ba nd ma y be symme trica l a bo ut the se tpoint or ab ove it (se e Fig. 32). In a dd ition,co ntrollers w ith a rising c harac teristic (direc t o perating se nse) are us ed for certain proc es se s. Forinstance, the manipulating variable in a cooling process must decrease as the process value in-creases.

XP1

KP------ 100%•=

KP

yHXP------

ma x. ma nipula ting ra ngeproportional band

--------------------------------------------------------==



49



Fig. 32: Position of the proportional band about the setpointThe ad vantag es of XP ba nds which are symme trica l or as ymmetrica l a bo ut the s etpoint will be d is-cussed in more detail under 3.2.2.

3.2.2 Permanent deviation and working point

A P controller only produces a manipulating variable when there is a control deviation, as we al-rea dy know from the c ontroller eq uation. This mea ns that the ma nipula ting varia ble b ec omes zerowhen the proces s va ria ble rea ches the se tpoint. This c a n be very useful in certain proc es se s, s ucha s level co ntrol. How ever, in our example of the furnac e, it mea ns tha t hea ting po we r is no long era pplied when the co ntrol devia tion is ze ro. As a co nseq uence, the temperature in the furnac e fa lls.

Now there is a deviation, which the controller then amplifies to produce the manipulating variable;the large r the de via tion, the large r the ma nipula ting va ria ble o f the c ontroller. The d evia tion no w





takes up a value such that the resulting manipulating variable is just sufficient to maintain the pro-ces s va ria ble a t a c onsta nt va lue.

A P controller always has a permanent control deviation or offset

This permanent d evia tion c a n be ma de sma ller by reducing the proportiona l ba nd XP . At first

glanc e, this might s eem to b e the o ptima l so lution. How ever, in prac tice , a ll co ntrol loo ps be co meunstable if the value of XP falls below a critica l value - the proc es s varia ble sta rts to o sc illa te.

If the sta tic cha ra cte ristic of the proce ss is know n, the res ulting c ontrol devia tion c a n be found di-rectly. Fig. 33 shows the characteristic of a P controller with an XP ba nd of 100° C. A setpoint of200° C is to b e held b y the co ntroller. The proc es s c hara cteristic of the furnac e s how s that a ma nip-ula ting va ria ble of 50% is req uired for a s etpo int of 200° C . How ever, the c ontroller prod uces z eroma nipula ting varia ble a t 200° C. The tempe ra ture w ill fa ll, a nd, a s the d evia tion increa se s, the c on-troller w ill deliver a highe r ma nipula ting va ria ble, c orres ponding to the XP band. A temperature willbe rea ched here, a t w hich the c ontroller prod uces the exa ct value o f ma nipula ting varia ble req uiredto ma inta in that tempe rature. The temperature rea ched , and the co rres pond ing ma nipula ting va ri-able, can be read off from the point of intersection of the controller characteristic and the static

proc es s c hara cteristic: in this c a se , a temperature of 150° C w ith a ma nipula ting va ria ble of 40%.

Fig. 33: Permanent deviation and working point correction

y /%Co ntroller cha rac teris tic

Permanent control deviation

S etpoint w

X = 100 ° C

T /° CT /° C

100

50

40

100 150 200 300 400

Static process characteristic

400

200

25 50 75 100 y /%

Working point correction

W

y /%

WP

100

50

50 100 150 200 250 300 T /° C

P



51



It is clear that in a furna ce , for insta nce, a ce rtain level of powe r must be supplied in order to rea chand maintain a particular setpoint. So it makes no sense to set the manipulating variable to zerow hen there is no control de via tion. The ma nipula ting va ria ble is us ua lly se t to a spec ific p ercenta geva lue for a c ontrol differenc e o f 0. This is c a lled w orking p oint correc tion, a nd c a n be a djuste d o nthe controller, normally over the ra nge of 0 — 100%. This me a ns tha t with a correc tion of 50%, the

controller wo uld p rod uce a ma nipula ting va ria ble of 50% for zero c ontrol de via tion. In the exa mplegiven, se e Fig. 33, this w ould lea d to the se tpoint w = 200° C b eing reac hed a nd held. We c a n seethat the proportiona l ba nd e xhibits a falling c hara cteristic tha t is symme trica l a bo ut the se tpoint. Ifthe proc es s a ctua lly req uires the ma nipula ting va ria ble se t a t the w orking point, a s in our example,the c ontrol operates without d evia tion.

Setting the working point in practice

In practice, the process characteristic of a process is not usually known. However, the workingpoint c orrec tion c a n be dete rmined by ma nually c ontrolling the proces s varia ble a t the s etpoint va l-ue tha t the controller is to ho ld la ter. The ma nipula ting va ria ble req uired for this is a lso the va lue forthe w orking point co rrec tion.

ExampleIn a furnace where a setpoint of 200°C is to be tracked, the controller would be set to manualmode and the manipulating variable slowly increased by hand, allowing adequate time after thecha nge for the e nd temp erature to be rea ched . A ce rtain value of ma nipula ting varia ble will be de-termined , for exa mple 50%, which is suffic ient for a proc es s va ria ble of 200° C . This ma nipula tingva ria ble is then fed in a s the va lue for the w orking p oint correc tion.

After feed ing in the va lue for the w orking po int c orrec tion, the co ntroller w ill only o pera te w ithoutcontrol difference at the particular setpoint for which the working point correction was made. Fur-thermore, the external conditions must not change. If other disturbances did affect the process,(for example, a fall in the temperature outside a furnace), a control difference would be set onceagain, although this time the value would be smaller.

We can summarize the main points about the control deviation of a P controller as follows(controller with falling characteristic, process with self-limitation):

Without working point WP

- The proc ess varia ble remains in a stea dy sta te below the setpoint.

With working point WP (see Fig. 33)

- below the working point (in this c a se 0 — 50% manipula ting varia ble)process variable is above the setpoint

- a t the working point (in this c a se 50% manipula ting va ria ble)proc ess varia ble = setpoint

- ab ove the wo rking point (in this c a se 50 — 100% manipulating varia ble)proc es s va ria ble is b elow the setpo int

In a P co ntroller, the output signa l has the sa me time c ourse a s the c ontrol devia tion, a nd be ca useof this it responds to disturbances very rapidly. It is not suitable for processes with a pure deadtime, as these start to oscillate due to the P controller. On processes with self-limitation, it is notpossible to control exactly at the setpoint; a permanent deviation is always present, which can besignificantly reduced by introducing a working point correction.





3.2.3 Controllers with dynamic action

As we sa w in the previous cha pter, the P co ntroller simply res pond s to the ma gnitude o f the devia -tion and amplifies it. As far as the controller is concerned, it is unimportant whether the deviationoccurs very quickly or is present over a long period.

When a la rge disturba nce o cc urs, the initia l res pons e o f a m a chine o perator is to increas e the ma -nipulating variable, and then keep on changing it until the process variable reaches the setpoint.He w ould c ons ider not o nly the ma gnitude o f the de via tion, b ut also its beha vior with time (dyna m-ic action).

Of course, there are control components that behave in the same way as the machine operatormentioned a bove:

- The D c omponent respo nds to c hanges in the proce ss varia ble. For exa mple, if there is 20% re-duc tion in the supply voltag e o f an electric furna ce, the furna ce temp erature w ill fall. This Dcomponent responds to the fall in temperature by producing a manipulating variable. In thisca se , the ma nipula ting va ria ble is propo rtiona l to the rate of cha nge of furnac e tempera ture, a nd

helps to c ontrol the proc es s varia ble at the se tpoint.- The I co mponent res ponds to the dura tion of the devia tion. It summa tes the de via tion a pplied to

its input over a p eriod of time. If this controller is use d o n a furna ce, fo r example, it w ill slow ly in-creas e the hea ting po we r until the furnac e tempe ra ture reac hes the req uired se tpoint.

In the pas t, dyna mic a ction wa s a chieved in ana log co ntrollers by feed ing pa rt of the ma nipula tingva ria ble ba ck to the c ontroller input, via timing c ircuits . The feed ba ck cha nge s t he input signa l (therea l control de via tion) so tha t the controller rec eives a simula ted de via tion s igna l tha t is m od ified bya time-dependent factor. In this way, using a D component, a sudden change in process variable,for exa mple, c a n be ma de to have e xac tly the sa me initia l effec t a s a much la rge r co ntrol devia tion.In this co nnection, b ec a use o f this reverse c oupling, w e o ften talk abo ut feed ba ck. In mode rn mi-croprocessor controllers, the manipulating variable is not produced via feedback, but derived

ma thema tica lly direc t from the se tpoint a nd proce ss varia ble.

We w ill a void using the te rm feedb a ck in this b oo k, a s fa r a s po ss ible.

The co mponents des cribed ab ove a re often co mbined w ith a P co mponent to give P I, P D or P IDcontrollers.



53



3.3 I controller

An I co ntroller (integ ral co ntroller) integ rates the d evia tion signa l app lied to its input over a period oftime. The longer there is a de via tion on the c ontroller, the la rger the m a nipula ting va ria ble o f theI controller bec ome s. Ho w q uickly the controller builds up its ma nipula ting va ria ble dep end s firstly

on the se tting o f the I co mponent, a nd s ec ondly on the ma gnitude o f the de via tion.The ma nipula ting varia ble cha nges a s long a s there is a de via tion. Thus, o ver a period of time, e vensmall deviations can change the manipulating variable to such an extent that the process variableco rres ponds to the required se tpoint.

In principle, an I controller can fully stabilize after a sufficiently long period of time, i.e. setpoint =proce ss va ria ble. The d evia tion is then ze ro a nd the re is no further increa se in ma nipula ting va ri-able.

Unlike the P controller, the I controller does not have a permanent control deviation

The s tep respo nse of the I controller show s the co urse of the ma nipula ting va ria ble over time, fol-low ing a ste p c ha nge in the c ontrol differenc e (see Fig . 34).

Fig. 34: Step response of an I controller

For a co nsta nt co ntrol devia tion ∆e, the equation of the I controller is as follows:

Here TI is the integral time of the I controller and t the duration of the deviation. It is clear that the

change in manipulating variable y is proportional not only to the change in process variable, buta lso to the time t.

∆y1TI---- ∆e• t•=





If the c ontrol de via tion is va rying, then:

The integ ra l time o f the I controller ca n a lso be eva lua ted from the s tep respons e (se e Fig . 34):

If the proc es s va ria ble is below the s etpoint on a n I co ntroller with a nega tive ope ra ting s ense , a sused, for example, in heating applications, the I controller continually builds up its manipulatingvaria ble. When the proc es s va ria ble rea ches the s etpoint, w e now have the pos sibility that the ma -nipula ting varia ble is too la rge , b ec a use of d ela ys in the proce ss . The proce ss varia ble w ill ag a in in-crea se slightly; how ever, the ma nipula ting va ria ble is now reduc ed, be ca use o f the sign reversa l ofthe proc es s varia ble (now a bo ve the s etpoint).

It is precisely this relationship that leads to a certain disadvantage of the I controllerIf the manipulating variable builds up too quickly, the control signal which arises is too large, andtoo high a proc es s va ria ble is reac hed. Now the proc es s va ria ble is a bo ve the setpoint and the s ignof the devia tion is reverse d, i.e. the c ontrol signa l dec rea se s a ga in. If the de creas e is to o s udde n, alow er proc es s value is a rrived a t, a nd s o o n. In other wo rds , w ith a n I controller, os cilla tions a bo utthe s etpoint c a n oc cur q uite freq uently. This is es pec ia lly the c a se if the I compo nent is too strong,i.e. when the selected integral time TI is to o s hort. The exc eption to this is the z ero-orde r proce sswhe re, b ec a use there are no energy sto ra ge pos sibilities , the proc es s va ria ble follow s the ma nipu-la ting va ria ble immediately, w ithout any d ela y; the c ontrol loop forms a sys tem w hich is not c a pa -ble of o scilla tion.

To develop a feel for the effect of the integral time TI, it can be defined as follows: The integ ral

time TI is the time that the integral controller needs to produce its constant control difference at itsoutput (without considering sign). Imagine a P controller for a furnace, where the response time TIis se t a t 60s ec a nd the c ontrol difference is co nsta nt a t 2° C. The c ontroller req uires a time TI =60s ec for a 2% increas e in manipula ting varia ble, if the c ontrol difference rema ins unchang ed a t2 ° C .

Summarizing the main points, the I controller removes the control deviation completely, in contrastto the P controller.

An I controller is not s ta ble whe n ope ra ting o n a proc es s w ithout s elf-limita tion, a nd is therefore un-suita ble for c ontrol of liq uid levels, for example. On proc es se s with long time c onsta nts, the I com-ponent must b e s et very low, s o tha t the proc es s va ria ble does not tend to os cilla te. With this sma llI component, the I controller works much too slowly. For this reason, it is not particularly suitable

for proce sse s w ith long time c ons ta nts (e.g . tempe ra ture control sy s tems ). The I type o f controlleris freq uently used for pres sure regula tion, a nd in suc h a ca se Tn is se t to a very low value.

3.4 PI controller

As w e ha ve found in the I co ntroller, it takes a relatively long time (de pend ing on Ti ) be fore the co n-troller has built up its manipulating variable. Conversely, the P controller responds immediately tocontrol differences by immediately changing its manipulating variable, but is unable to completelyremove the c ontrol difference . This wo uld se em to sug ge st co mbining a P co ntroller with an I co n-troller. The res ult is a P I co ntroller. S uch a co mbina tion c a n co mbine the a dva ntag e o f the P co n-troller, the rapid response to a control deviation, with the advantage of the I controller, the exact

co ntrol at the se tpoint.

y1TI---- e∫ d t•

sK---•=

TI∆e ∆t•

∆y-----------------=



55



We c a n obta in the step res pons e of a P I co ntroller simply by s uperimpos ing the s tep respons es ofa P a nd a n I co ntroller, a s show n in Fig. 35.

Fig. 35: Step response of a PI controller

If the diagonally rising straight line of the PI manipulating variable is projected back to its point ofintersec tion S w ith the t ime a xis , it intercept s a leng th of time there. With a P I co ntroller, this corre-spo nds to the res et time Tn .

For a control deviation e = ∆e = co nsta nt, we ob ta in the follow ing e q uation for the P I co ntroller:

The res et time is a m ea sure of the extent to w hich the duration of the co ntrol de via tion a ffec ts thecontrol function. A long reset time means that the I component has little influence, and vice versa.From the eq uation ab ove, it is evide nt that the rea l a mplifica tion o f the I co mponent is the fac tor

With a PI controller, therefore, a change in proportional band XP a lso ca uses a cha nge in the inte-g ral a c tion. If the proportiona l g a in of a P I controller is increa se d b y red ucing XP , the integral actionwill also be increased, so the controller will make a faster integration of the control difference.

It is also possible to interpret Tn as the time interval required for the I component to produce thesa me ma nipula ting va ria ble y (for a given de via tion), a s that a lrea dy produce d b y the P co mponent

t

t

y

y

t

S

P I co ntroller

P controller

e

I co ntroller

De

t 0

Tn

∆y1

XP------ 100%• ∆e

1Tn------ ∆e t••+⎝ ⎠

⎛ ⎞•1

XP------ 100%• ∆e 1

1Tn------ t•+⎝ ⎠

⎛ ⎞••==

1XP------ 100%•

1Tn------•





(follow ing a s tep c ha nge ). The formula g iven a bo ve is o nly va lid w hen the de via tion rema ins c on-stant during the time interval t. If this is not the case, the relationship is as follows:

As mentioned ea rlier, a P I co ntroller ca n, in principle, b e b uilt up b y c omb ining a P controller and a nI controller. With a sudden deviation, the manipulating variable is initially formed by the P compo-nent (see Fig. 36). Because of the changed manipulating variable, the process variable moves to-wa rds the s etpoint, i.e. the devia tion is reduc ed, a nd w ith it the ma nipula ting va ria ble prod uced bythe P controller. Now the manipulating variable produced by the I component ensures exact con-trol. Whereas the P component of the manipulating variable steadily decreases as the setpoint isapproached, the I component continues to build up. Here, however, the increase is also smaller,because of the reducing deviation, until finally, when the setpoint is reached, nothing more is add-ed to the current manipulating variable. When the system has stabilized, the manipulating variableof the PI controller is produced solely by the I component.

Fig. 36: Formation of the manipulating variable in a PI controller

∆y1

XP------ 100%• e

1XP------ 100%

1Tn------ e∫ d t••••+•=

T /° C

S etpoint w

tT /° C

50 % power required

y /%

P c o mpo ne nt I c ompo ne nt

t

t

50

100

100

400

300

200

400

300

200

100



57



Summarizing the main points:

In a P I co ntroller, the P co mponent c a uses the ma nipula ting varia ble to res pond immediately to thecontrol devia tion. The P I controller is therefore muc h fas ter tha n an I controller. The I com ponent in-teg ra tes the co ntrol devia tion a t the output of the c ontroller, s o tha t the closed co ntrol loo p a cts toreduce the remaining deviation.

3.5 PD controller

If a large disturba nce oc curs in a c ontrol loo p w hic h is be ing c ontrolled ma nually, b ringing w ith it acha nge in the ma nipula ting va ria ble, the ope ra tor will try to cus hion the e ffec t of the disturba nce byma king a la rge initia l adjustment o f the a ctua tor. He then q uickly reduces the a djustment, s o thatthe new equilibrium of the control loop can be approached gradually. A controller which respondsin a similar way to the above operator is the PD controller: it consists of a P component with aknow n propo rtiona l ac tion, a nd a D c ompo nent with a derivative a ction. This D c omponent re-sponds not to the magnitude or duration of the control deviation, but to the rate of change of theproc es s varia ble. Fig. 37 show s how suc h a P D c ontroller builds up its ma nipula ting va ria ble.

Fig . 37 explains how the P D co ntroller works . If a new se tpo int is a pplied , the ma nipula ting va ria bleis increased by the P component; this component of the manipulating variable is always propor-tiona l to the d evia tion. The proce ss varia ble res ponds to the increas ed ma nipula ting varia ble, forexample, a furnace temperature rises. As soon as the process variable changes, the D componentstarts to take effect: while the process variable increases, the D component forms a negative ma-nipulating variable, which is subtracted from the manipulating variable of the P component, finallyproduc ing the ma nipula ting va ria ble a t the c ontroller output. When the p roc es s va ria ble is tra ckingthe setpoint, the D component “brakes”, thus preventing the manipulating variable overshootinga bove the s etpoint.

If the process variable has reached its maximum value after an overshoot above the setpoint, andis now reducing, the D component gives out a positive manipulating variable. In this case, the D

co mponent countera cts the cha nge in proc ess variab le.

The D co mpo nent only intervenes in the proce ss w hen there is a cha nge in proc es s va ria ble. Thesize of the manipula ting va ria ble of the D co mponent dep ends on the ra te of cha nge o f the proc es svaria ble, tha t is o n the mag nitude of ∆x/∆t (see the g rad ient tria ngle in Fig . 37). In ad dition, the e f-fect c a n be c hang ed a t the controller via the time Td (derivative time), which we will get to know inthis chapter. A pure D controller is not suitable for control, as it does not intervene in the processwhen there is a co nsta nt devia tion, or when the proc es s va ria ble rema ins c ons ta nt.





Fig. 37: Formation of the manipulating variable in a PD controller

T /° C

S etpoint w

t

T /° C

t

t100

400

300

100

400300

200

100

Dx

Dt

200

-100

py /%

t

100

-100

Dy /%

P roc ess value X

P component

D component



59



Fig. 38 show s the ramp function respons e for a P D c ontroller, w here w e c a n ima gine the increas ingcontrol de via tion resulting from a falling p roc es s va ria ble.

Fig. 38: Response of a PD controller to a ramp function

From Fig. 38 we c a n see that there is a notice a ble manipula ting va ria ble from the D c ompo nent atthe s ta rt of the ramp function, s ince this ma nipula ting varia ble is propo rtiona l to the ra te o f cha ngeof the proces s value. The P co mponent need s a ce rta in time, na mely the d erivative time Td , torea ch the sa me va lue ma nipula ting va ria ble a s the D co mponent ha s b uilt up. The de riva tive time isobtained by projecting the diagonally rising line back to its point of intersection S with the timeaxis.

Mathema tica lly, the rate of cha nge v is ob ta ined from the c hang e in co ntrol devia tion “ de” per unit

time “ dt”:

For the PD controller, this leads to the following control equation:

In principle, the D component has the following effects:

As s oon a s the proc ess varia ble c hanges , the D component counterac ts this c hange.

For a controller with an inverse operating sense (i.e. for heating), this means for example:

vded t-----=

y1

XP------ 100%• e Tv

ded t-----•+⎝ ⎠

⎛ ⎞•=





- If the proc ess variab le reduces a s a result of a d isturba nce in the proc ess , the D componentforms a pos itive ma nipula ting varia ble, which c ountera cts the red uction in the proc es s varia ble.

- If the proc ess variab le increa ses a s a result of a d isturba nce in the proc ess , the D componentforms a nega tive ma nipula ting va ria ble, which c ountera cts the increas e in the proc es s va ria ble.

3.5.1 The practical D component - the DT1 element

In principle, w e c ould a lso co nsider the s tep respons e o f a P D c ontroller in the sa me w a y a s previ-ously for P and PI controllers. Now, however, the rate of change at a step is infinitely large. In theo-ry, the D s igna l de rived from a st ep w ould therefore b e a n infinitely high a nd infinitely narrow sp ike(se e Fig . 39). Theoretica lly, this me a ns tha t the ma nipula ting va ria ble ha s to ta ke up a n infinitelyhigh value for an infinitely short time, and then return immediately to the value produced by the Pcomp onent. This is simply not pos sible, for both elec tric a l a nd mec ha nica l rea sons . Furthermore,such a short pulse would scarcely influence the process. In practice, the immediate decay is pre-vented b y forming the D c ompo nent through a DT1 element. This element c ons ists of a D c ompo -nent, which we have already met in this chapter, in series with a T1 element. The T1 element be-

haves like a first-order process with a transfer coefficient of 1.

Fig. 39 show s the step respo nse of the “prac tica l” D co mponent. T1 is the time cons ta nt of the T1element. In pra ctice, this time c ons ta nt is se t a t Td /4, a nd w hen Td is cha nged, the time c onsta nt ischa nged by the sa me ratio. The d erivative time Td can be determined from the step response ofthe “practical” D component, on the basis of the ratio T1 = Td /4.

T1 is s pec ified by the ma nufa cturer, a nd c a nnot be a ltered by the us er.

Fig. 39: Step response of a DT1 element

Narrow sp ike Theory

t

t

yPractice

y

e

t

De

T1

yh

TeD

d

T1



61



Summarizing the main points:

A pure D c ontroller has no prac tica l importanc e s ince it ta kes no a cc ount of a permanent de via tion,a nd simply res ponds to the ra te of cha nge o f the proc es s va ria ble. B y co mpa riso n, the P D co ntrol-ler is very widely used . The D co mponent ens ures a fas t res pons e to disturba nces , w herea s its“b ra king b eha vior” a lso sta bilizes the c ontrol loo p. The D c ompo nent is not s uita ble for proc es se s

with pulsa ting va ria bles, s uch a s pres sure a nd flow co ntrol.The ma in applica tion for the P D co ntroller is w here tools o r prod ucts a re prone to d a ma ge if these tpoint is exce ede d. This a pplies pa rticula rly to plas tics proc es sing ma chines . How ever, P D c on-trollers, like the P controller, always have a permanent deviation, when controlling processes withself-limitation.

3.6 PID controller

We ha ve s een ea rlier tha t the co mbina tion o f a D co mponent or a n I co mponent w ith a P co ntrolleroffered certain advantages in each case. Now it seems logical to combine all three structures, re-sulting in the P ID c ontroller.

With this controller, the XP , Tn , Td pa ra meters a re a djusted for the P, I a nd D a ction. These threeco mponents ca n be s een in the ste p res pons e o f a P ID co ntroller (se e Fig. 40).

Fig. 40: Step response of a PID controller

According to DIN 19 225, suc h a controller ob eys the follow ing c ontroller eq ua tion:

(ide a l PID c ontroller)

As already discussed in the previous section, the individual parameters (KP , Td , Tn ) have differenteffec ts on the individua l components .

t

y

t

D componentI component

P component

e

K • eD

T

Td /4

t

De

0

n

P

∆y KP e1

Tn------ e dt Td

dedt-----•+•∫+⎝ ⎠

⎛ ⎞•=





On some controllers with PID action, Td and Tn cannot be adjusted separately. Practical experi-ence ha s s how n tha t optimum performa nce is obta ined w ith a ra tio Td = Tn /4 to 5. This rat io is fre-quently a fixed setting on the controller, and only one parameter can be varied (usually Tn ).

We c a n summa rize b y noting tha t the P ID c ontroller brings tog ether the b es t c harac teristics of theP, I and D c ontrollers. The P co mponent respond s w ith a suita ble manipula ting va ria ble when a de-via tion oc curs. The D comp onent co untera cts cha nges in the proc es s va ria ble, and increa se s thes ta bility of the control loo p. The permanent de via tion is removed by the I com pone nt. The P ID typeof controller is used for most applications.

3.6.1 Block diagram of the PID controller

Fig. 41: Block diagram of the PID controller

As we have a lrea dy s een in this cha pter, from the c ontroller eq uations for the P I, P D a nd P ID co n-trollers, the I and D actions of a PID controller are influenced not only by the adjustment of the Tnand Td pa ra meters, b ut also b y the propo rtiona l ga in with XP . If the proportional gain of a PID con-troller is doubled (by halving XP ), the controller not only has double the proportional action, but theI and D components are also increased to double the value.

An example

The P ID c ontroller show n in Fig . 41 has se ttings Tn = 10sec and XP = 100 (the D c ompo nent shouldbe disrega rde d in this exa mple). The c ontrol de via tion is 2.

When KP and XP a re given as percenta ge va lues, the P c omponent has a ga in of:

The c ontrol de via tion is thus o ffered d irec tly to the I co mponent. We a lrea dy know from C hapter 3.3“ I controller”, tha t a n I co ntroller req uires a time e q ua l to Tn to fully reprod uce the input signa l a t itsoutput (percenta ge va lues ). The I co mpo nent w ould thus req uire 10s ec be fore it has increa se d itsmanipulating variable by 2%. XP is now set to 50, so that the ga in of the P co mponent is 2.

Now the control difference is first amplified by a factor of 2, before it is offered to the I component.The I comp onent now increas es its ma nipula ting va ria ble by 4% every 10 s ec ond s . The effect o fthe I co mponent wa s a lso a mplified b y a fa cto r of 2.

Changing the proportional gain in a PID controller

changes the I and D action to the same extent

larger X P (corresponds to smaller K P ): corresponds to smaller P component

larger Tn : corresponds to reduced I component

larger Td : corresponds to increased D component

1 KP1

XP------ 100 %•=⎝ ⎠

⎛ ⎞




4 Control loops with continuous controllers

4.1 Operating methods for control loops with continuous controllers

The previous cha pters d ea lt with the individua l elements o f a c ontrol loo p, the proce ss a nd theco ntroller. Now we co nsider the intera ction betw een thes e tw o e lements in the c los ed co ntrol loop .

Among st o ther things , the sta ble and unsta ble beha vior of a c ontrol loop sho uld b e exa mined, to-

gether with its response to setpoint changes and disturbances. In the section on “Optimization”,we w ill come a cross the va rious criteria for ad justing the c ontroller to the proc es s.

We a lso o ften refer to the s ta tic a nd d yna mic b eha vior of the co ntrol loo p. The s ta tic b eha vior of acontrol loop characterizes its steady state on completion of all dynamic transient effects, i.e. itssta te long a fter a ny ea rlier disturbanc e o r setpoint cha nge. The dyna mic beha vior, on the otherhand, shows the behavior of the control loop during changes, i.e. the transition from one state ofres t to a nother. We ha ve a lrea dy d isc uss ed this kind o f dyna mic b eha vior in Cha pter 2 “The pro-cess” .

When a co ntroller is c onnec ted to a proc es s, we expec t the proc es s varia ble to follow a co urse likethat shown in Fig. 42.

Fig. 42: Transition of the process variable in the closed control loop





- After the co ntrol loo p is c los ed, the proces s va ria ble (x) sho uld rea ch a nd hold the prede ter-mined se tpoint (w ) a s q uickly a s pos sible, w ithout a pprec ia ble overshoot. In this co ntext, therun-up to a new se tpoint value is a lso ca lled the se tpoint res pons e.

- After the sta rt-up phase, the proc ess varia ble s hould ma intain a stea dy value w ithout a ny appre-ciab le fluctua tions, i.e. the c ontroller should ha ve a sta ble effec t on the proces s.

- If a disturba nce oc curs in the proce ss , the controller should a ga in be a ble to c ontrol it with theminimum pos s ible oversho ot, a nd in a rela tively sho rt res ponse t ime. This me a ns tha t the co n-troller should a lso exhibit a g oo d d isturbanc e respo nse.

4.2 Stable and unstable behavior of the control loop

After the end o f the sta rt-up phas e, the proces s va ria ble s hould ta ke up the ste a dy va lue, prede ter-mined b y the s etpoint, a nd enter sta ble operation. How ever, it c ould ha ppen tha t the c ontrol loopbe co mes unsta ble, and that the ma nipula ting va ria ble and proc es s va ria ble perform period ic os cil-lations. Under certain circumstances, this could result in the amplitude of these oscillations not re-

maining constant, but instead increasing steadily, until it fluctuates periodically between upper andlow er limit values . Fig. 43 show s the two ca se s of a n unsta ble control loop .

Here, we often talk about the self-oscillation of a control loop. Such unstable behavior is mostlyca used by low noise levels pres ent in the c ontrol loo p, w hich introd uce a ce rta in res tles snes s intothe loop. Self-oscillation is largely independent of the construction of the control loop, whether itbe mec hanica l, hydraulic or elec trica l, a nd o nly o cc urs when the returning os cilla tions have a la rge ra mplitude than thos e se nt out, and a re in phas e with them.

Fig. 43: The unstable control loop

If certa in operating c ond itions , (e.g . new co ntroller settings ), a re c ha nge d in a c ontrol loo p tha t is insta ble opera tion, there is a lwa ys a pos sibility of the co ntrol loop b ec oming unsta ble. Howe ver, inprac tic a l co ntrol engineering, the s ta bility of the c ontrol loo p is a n ob vious req uirement. We c a ngeneralize by stating that stable operation can be achieved in practice by choosing a sufficientlylow gain in the control loop and a sufficiently long controller time constant.



65



4.3 Setpoint and disturbance response of the control loop

As already mentioned, there are basically two cases which result in a change in the process vari-a ble. When des cribing the be havior of a proc es s in the c ontrol loo p, w e use the terms s etpoint re-spons e or disturba nce respo nse, depending o n the c a use of the change :

Setpoint responseThe se tpoint has bee n ad justed a nd the proc es s ha s reac hed a new e q uilibrium.

Disturbance response

An external disturbance affects the process and alters the previous equilibrium, until a stable pro-ces s value has d eveloped once a ga in.

The s etpoint res pons e thus co rres ponds to the beha vior of the c ontrol loo p, follow ing a cha nge inse tpoint. The disturbanc e res pons e de termines the respons e to external chang es , suc h as the in-trod uction of a co ld c harge into a furna ce . In a c ontrol loo p, the se tpoint and disturba nce respo ns-es are usually not identical. One of the reasons for this is that they act on different timing elementsor a t va rious intervention po ints in the c ontrol loo p.

In many cases, only one of the two types of process response is important.

When a motor subjected to continuously variable shaft loading still has to maintain a constantspe ed, it is clearly only the disturba nce res pons e w hich is of importa nce. Co nverse ly, in the c a se ofa furnace, where the charge has to be brought to different temperatures over a period of time, ina cc orda nce with a s pec ific se tpoint profile, the s etpoint res pons e is o f more interes t.

The purpose of co ntrol is to influenc e the proces s in the de s ired m a nner, i.e. to cha nge the setpo intor disturbance response. It is impossible to satisfactorily correct both forms of response in thesa me w a y. A dec ision must therefore b e ma de whether to o ptimize the c ontrol for disturbanc e re-spo nse or setpo int res pons e. More a bo ut this in the se ction on “Optimiza tion” .





4.3.1 Setpoint response of the control loop

As already explained, the main objective in a control loop with a good setpoint response is that,when the s etpoint is c hang ed , the proc es s va ria ble should reac h the new se tpoint value a s q uicklya s pos sible a nd w ith minima l overshoot. Overshoo t ca n be prevented by a different co ntroller set-

ting, but only at the expense of the stabilization time (see Chapter 4.1, Fig. 42). After closing theco ntrol loo p, it takes a ce rta in time for the proces s varia ble to reac h the s etpoint va lue prede ter-mined a t the c ontroller. This a pproa ch to the s etpo int ca n be m a de either g ra dua lly (c reep) or in a noscillatory manner (see Fig. 44).Which pa rticula r control loo p respons e is c ons idered mo st important va ries from one ca se to a n-other, and depends on the process to be controlled.

Fig. 44: Approach to the setpoint



67



4.3.2 Disturbance response

When the sta rt-up phas e is co mplete a nd the c ontrol loop is sta ble, the c ontroller now ha s the ta skof suppress ing the influence o f disturbanc es , a s fa r a s po ss ible. When a d isturbanc e do es oc cur, ita lw a ys results in a tempo ra ry control de via tion, w hich is o nly co rrec ted a fter a certain time. To

achieve good control quality, the maximum overshoot, the permanent control deviation and thestabilization time should be as small as possible (see Chapter 1.4, Fig. 3). As the size of distur-ba nces of the cha rac teristics in a co ntrol loop normally has to b e a cc epted a s given, g ood co ntrolquality can only be achieved by a suitable choice of controller type and an appropriate optimiza-tion.

The disturba nces ca n a ct a t different points in the proc es s. Depending on the point of a pplica tionof the disturbance, its effect on the dynamic transition of the process variable will differ. Fig. 45shows the course of a disturbance step response of the process, when a disturbance acts at thebe ginning, in the midd le a nd a t the end of the proc es s.

Fig. 45: Disturbance step response of a process





4.4 Which controller is best suited for which process?

After selecting a suitable controller according to type, dimensions etc. (see Chapter 1.5), the prob-lem now arises of deciding which dynamic response should be employed to control a particularprocess. With modern microprocessor controllers, the price differentials between P, PI and PID

controllers have been eroded. Hence it is no longer crucial nowadays, whether a control task cans till be s olved w ith just a P co ntroller.

Regarding dynamic action, the following general points can be made:P co ntrollers ha ve a permanent de via tion, w hich c a n be removed by the introd uction of a n I co m-ponent. However, there is an increased tendency to overshoot, because of this I component, andthe co ntrol bec omes a little more slugg ish. Acc ura te sta ble control of proc es se s a ffec ted b y delaysca n be a chieved by a P co ntroller, b ut only in c onjunction w ith a n I compo nent. With a dea d time,an I component is always required, since a P controller, used by itself, leads to oscillations. An Icontroller is not s uita ble for proc es se s w ithout s elf-limita tion.

The D c omp onent ena bles the controller to res pond more q uickly. How ever, w ith strong ly puls a tingprocess variables, such as pressure control etc., this leads to instabilities. Controllers with a D

component are very suitable for slow processes, such as those found in temperature control.Where a permanent d evia tion is unac ce pta ble, the P I or P ID c ontroller is used .

The relationship betw een proc es s o rde r a nd c ontroller structure is a s follow s :For processes without self-limitation or dead time (zero-order), a P controller is adequate. Howev-er, even in apparently delay-free processes, the gain of a P controller cannot be increased indefi-nitely, as the control loop would otherwise become unstable, because of the small dead times thata re a lw a ys present. Thus, a n I co mponent is a lwa ys required for a cc ura te c ontrol a t the setpo int.

For first-order processes with small dead times, a PI controller is very suitable.

Second-order and higher-order processes (with delays and dead times) require a PID controller.When very high standards are demanded, cascade control should be used, which will be dis-

cus se d in more deta il in C hapte r 6. Third-order a nd fourth-order proc es se s ca n so metimes be co n-trolled sa tisfa cto rily w ith P ID co ntrollers, but in mos t ca se s this ca n only be a chieved w ith ca sc a decontrol.

On processes without self-limitation, the manipulating variable must be reduced to zero after these tpoint has bee n rea ched . Thus, they ca nnot be co ntrolled by a n I controller, s ince the ma nipula t-ing variable is only reduced by an overshoot of the process variable. For higher-order processesw ithout s elf-limita tion, a P I or P ID c ontroller is suita ble.

S umma rizing the selection criteria res ults in the follow ing ta bles:

Table 3: Selection of the controller type for controlling the most important process variables

Permanent deviation No permanent deviation

P PD PI PID

Temperature simple proces sfor low dema nds

simple proces sfor low dema nds

suita ble hig hly s uita ble

Pressure mos tly uns uita ble mos tly uns uita ble hig hly s uita ble; for pro-cesses with long delaytime I co ntroller a s we ll

suitable, if process val-ue pulses not too much

Flow unsuita ble unsuita ble suita ble, but I controllerfreq uently b etter

suitable

Level with short dea d timesuitable

suita ble suita ble hig hly s uita ble

Conveyor unsuitable beca use ofdea d time

unsuita ble suita ble, but I controllermostly bes t

nearly no ad vantag escompa red w ith PI



69



Table 4: Suitable controller types for the widest range of processes

4.5 Optimization

Controller optimiza tion (or “tuning ” ) mea ns the a djus tment o f the c ontroller to a g iven proce ss. Thecontrol parameters (XP , Tn , Td ) have to be selected such that the most favorable control action ofthe control loop is achieved, under the given operating conditions. However, this optimum actioncan be defined in different ways, e.g. as a rapid attainment of the setpoint with a small overshoot,or a so mew hat long er sta biliza tion time w ith no overshoot.

Of course, as well as very vague phrases like “stabilization without oscillation as far as possible”,co ntrol engineering ha s mo re prec ise des criptions , suc h as exa mining the a rea enclosed by the os -cillations and other criteria. However, these adjustment criteria are more suitable for comparing in-dividual controllers and settings under special conditions (laboratory conditions). For the practicalengineer working on the installation, the amount of time taken up and the practicability on site areof g rea ter sig nifica nce.

The formula e a nd c ontrol settings g iven in this cha pter a re e mpirica l va lues from very differentso urce s. They refer to ce rtain idea lized proce ss es a nd ma y not alwa ys a pply to a s pec ific c a se .Howe ver, a nyone w ith a know ledg e o f the various a djustment pa ra meters, on a P ID c ontroller, forexa mple, sho uld b e a ble to a djust the co ntrol a ction to sa tisfy the releva nt dema nds .

Apa rt from the ma thema tica l derivation of the proc es s pa ra meters a nd the c ontroller da ta derivedfrom them, there a re va rious empirica l methods . One me thod co nsists of period ica lly c hang ing thema nipula ting va ria ble and investiga ting how the proc es s va ria ble follow s thes e c ha nges . If this te stis ca rried out for a rang e of o sc illa tion freq uencies of the s etpoint, the a mplitude a nd pha se shift ofthe res ulting proce ss varia ble fluctuations c a n be used to d etermine the freq uency respons e c urveof the process. From this it is possible to derive the control parameters. Such test methods are

very expens ive, involve increas ed ma thema tica l trea tment, a nd a re not suita ble for pra ctica l use.

Other controller settings are based on empirical values, obtained in part from lengthy investiga-tions . S uch method s of s elec ting co ntroller se ttings (es pec ia lly the Zieg ler a nd Nicho ls method a ndthat of Chien, Hrones and Reswick) will be discussed in more detail later.

Process Controller structure

P PD PI PID

pure dead time unsuita ble unsuita ble very suita ble, orpure I controller

first-order withshort dead time

suitab le if deviation isacceptable

suitab le if deviation isacceptable

hig hly s uita ble hig hly s uita ble

second-order with

short dead time

deviation mostly toohigh for nece ss a ry XP

deviation mostly toohigh for nece ss a ry XP

no t a s g o od a s P ID hig hly s uita b le

higher-order unsuita ble unsuita ble not a s good a s P ID highly suita ble

without self-limitation

with delay

suita ble suita ble suita ble suita ble





4.5.1 The measure of control quality

Standard text book instructions for controller optimization are usually based on step changes in,for example, a disturbance or the setpoint. Disturbances are usually assumed to act at the start ofthe proc ess .

Fig. 46: The measure of control quality

This type o f disturba nce is a lso the mos t important o ne, a s it freq uently oc curs in norma l opera tion,tes ting is very fea sible a nd be ca use of its c lea r ma thema tica l a nalysis. Fig. 46 show s tha t for a s tepchange disturbance, the overshoot amplitude Xo and the stabilization time Ts offer a measure of

q uality. For a more exa ct definition o f the sta biliza tion time, we have to e sta blish w hen the c ontrol

x

t

x

t

y = 10 % of y

Disturbance change

w

w 0

w 1

A1

A1

X

X

A3

A3

A4

A4

A2

A2

T

T

t

t

∆x = ± 1 % of w

∆x = ± 1 % of w

Setpoint change

ma x

ma x

s

s

0

0

z H



71



a ction is rega rde d a s c omplete. It is c onvenient to reg a rd s tab iliza tion a fter a disturba nce a s b eingco mplete, when the c ontrol difference rema ins within ± 1% of the se tpoint w. For exped iency, thesize of the disturba nce is ta ken a s 10 % of yH .

In addition to the overshoot amplitude and the stabilization time, for mathematical analysis, thea rea of the c ontrol error is a lso used a s a mea sure o f co ntrol qua lity (se e Fig. 46).

Linea r c ontrol a rea (linea r optimum): [A]min = A1 - A2 + A3 .. .

Magnitude contro l area (magnitude optimum): [A]min = | A1 | + | A2 | + | A3 | + . ..

S q ua red c ontrol a rea (s qua red optimum): [A]min = A12 + A22 + A32 + . ..

Without doubt, q uite a pa rt from a ny other cons idera tions , one co ntroller setting c a n be s a id to ex-hibit better control quality than another, if the resulting overshoot amplitudes are smaller and thestabilization time is shorter. Some tests indicate, however, that within certain limits it is possible toha ve a sma ll overshoot a t the expense of a long er sta biliza tion time, a nd vice versa . For the givencontrol error area, there is a definite controller setting at which the areas are at a minimum.

As mentioned several times previously, differing levels of importance are attached to the variousmea sures of co ntrol qua lity, de pending on the type o f proc es s varia ble and the purpos e of the in-sta lla tion (se e a lso Cha pter 4.3 “S etpoint a nd d isturbanc e respons e of the c ontrol loo p”).

4.5.2 Adjustment by the oscillation method

In the o sc illa tion (or limit c yc le) metho d, de vise d by Zieg ler a nd Nichols, the control pa rame ters a readjusted until the stability limit is reached, and the control loop formed by the controller and theprocess starts to oscillate, i.e. the process variable performs periodic oscillations about the set-point. The controller setting va lues c a n be d ete rmined from the pa rameters found from this tes t.The proce dure c a n only be use d in proc ess es that ca n ac tua lly be mad e unstab le a nd where a n

overshoot do es not c a use da nger. The proces s varia ble is ma de to o sc illa te b y initia lly reducing thecontroller gain to its minimum value, i.e. by setting the proportional band to its maximum value.The c ontroller must be op erating a s a pure P controller; for this rea son, the I component (Tn ) andthe D c ompo nent (Td ) a re s w itched off. Then the p ropo rtiona l ba nd XP is reduced until the processvaria ble performs unda mped os cilla tions of co nsta nt amp litude.

This te st prod uce s :

- the critica l proportional ba nd XP c , and

- the oscillation time Tc of the proce ss varia ble (se e Fig. 47)





Fig. 47: Oscillation method after Ziegler and Nichols

The c ontroller ca n then be s et to the follow ing va lues :

Table 5: Adjustment formulae based on the oscillation method

Without doubt, the advantage of this process is that the control parameters can be studied underoperational conditions, as long as the adjustments described succeed in achieving oscillationsa bo ut the setpo int. There is no need to o pen the c ontrol loo p. Rec order data is ea sily eva luated ;with slow proces se s, the values c a n even be d etermined by ob se rving the proc es s va ria ble a nd us-ing a stopw a tch. The disa dvanta ge of this method is that it ca n only be used on proce ss es which

ca n be made unstab le, as mentioned a bove.The Zieg ler a nd Nicho ls a djustment rules a pply ma inly to proc es se s with short dea d times a nd w itha ra tio Tg /Tu grea ter than 3.

4.5.3 Adjustment according to the transfer function or process step response

Another method o f dete rmining the pa ra meters involves mea suring p roc es s-rela ted pa ra meters b yrecording the step response, as already described in Chapter 2.6. It is also suitable for processeswhich ca nnot be ma de to o sc illa te. How ever, it do es req uire o pening the c ontrol loo p, for insta nce,by switching the controller over to manual mode in order to exert a direct influence on the manipu-la ting va ria ble. If pos sible, the s tep c hang e in manipula ting va ria ble should b e ma de when the pro-

ce ss varia ble is c los e to the se tpoint.

Controller structure

P XP = XP c /0.5

P I XP = XP c /0.45

Tn = 0.85 · Tc

P ID XP = XP c /0.6

Tn = 0.5 · Tc

Td = 0.12 · Tc

X < X

X = X

T

x

x x

t

tt

P

P

P c

P c

X > XP P c

c

w w

w



73



A method that can be used to determine the control parameters when the process parameters areknow n ha s bee n de veloped by Chien, Hrones , a nd Re sw ick (CHR). This a pproxima tion me thodyields good control parameters, not only for disturbances, but also for setpoint changes, and issuita ble for proc es ses w ith P Tn structure (with n eq ua l to 2 o r grea ter).

The s tep respo nse is used to d etermine the d ela y time Tu, the response time T

gand the process

transfer coefficient KS (see Fig. 46).

Fig. 48: Adjustment according to the step response

The va lues found a re a pplied using the follow ing setting rules :

Table 6: Formulae for adjustment according to the step response

Example:Tn, Td and XP have to b e d etermined for a temperature c ontrol proc es s. The future o perating ra ngeis a t 200° C. The hea ter powe r ca n be co ntinuously a djusted using a varia ble tra nsformer, a nd the

ma ximum output is 4kW. The d isturbanc e respo nse pa ra meters for a P ID s tructure ha ve to be eva l-uated.

Controller structure Setpoint Disturbance

P XP

= 3.3 · KS

· (Tu/T

g) · 100% X

P= 3.3 · K

S· (T

u/T

g) · 100%

P I XP = 2.86 · KS · (Tu/Tg ) · 100%

Tn = 1.2 · Tg

XP = 1.66 · KS · (Tu/Tg ) · 100%

Tn = 4 · Tu

P ID XP = 1.66 · KS · (Tu/Tg ) · 100%

Tn = 1 · Tg

Td = 0.5 · Tu

XP = 1.05 · KS · (Tu/Tg ) · 100%

Tn = 2.4 · Tu

Td = 0.42 · Tu

KSchange in process variable

change in manipulating variable------------------------------------------------------------------------

∆x∆y------= =

y

t

Dy

Dx

t

x

Point of inflection

Inflection tangent

T

Tg

u





First the heater power is set to give a temperature close to the future working point, for example180°C at 60% heater power. Now the heater power is suddenly increased to 80% and the varia-tion in tempera ture recorded . The inflec tion ta nge nt is then d raw n, g iving Tu a s 1 min and Tg as 10min. If it is d ifficult to de termine the point of inflec tion, the c ha nge in ma nipula ting va ria ble mus t b eincreas ed , i.e. b y sta rting the te st a t a low er heater pow er and e nding a t a higher hea ter pow er. The

final temperature in the case illustrated here is 210°C.

This g ives the follow ing va lues :

Using the values ob ta ined for Tu and Tg , the pa ra meters a re c alcula ted a s follow s:

We s hould not o verloo k a c ertain disa dva ntag e o f this proc es s. In prac tice , the g ra ph very rarelysho ws a very c lea r point of inflec tion. Hence , d ra wing the ta ngent a t the po int of inflec tion c a n lea dto e rrors in det ermining the va lues of Tu and Tg , w hich ma y or ma y not b e s ignifica nt. The method

illustrated is still very useful for forming a first impression of the controller settings. Other criteriaca n then be used to tune the settings.

KS∆x∆y------

210 ° C 180 ° C–

80 % 60 %–------------------------------------

30 ° C20 %------------ 1.5 ° C/%= = = =

Tn 2.4 Tu 2.4 1 2.4 min 144 sec≈ ≈•≈•≈

Td 0.42 Tu 0.42 1 0.42 min 25 sec≈ ≈•≈•≈

XP 1.05 KS

TuTg------ 1.05 1.5

° C%-----

1 min10 min---------------• 100 % 15,75 ° C≈••≈••≈



75



4.5.4 Adjustment according to the rate of rise

In s ome ca se s, there c a n be difficulties in de termining the res pons e time Tg when us ing the me th-od s des cribed a bo ve. Very often, the ma nipula ting va ria ble ca n only be se t to e ither 0 or 100%.Operating the proces s co ntinuously a t 100% manipula ting varia ble ca n be highly d es tructive.

A more constructive alternative is to avoid determining Tg , and instead to evaluate the maximumrate of rise Vma x . To d o this, the ma nipula ting va ria ble is s udd enly set to 100% and the output o fthe proces s ob se rved (se e Fig. 49). The proces s varia ble will only s ta rt to cha nge a fter a ce rta intime, fo llow ing the cha nge in ma nipula ting va ria ble. The rate of c ha nge w ill increa se c ontinuous lyuntil the po int of inflec tion is reac hed . At the po int of inflec tion, the proc es s va ria ble a pproa ches itsfina l va lue more a nd mo re s low ly. Us ing this me thod , it is nec es sa ry to wa it until the po int of inflec -tion is rea ched , a nd then s et the ma nipula ting va ria ble ba ck to 0 % ag a in. It is important to remem-ber that, es pecia lly in proce ss es with long dela ys, s uch a s furnac es, the proce ss variab le c a n co n-tinue to increas e c ons iderab ly, even a fter the hea ting ha s b een s witched o ff.

Fig. 49: Adjustment according to the rate of riseThe ta ngent a t the po int of inflec tion is now draw n, a nd Vma x de termined from the gra dient tria ng le.Using the delay time determined in a similar manner from the step response, the controller settingsca n be implemented in a cc ordanc e w ith the tab le w hich follow s la ter.

The metho d d es c ribe d yields even b ette r va lues if the controller and a ny manipula ting d evice tha tmight be present allow the manipulating variable to be set to any value. In this case, the stepcha nge in ma nipula ting va ria ble should be ma de close to the s etpoint req uired la ter:

Example:The future s etpo int value of a furna ce is 300° C . The existing c ontroller is s et to ma nua l mod e a ndthe ma nipula ting varia ble ma nually increas ed until the furna ce temperature rea ches 280° C; this

temperature is reac hed a t, s a y, 60% ma nipula ting va ria ble. Now the ma nipula ting varia ble is sud -de nly s et to 100%, a nd the point of inflec tion a w a ited. To a pply the a djustment a cc ording to the





ra te of rise , for this exa mple, w e a lso need the height o f the ste p c hang e in manipula ting va ria ble∆y (40%) a nd the ma nipula tion ra nge yH (in this ca se 100%, a s the co ntroller ca n be se t to 100%ma nipula ting va ria ble) for use in the formula e w hich follow la ter.

Table 7: Formulae for adjustment according to the slew-rate response,

for processes with self-limitation

4.5.5 Adjustment without knowledge of the process

Occ a sionally, a co ntroller has to b e a djusted to a proc es s w here it is s imply not pos sible to reco rda trans fer function or to o pen the control loo p. If the proc es s is not o verly s low, the c ontroller is ini-tia lly s et to a pure P structure w ith the la rge st propo rtiona l ba nd po ss ible, so that p ure P a ction isachieved.

The s etpoint is se t c los e to the future o perating p oint, a nd the p roc es s value indica tion o n the c on-troller is observed. After some time, the process value will stabilize at a value quite some way fromthe s etpoint. This is bec a use of the low g a in through the la rge propo rtiona l ba nd s etting. XP is now

red uced , a s a res ult of w hich the devia tion from the se tpoint bec omes sma ller and sma ller. As XP isfurther reduc ed , a point is eventually rea ched a t w hich the proc es s value s ta rts to o sc illa te period i-ca lly. There is no po int in red ucing XP any further, as it would only increase the amplitude of theseos c illa tions . Thes e os cilla tions a re not us ua lly sy mmetrica l a bo ut the se tpoint; their mea n value iseither a bo ve or below the se tpoint. The rea so n for this, a s w e ha ve a lrea dy es ta blished , is the c on-tinued pres ence of the permanent d evia tion tha t a P co ntroller prod uces .

The propo rtiona l ba nd is no w increa se d o nce a ga in, until the proc es s va lue bec omes sta ble.

Next, the I component is added (PI structure), and the reset time Tn is reduc ed step by s tep. Theprocess variable slowly approaches the setpoint, as a result of the I component. Reducing Tn stillfurther a cc elerates the app roa ch, b ut also lea ds to os cilla tions . We now a pply a d isturbanc e to theproc es s, e ither by c hang ing the s etpoint or an external disturba nce. The a pproa ch to the new se t-

point is monitored. If the proc es s value overshoo ts, w e ha ve to increas e Tn . If the approach is onlyvery slow, the reset time setting ca n be red uce d s till further.

The D comp one nt ca n be a c tiva ted next, if req uired , (P ID structure), by s etting Td to a value of ap-proxima tely Tn /4.5.

The proc edure de sc ribe d a bo ve is a widely used pra ctica l method , suita ble for simple proc es se s.

Controller structure 100% step change in MV Any changes in MV

P XP = Vma x · Tu XP = Vma x · Tu · yH/∆y

P I XP = 1.2 · Vma x · Tu

Tn = 3.3 · Tu

XP = 1.2 · Vma x · Tu · yH/∆y

Tn = 3.3 · Tu

P ID XP = 0.83 · Vma x · Tu

Tn = 2 · Tu

Td = 0.5 · Tu

XP = 0.83 · Vma x · Tu · yH/∆y

Tn = 2 · Tu

Td = 0.5 · Tu

P D XP = 0.83 · Vma x · Tu

Td = 0.25 · Tu

XP = 0.83 · Vma x · Tu · yH/∆y

Td = 0.25 · Tu



77



4.5.6 Checking the controller settings

We ca nnot expec t the co ntrol loo p to a chieve optimum performa nce w ith the initia l pa rameter set-tings. Some readjustment will usually be required, particularly on processes that are difficult tocontrol, with a Tg /Tu ra tio less than 3. The ste p res pons e o f the proces s varia ble to a se tpoint

cha nge clearly sho ws a ny misma tch o f the control pa ra meters. The res ulting trans ient res pons ecan be used to draw conclusions about any necessary corrections. Alternatively, an external dis-turbance can be applied to the process, for example, by opening a furnace door, and then analyz-ing the effects of the disturba nce. A rec order is us ed to monitor the proces s va ria ble, and the co n-troller se tting a djuste d if nec es sa ry (se e Fig . 50).

Increas ing the proportiona l band XP - corresponding to a reduction in controller gain - leads to amore stable transient response. Without an I component, a permanent deviation can be detected.Reduc ing XP reduces the deviation, but a further reduction in proportional band eventually leads tounda mped os cilla tions . S etting the c ontroller se tting just b elow se lf-os cilla tion, by se tting a sma llXP , lea ds to a sma ll devia tion, b ut this is not the optimum s etting, a s in this ca se the co ntrol loop isonly very lightly damped. As a consequence, even small disturbances cause the process variable

to o sc illa te.The I compo nent reduc es the permanent d evia tion in ac co rda nce with the reset time Tn . If the Ico mponent is too low (Tn too la rge ), the visible e ffec t is that the proc es s varia ble o nly c reeps gra d-ually towards the setpoint. A larger I component (Tn sma ll) a c ts like a n exc es sive c ontrol ga in, a ndmakes the control loop unstable, resulting in oscillations.

A large derivative time Td has an initial stabilizing effect, but, with a pulsating process variable, itca n also ma ke the co ntrol loo p unstab le.

Fig. 50 indica tes pos sible inco rrec t se ttings . It uses a s a n exa mple the s etpoint res ponse of a third-order process with a PID controller.

When optimizing a controller, only one parameter should be adjusted at a time, then the effect of

this c hang e a wa ited b efore c hang ing further pa ra meters. Furthermore, we ha ve to co nsider wheth-er the c ontroller should be optimized for disturbanc e respo nse or se tpoint res pons e.

It is found, for example, that a “tight” controller setting with a high controller gain may indeed givea fast approach to the setpoint, but the control loop is poorly damped because of the high gain.This co uld me a n that a sho rt duration d isturba nce prod uces os cilla tion. In other words, a low ercontroller gain slows down the approach to the setpoint somewhat, but makes the entire controlloo p more s ta ble.





Fig. 50: Indications of possible incorrect adjustments

t

t t

t

t

w

w w

w

w

x

x x

x

x

X too la rgeP

X too sma llP

optimum a djustment

T , T to o largen d

T , T too sma lln d




5 Switching controllers

5.1 Discontinuous and quasi-continuous controllers

With the co ntinuous c ontrollers de sc ribe d previous ly, with P, P D, I, P I a nd P ID a c tions , the ma nipu-la ting va ria ble y ca n take on a ny value betw een the limits y = 0 and y = yH. In this w a y, the c ontrol-ler is a lwa ys a ble to keep the proces s va ria ble eq ual to the setpo int w.

In co ntra st to c ontinuous c ontrollers, d isc ontinuous a nd q uas i-continuous c ontrollers d o no t ha vea co ntinuous output s igna l, but o ne tha t c a n only ha ve the s ta te ON or OFF. The outputs from s uchcontrollers are frequently implemented as relays, but voltage and current outputs are also com-mon. However, unlike the continuous controller, these are binary signals that can only have a valueof 0 or the ma ximum value. These signa ls c a n be us ed to co ntrol device s s uch a s s olid-sta te re-lays.

Fig. 51: Continuous, discontinuous and quasi-continuous controllers

In a dd ition to these co ntroller types with binary outputs, there a re a lso 3-sta te a nd multi-sta te c on-trollers, where the manipulating variable output can have 3 or more levels. A tri-state controllerwo uld, for insta nce, be used for hea ting a nd c oo ling ta sks, or humidifica tion a nd d ehumidifica tion.

It might b e a ss umed tha t c ontrollers w ith outputs w hich c a n only be in the ON or OFF sta te w ouldonly produce a n unsa tisfa cto ry c ontrol ac tion. B ut surprisingly eno ugh, s a tisfa cto ry results for theintended purpos es ca n be a chieved in ma ny co ntrol proc es se s, pa rticula rly with qua si-co ntinuousco ntrollers. Disc ontinuous a nd q uas i-continuous c ontrollers a re very widely use d, bec a use of thesimple construction of the output stage and the actuators that are required, resulting in lowerco sts . They a re found universa lly in those a rea s o f proc es s co ntrol where the proce ss es a re rela -tively slow a nd c a n be rea dily co ntrolled with switching a ctua tors.

The s implest co ntroller w ith a binary o utput is the disc ontinuous controller, w hich is e ffec tively a li-mit s witch tha t s imply s witches the ma nipula ting varia ble on o r off, dep ending on w hether the pro-ce ss varia ble goe s b elow or ab ove a prede termined s etpoint. A simple exa mple of suc h a c ontrol-ler is the tw o-sta te b imeta llic tempe rature c ontroller in a n electric iron, o r a refrigerato r thermos ta t.

Quas i-continuous co ntrollers ca n be put tog ether, for exa mple, by a dd ing a sw itching s ta ge to theoutput of a continuous controller (see Fig. 51), thus converting the continuous output signal into aswitching sequence. P, PD, I, PI and PID actions can also be implemented for these controllers

(Fig. 51) and the foregoing remarks about continuous controllers are also applicable.

fine g ra dua tionof ma nipula ting va ria ble

( 0 – 100 %)

coarse graduationof ma nipula ting va ria ble

( 0 or 100 %)

continuouscontroller

switchings tagefine grad uation

of ma nipula ting va ria ble

( 0 – 100 %)


y

yyR

w

-x

comparatorwith hysteresis

y


discontinuouscontroller

quasi-continuouscontroller

w

-x

w

-x





5.2 The discontinuous controller

The d isc ontinuous c ontroller has only 2 s witching s ta tes, i.e. the output signa l is sw itched on a ndoff, depending on w hether the proc es s varia ble goe s below o r abo ve a prede termined limit or set-point. These device s a re a lso often used a s limit monitors, w hich initia te a n a la rm mes sa ge when a

setpoint is exceede d.A simple exa mple of a me cha nica l disc ontinuous controller is , a s p revious ly explained, the b imeta l-lic sw itch o f a n electric iron, w hich sw itches the hea ting element o ff when the s et te mperature is re-a ched a nd s w itches it on a ga in whe n the temperature fa lls by a fixed sw itching d ifferentia l (hystere-s is ). There a re o ther exa mples in the field of e lec tronic c ontrollers. For example, a res ist a nce ther-momete r (P t 100), w hos e elec tronic circuitry s witches hea ting on if the temperature fa lls below acertain value, say 5°C, to provide frost protection for an installation. In this case, the resistancethermometer together with the necessary electronic circuitry takes the place of the bimetallicswitch.

Fig. 52: Characteristic of a discontinuous controller

The d isco ntinuous c ontroller show n here supplies 100% pow er to the p roc es s until the s etpo int isreached. If the process variable rises above the setpoint, the power is taken back to 0%. Apartfrom the hys teres is, we se e tha t the disco ntinuous co ntroller corres ponds to a co ntinuous c ontrol-ler with no p ropo rtiona l ba nd (XP = 0) a nd the refore “ infinite” g a in.



81



5.2.1 The process variable in first-order processes

If we co nnect a disco ntinuous c ontroller, s uch a s a rod thermos ta t, to a first-orde r proc es s (e.g . athermos ta tic ba th with wa ter circula tion, w a rmed by a n immersion hea ter), w e find tha t the c ourseof the process variable and manipulating variable is as shown in Fig. 53. In theory, the controller

should switch off the energy when the setpoint is reached, the process variable would fall imme-diately a nd o nce a ga in go below the s etpoint. The c ontroller would immedia tely s witch on a ga in,and so on. Because an idealized first-order process has no delay time, the relay would switch ona nd off co ntinuously, a nd w ould b e d es troyed in a very s hort time.

For this reason, a discontinuous controller usually incorporates a switching differential XS d (alsoknown as hysteresis) about the setpoint, within which the switch status does not change. In prac-tice, the switching differential is often to one side of the setpoint, either below (for example withheating) or above (for example with cooling). Fig. 53 shows a case where the switching differentiallies be low the s etpo int. The s w itch-off point of the c ontroller is t he s etpo int w. In pra c tice , a s theproc es s is not idea l (it ha s so me d ela y time), the higher a nd lower values of the proces s varia ble donot coincide exactly with the switching edges of the differential (XS d).

Fig. 53: Discontinuous controller in a first-order process





What matters however, is that the controller only switches when the process variable has movedoutside the d ifferentia l ba nd tha t ha s bee n se t. The proc es s varia ble co ntinually fluctuate s, a t lea stbetw een the values Xhi and Xlo. The fluctua tion b a nd o f the proce ss va ria ble is therefore influencedby the s w itching differential.

In a process with delay, the discontinuous controller can only maintain the process variable con-sta nt between the values Xhi and Xlo. The on-off sw itching is due to the ma nipula ting va ria ble be ingtoo large to ma intain the proc es s va ria ble c ons ta nt when it is s witched o n, a nd too sma ll when it issw itched off. In a la rge number of co ntrol ta sks, w here the proc es s va ria ble only needs to b e ma in-ta ined a pproxima tely c ons ta nt, these fluctuations a re not a prob lem. An exa mple o f this is a do me-stic elec tric o ven, where it do es not ma tter if the ac tual tempera ture fluctuate s b etw een 196° C a nd204° C for a ba king tempe ra ture of 200 ° C.

If these c ontinuous fluctuations o f the proc es s varia ble do c a use prob lems , they c a n be minimizedto a limited extent b y selecting a sma ller sw itching d ifferentia l Xsd. This a utoma tica lly lea ds tomore sw itching opera tions per unit time, i.e. the sw itching freq uency increa ses . This is not a lw a ysde sira ble, a s it a ffec ts the life of the c ontroller relay.

It can be shown (mathematical details are not entered into here) that the following relationshipexists betw een the s witching freq uency (fsw ) a nd the pa ra mete rs T, Xma x and XS d :

fsw : sw itching freq uencyTosc : period of oscilla t ionXma x : max. proc ess varia ble rea ched w ith the controller output permanently sw itched onXS d : switching different ialT : time c ons ta nt o f the firs t-o rd er pro c es s

We c a n s ee from this relationship tha t the sho rter the time c ons ta nt (T), the higher the s w itchingfrequency. A control process with short time constants will therefore produce a high switching fre-q uency, which w ould co ntribute to ra pid we a r of the sw itching s ta ge of the c ontroller. For this rea -so n, a disco ntinuous c ontroller is unsuita ble for this type o f proc es s.

va lid forfs w 1

To s c------------

14--

Xm a xXS d

-------------•1T--•= = x

Xm a x2

-------------≈



83



5.2.2 The process variable in higher-order processes

In a proc es s with dela y, w e ha ve s een tha t under ide a l co nditions the fluctuation ba nd is dete rmi-ned only by the sw itching differential XS d of the c ontroller. The proc es s itself ha s no e ffec t here. Ina proc ess with severa l delays , which c an b e de sc ribed a s d elay time, respo nse time a nd tra nsfer

coefficient, this is no longer the case. As soon as there are any delays the process variable willcontinue to rise o r fa ll a fter sw itch-off and w ill only return afte r rea ching a ma ximum. Fig . 54 sho w show the proc es s varia ble o versho ots the res pons e thres hold o f the rela y w hen the ma nipula ting va -ria ble is sw itched on a nd off.

Fig. 54: Discontinuous controller in a higher-order process

This prod uce s a n overshoot o f the proce ss va ria ble, with limits g iven by the va lues Xhi and Xlo. Thismea ns tha t the proc es s varia ble fluctua tes even when the c ontroller ha s zero sw itching differentia l,a s the proce ss only rea cts to the c hang e in manipula ting va ria ble after the end o f the d ela y time.

Once a ga in, take the elec trica lly hea ted furnac e a s a n example. If the energy s upply is s witched o ffw hen the setpo int is rea ched , the tempe ra ture s till continues to rise. The rea so n is tha t the tempe-ra ture in the furna ce only permea tes slowly, a nd w hen the s etpoint is rea ched , the hea ter rod is a l-rea dy a t a higher temperature tha n that reported by the se nsor. The rod a nd furnac e ma teria l co nti-nue to supply a dd itiona l hea t. S imila rly, w hen the hea ting is sw itched on a ga in, hea ting-up is ra ther

s lugg ish a nd initia lly the tempe rature c ontinues to fall a little further a fter sw itch-on.





The more po w erful the hea ter, the g rea ter is the temp erature difference b etw een the hea ter rod a ndthe sensor during heating-up, because of the process delay, and the process variable will overs-hoot the s etpoint even mo re d uring hea ting-up. We us e the term exces s pow er in this co nnection,meaning the percentage by which the maximum power of a furnace is greater than the power re-quired to approach a setpoint.

Example: A furna ce which req uires a ma nipula ting varia ble of 2kW on a vera ge to sta bilize a t a se t-point of 200° C, but ha s a 4 kW co ntinuous output ra ting, ha s a n exces s pow er of 100% a t the wo r-king p oint of 200° C .

This mea ns that the higher the exc es s pow er, the w ide r is the fluctuation ba nd ∆x of the processvaria ble ab out the se tpoint.

Now the pres ent (but unwa nted) fluctuation ba nd of the proc es s va ria ble ca n be e stimated for thecase where 100% excess power is available:

It is a s sume d tha t the sw itching differentia l XS d = 0

As we can see, the fluctuation band is dependent not only on Xma x (with a linear process this ispropo rtiona l to the e xces s pow er) but a lso on the ratio Tu/Tg , whose reciprocals we are already fa-milia r with from C hapte r 2, a nd w hich g ive a mea sure o f how go od the c ontrolla bility of a proc es sis. The s horter the delay time in co mpa rison w ith the res ponse time, the na rrow er is the fluctua tionba nd. The formula g iven for the fluctua tion b a nd ∆x is valid for XS d = 0. If there is a sw itching d iffe-rentia l, this is a lso a dd ed to the fluctuation ba nd.

This g ives us the formula:

The fo rmula for the pe riod of o sc illa tion is: Tosc = 4Tu (valid for XS d = 0)

If a switching differential XS d ha s be en s et, the n the period of os cilla tion is s lightly long er. From thiswe ca n derive the maximum sw itching freq uency, which c a n be used to predict the expected co n-ta c t life:

va lid for∆x Xm a x

Tu

Tg------•= x

Xm a x2

-------------≈

∆x Xm a x

TuTg------ XS d+•=

fo s c1

4Tu--------=



85



5.2.3 The process variable in processes without self-limitation

Because the step responses of an integrating process are linear, the behavior of a discontinuouscontroller is easy to describe and calculate. Here again the process value fluctuates between theg iven limits Xhi and Xlo (Fig . 55). In an idea l proc es s w ithout d elay time Tu, the limit values are equal

to the s w itching differentia l XS d .

Fig. 55: Discontinuous controller in a process without self-limitation

The s w itching freq uency fsw is given by:

Kp : proportionality factor of the processyH : ma ximum value o f the ma nipula ting varia ble

An exa mple of such an ap plica tion is a disc ontinuous controller used a s a limit sw itch for level con-trol of a w a ter tank. The ta nk is used a s a sto ra ge res ervoir, from w hich w a ter is draw n to mee t de-mand or into which a co nstant a mount flow s.

Summarizing, we can say that the discontinuous controller offers the advantage of simple con-struction a nd few pa ra meters w hich ha ve to be se t. The disa dva ntag e is the fluctua tion of the pro-ce ss varia ble ab out the se tpoint. In non-linea r proc es se s these fluctua tions ca n be wider in the lo-wer operating range than in the upper, because the process has excess power here. Approaching

the setpoint in the lower operating range will often result in wider fluctuations than in the upperope rating ra nge . The a rea of applica tion for such disc ontinuous co ntrollers is limited to a pplica ti-ons where precise control is not required. In practice, these controllers are implemented throughmechanical thermostats, level switches etc. If an electronic controller with a sensor is used, theco ntroller is a lmos t a lwa ys provide d w ith a dyna mic a ction.

fs w 1

To s c------------

Kp yH•

2 X

S d

•-------------------= =





5.3 Quasi-continuous controllers: the proportional controller

As we have already seen, a quasi-continuous controller consists of a continuous controller and aswitching stage. If this controller is operated purely as a proportional controller, then the characte-ristics which w e ha ve a lrea dy met in Cha pter 3.2.1 “The proportiona l band ” a pply eq ually here.

Fig. 56 : Proportional band of a quasi-continuous proportional controller

The q uas i-co ntinuous c ontroller whos e c hara cteristic is s how n in Fig. 56 alwa ys gives out a 100%ma nipula ting varia ble, a s long a s the proc es s value lies below the proportiona l ba nd. As the pro-ce ss value enters the propo rtiona l ba nd a nd a pproa ches the setpo int, so the ma nipula ting va ria blebe co mes prog res sively lower.

How ca n a co ntroller with a sw itched output provide a virtually c ons ta nt energy supply i.e. s teples sdosage?

In the end it is immaterial whether a furnace is operated at 50% heating power all the time or at100% hea ting p ow er for only half the time. The q uas i-continuous c ontroller cha nge s the sw itch-on

ratio or ON-time ratio (also known as duty-cycle) of the output signal instead of changing the sizeof the o utput signa l. An ON-time ra tio o f 1 co rres pond s to 100% of the m a nipula ting va ria ble, 0.25co rres ponds to 25% of the ma nipula ting va ria ble, a nd s o o n.

The ON-time ra tio, or duty-cyc le R is de fined a s follow s :

Ton = ON timeToff = OFF time

Multiplying the ra tio R b y 100 gives the rela tive ON-time in % of R, w hich corres pond s to the ma ni-pulating variable in %.

With a q uas i-co ntinuous c ontroller the c harac teristic o f the proce ss (es pec ia lly the time co nsta nts)exerts a strong influence o n the course of the proc es s va ria ble. In a proce ss where a d isturba nce istra nsmitted rela tively s low ly (a proc es s with long time c ons ta nts) a nd w here energy c a n be sto red,there is a smo othing effect o n a ny pulse s. With a suita ble sw itching freq uency, the use of a q uas i-

co ntinuous c ontroller with these proc es se s a chieves a simila r res ult to that a chieved using a co nti-nuous controller.

y

100 %X

w x

P

RTo n

Ton To ff+------------------------=

R(%) y R 100%•= =



87



The s itua tion is d ifferent with a very fa st p roc es s , w here there is ha rdly any s moothing of the con-sta ntly cha nging flow of energy, and the proce ss varia ble fluctuate s a cc ordingly. Hence q uas i-co n-tinuous c ontrollers a re prefera bly use d where the proc es s is co mpa ra tively s low, a nd a re es pec ia llypopular in temperature control systems.

Fig. 57: Power control

The d efinition of ON-time ra tio (or d uty-cyc le) mea ns the ra tio of the sw itch-on time of a co ntrolleroutput to the s um of the sw itch-on a nd s witch-off times , e.g . a n ON-time ratio of 0.25 mea ns tha tthe power supply is switched on for 25% of the total time. It gives no information on the actual du-ra tion o f the period s during w hich the sw itching c ycles ta ke plac e.

For this reason, the so-called cycle time (C y) is defined, which fixes this time period. It represents

the period during which switching on and off takes place once, i.e. it is equal to the sum of the

s w itch-on a nd s w itc h-off times (Fig . 57). The s w itching freq uenc y is the reciproc a l of the c yc le time.

Fig . 57 show s the s a me ON-time ra tio (R = 0.25) for different cyc le times.





For a g iven ON-time ra tio o f 0.25 a nd a cy c le time of C y = 20 sec , this mea ns that the energy s up-ply is switched on for 5 seconds and switched off for 15 seconds. If the cycle time is 10 sec, theenergy supply is switched on for 2.5 seconds and switched off for 7.5 seconds. In both cases, thepow er supplied is 25 %, b ut with a finer dos a ge with Cy = 10 sec . The fluctua tions of the proces svaria ble are sma ller in the sec ond c a se .

Theo retica lly, the ON-time o f the c ontroller is g iven b y the fo llow ing rela tions hip:

Ton = ON timey = manipula ting va ria ble in %C y = cyc le time

This mea ns tha t a sho rter cyc le time res ults in a finer dos a ge of the energy s upply. On the o therha nd, there is increa se d s w itching o f the ac tua ting d evice (relay or co nta c tor). The sw itching fre-q uency c a n ea sily be d etermined from the c ycle time.

Example:

The c yc le time of a controller used for tempera ture c ontrol is C y = 20 se co nds . The rela y used ha sa conta c t life of 1 million s w itching o pera tions . The va lue g iven for Cy results in 3 switching opera-tions p er minute, i.e. 180 pe r hour. For 1 million o perat ions , this g ives a life of 5555 hours = 231da ys. B a se d o n an o perating time of 8 hours pe r da y, this repres ents a pprox. 690 da ys. Ass uminga round 230 working d a ys per year we a rrive a t a n operating life of a pprox. 3 yea rs.

Generally, the cycle time is selected so that the control process is able to smooth out the energybursts supplied , to eliminate period ic fluctua tions of the proce ss varia ble a s far a s pos sible. At thesa me time, the number of sw itching o perations must a lwa ys b e ta ken into a cc ount. With a micro-processor controller however, the value set for the cycle time C y is not held co nsta nt over the who-

le o f its wo rking ra nge. A deta iled discus sion of this point is ra ther complica ted a nd w ould b e to oa dva nced a t this sta ge . If it is pos sible to o perate a sw itching P co ntroller in ma nual mode, the in-fluence on C y ca n be o bs erved by d irec t input of a ma nipula ting va ria ble.

When C y is ma tched to the dyna mic a ction of the proc es s, the be havior of a q uas i-co ntinuous co n-troller (as a proportional controller with dynamic action) can definitely be comparable with that of acontinuous controller, w hich a lso e xpla ins its na me. With q uas i-continuous co ntrollers the differentmanipulating variables are the result of a variation of the ON-time ratio, but there is no discernibledifference in the c ourse of the proces s varia ble w hen c ompa red to tha t of a co ntinuous c ontroller.

Ton

y C y•

100 %----------------=



89



5.4 Quasi-continuous controllers: the controller with dynamic action

A quasi-continuous controller, operated as a pure proportional controller and with C y suita bly mat-

ched, show s a lmost the s ame beha vior in a proce ss a s does a continuous c ontroller with P ac tion.Although it reacts very quickly to changes in the control deviation, it cannot reduce the control de-

viation to zero, which is also the case with a proportional controller. A quasi-continuous controllercan also be configured as a PID controller, which means that it slows down as the setpoint is ap-proa ched a nd s tab ilizes a cc ura tely a t the s etpoint.

A q uas i-continuous c ontroller (a nd a lso a P co ntroller) ca n be pictured a s a co mbina tion o f a co nti-nuous c ontroller and a sw itching s ta ge co nnected to the o utput. The c ontinuous c ontroller ca lcula -tes its manipulating varaible from the course of the process variable deviation and controls thesw itching sta ge a cc ordingly. The s witching sta ge ca lcula tes the rela tive ON-time o f the sw itchingsta ge output. The output of the sw itching s tag e is pulse d in ac co rda nce w ith the ON-time ratio a ndthe va lue se t for Cy.

Fig. 58: Quasi-continuous controller with dynamic action,

as a continuous controller with a switching stage

Example: The c ontinuous controller prod uce s a ma nipula ting va ria ble of 50%. Likewise, for thesw itching s ta ge , 50% ma nipula ting va ria ble me a ns a n ON-time o f 50%. Let us a ss ume that the va-lue s et fo r Cy is 10 se co nds , then the sw itching s ta ge will turn the input on and off every 5 sec onds .





5.4.1 Special features of the switching stages

As already described, the switching stage calculates the relative ON-time of the switching stageoutput on the b a s is o f the ma nipula ting va ria ble of the c ontinuous co ntroller. The output o f thesw itching s ta ge is pulse d in ac co rda nce with the rela tive ON-time a nd the va lue se t for Cy.

Using analog technology, an additional D action stems from the technical implementation of thesw itching s ta ge . If the c ontinuous co ntroller prec eding the sw itching s ta ge is o perated a s a propo r-tiona l controller, then co mb ining it with the sw itching s ta ge res ults in a P D a c tion. This me a ns tha tif the analog controller referred to has a pure P action (i.e. only a proportional band XP ca n be se t),then the quasi-continuous controller, obtained by connecting a switching stage to its output, hasa n a dd itiona l D a ction which is not a djusta ble. As this cha ra cte ristic proved to be very useful, it ha sa lso b een reta ined in the microproc es sor co ntrollers. The va lue of the de riva tive comp onent is no r-ma lly set b y the manufac turer and therefore c a nnot be a djusted .

The follow ing tab le s how s the c ontrol configurations of a q uas i-co ntinuous c ontroller a nd the a ctu-a l co ntrol ac tions :

Table 8: Control configurations of a quasi-continuous controller, and actual control actions

Example: A q uas i-co ntinuous c ontroller co nfigured a s a proportiona l controller a ctua lly ha s P D a c-tion, bec a use of the sw itching s tag e c onnected to its o utput.

If the continuous controller has a PID structure, the existing switching controller has a P, an I andtwo D c ompo nents. P D/P ID c ontrollers o f this type a re w idely used with sw itched temperatureco ntrollers, a s they offer the b es t s ta rt-up a ction for this a pplica tion.

Depending on the c ontroller structure, va rious se tting pa ra meters a re o bta ined for the q uas i-conti-nuous controller:

Table 9: Setting parameters for differing dynamic actions

5.4.2 Comments on discontinuous and quasi-continuouscontrollers with one output

In prac tice , d isc ontinuous a nd q uas i-co ntinuous c ontrollers a re o ften brought tog ether under theco ncept o f the 2-sta te c ontroller, on the b a sis that the o utput of the co ntroller ca n only a ss ume twocond itions , either on o r off. Thes e c ontrollers a re c onfigured a t the ins trument b y de fining the co n-trol type, na mely 2-sta te c ontroller. If the propo rtiona l ba nd XP is now set to 0, we ha ve a d isc onti-nuous controller. If a proportiona l ba nd X

Pgrea ter tha n 0 is se lec ted, a q uas i-continuous c ontroller

is ob ta ined, o n which the a ppropria te c ontrol paramete rs (Tn, Td and C y) ca n be s et.

Set control action Actual control action

P P D

P D P DD

I P I

P I P ID

P ID P D/P ID

PD PDD PI PID PD/PID

XP XP - XP XP

C y C y C y C y C y

- - Tn Tn Tn

- Td - - Td



91



5.5 Controller with two outputs: the 3-state controller

5.5.1 Discontinuous controller with two outputs

A disco ntinuous c ontroller with two outputs c a n be thought of in simple terms a s a co mbina tion of

tw o d isco ntinuous c ontrollers, e a ch with one output, b ut linked tog ether, one b elow the o ther. Theyca n be used for heating, for exa mple, when be low the setpo int and for co oling w hen ab ove the set-point. Other applications are, for instance, humidification and dehumidification in a climatic cabi-net, etc. Each output of the controller is assigned a manipulating variable. E.g. the first controlleroutput would o ften be used for hea ting a nd the se co nd output for co oling. All pa ra meters a ss oc ia -ted with the “heating controller” are identified by an Index 1 and those a ss oc ia ted w ith the “c oolingcontroller” b y Inde x2.

If the proc es s varia ble va ries within a fixed interva l a bo ut the setpo int – the c onta ct s pa cing XS h –then neither output is a c tive (Fig . 59). This c onta c t sp a cing is nec es sa ry to prevent continua l sw it-ching b etw een the two ma nipula ting va ria bles e.g . hea ting a nd co oling reg iste rs, w hen the proc es svaria ble is unstea dy.

As well as the contact spacing, discontinuous controllers with two outputs also have a hysteresisfor each of the heating and cooling contacts, which are normally indicated by the switching diffe-rentials XS d1 and XS d2 . These tw o pa ra meters eliminate a ny conta ct “ cha tter”, w hen the proc es svaria ble moves from the hea ting zo ne or cooling zo ne into the conta ct s pa cing.

With regard to the switching differentials XS d1 and XS d2, and the related switching frequency andcontrol quality in connection with the process characteristics, the same considerations apply asfor a d isc ontinuous controller with only one output.

With this c ontroller, the c ontrol a ccuracy which c a n be a chieved is limited by the sw itching hys tere-sis values a nd the conta ct s pa cing (Fig. 59).





Fig. 59: Characteristic of a discontinuous controller with two outputs

Discontinuous controller with 2 outputs

w

Characteristicy

100 %

X

w x

X

- 100 %

HeatingX

XCooling S d2

S d1

S w itc hing differential (X , X ) a nd conta c t spa c ing (X )S d1 S d2 S h

S d1

XS h

S d2

XS h



93



5.5.2 Quasi-continuous controller with two outputs, as a proportional controller

Even a q uas i-co ntinuous 3-sta te c ontroller, where ea ch o f the two o utputs is ope ra ted b y a propo r-tiona l co ntroller, c a n be thought of in simple terms a s a co mbina tion of tw o q uas i-continuous c on-trollers linked tog ether, o ne b elow the o ther. The s w itching differentia l do es not a pply here, b ut a

contact spa cing ca n be set.Fig. 60 show s the cha ra cteristic of a q uas i-co ntinuous 3-sta te c ontroller use d to co ntrol an c lima ticcabinet.

Fig. 60: Characteristic of a quasi-continuous controller with two outputs,

as a proportional controller

As sho wn in Fig . 60, the two propo rtiona l ba nds XP 1 and XP 2 can be adjusted independently for aq uas i-co ntinuous c ontroller with two outputs. This is nece ss a ry, b ec a use in g eneral the proce ssgain is different for the two manipulating variables, as the heating register influences the processdifferently from the c oo ling (throug h a fan, for example).

The w a y in which this c ontroller wo rks is d es cribe d b elow : The proce ss va lue in the ca binet is26°C . Now the co ntrol is s witched o n:

Heating: The hea ting rela y is e nerg ized a nd the hea ting hea ts up w ith 100% ma nipula ting va ria ble,w hereupon the proc es s va lue increa se s . The hea ting ma nipula ting va ria ble continually red uce sfrom a process value above 27°C (on reaching the proportional band), the heating relay starts topulse a nd the sw itch-on times bec ome prog res sively s horter. The c ontrol devia tion a nd hence themanipulating variable become smaller, until a manipulating variable is obtained which is just suf-ficient to ma intain the proc es s va lue. Now the proc es s va lue is b elow the conta ct s pa cing, a nd weob ta in a pos itive ma nipula ting va ria ble (for example 28.5° C a nd 25% ma nipula ting va ria ble).

Cooling: Now the a mbient tempera ture increa se s (disturba nce), w hereupon the inner chamb er ofthe ca binet is hea ted. The proc es s va lue increas es – on entering the c onta ct s pa cing (29°C ) thema nipula ting va ria ble is 0%, a nd the re is neither hea ting nor co oling. The c oo ling relay o nly s ta rtsto p ulse a bo ve a temperature o f 31° C (the ma nipula ting varia ble b ec omes nega tive). Likew ise , thecontrol deviation reaches a value such that the manipulating variable produced is just sufficient tomaintain the resulting process value.





5.5.3 Quasi-continuous controller with two outputs and dynamic action

With a quasi-continuous controller with two outputs and dynamic action, it is additionally possibleto set a Tn and a Td for each of the two manipulating variables. With this controller, the three con-trol compo nents (P, I and D) a re o nly a ll effec tive tog ether outside the c onta ct spa cing. If a proc es s

enters the conta ct s pa cing, the P co mponent is not effective. In the co ntac t spa cing, only the I a ndD c ompo nents a re effective, a nd in principle the proce ss varia ble ca n therefore b e s ta bilized exa ct-ly.

Because there is no P component in the contact spacing, a large contact spacing has an adverseeffec t on the dyna mic respo nse, a s the c ontroller is s low in the conta ct s pa cing. The size of thecontact spacing chosen should be no larger than necessary, but no unwanted changeover bet-ween the manipulating variables should occur, either when approaching the setpoint (possibleoversw ing of the proc es s varia ble a bo ve the se tpoint) or when sta bilizing a t the s etpoint (fluctuati-ons of the proc es s va ria ble a bo ut the s etpoint).

Too small a contact spacing can lead to a pointless waste of energy in an installation

Ta ble 10 show s the s etting pa ra meters of a q uas i-co ntinuous c ontroller with two outputs a nd d yna-mic a ction:

Table 10: Setting parameters for a quasi-continuous controller,

with two outputs and dynamic action

5.5.4 Comments on controllers with two outputs

In C hapte r 5.5 we met c ontrollers w here the two outputs c ould influence a proc es s varia ble in twodirec tions. The c ontrollers de sc ribed a lw a ys had two outputs of the s a me type (disco ntinuous orquasi-continuous). In practice, it may turn out that the outputs are different. It may well be, for ex-a mple, tha t a co ntroller has to provide a disco ntinuous output for co oling a nd a q uas i-co ntinuousoutput for heating. Such a controller cannot be classified under any of Chapters 5.5.1, 5.5.2 or

5.5.3.

To limit the numb er of na mes , in prac tice a ll co ntrollers w ith tw o o utputs w hich ca n influence a pro-ce ss varia ble in two direc tions a re referred to a s...

. . . 3-state controllers

irres pec tive o f whether the o utputs a re d isc ontinuous, q uas i-co ntinuous o r continuous. As a n ex-ample, mention should be made of a controller which operates a thyristor-controlled power unita nd a refrige ra tor unit a nd thus ma intains co nsta nt temperature in a clima tic ca binet. The tw opla nts req uire tw o c ontroller outputs – but the c ontroller must provide a continuous output for hea -ting a nd a sw itched output for coo ling.

set effec tive

Discontinuous XP 1 = 0XP 2 = 0

– – – XS h Xd1; Xd2

Quasi-continuous P P D XP 1; XP 2 – – C y1 ; C y2 XS h –

P I P ID XP 1; XP 2 Tn1; Tn2 – C y1 ; C y2 XS h –

P ID P D/P ID XP 1; XP 2 Tn1; Tn2 Td1; Td2 C y1 ; C y2 XS h –

P D P DD XP 1; XP 2 – Td1; Td2 C y1 ; C y2 XS h –

I P I – Tn1; Tn2 – C y1 ; C y2 XS h –



95



5.6 The modulating controller

Actuators (actuator drives) have three operating conditions: opening, holding, closing.

Electrical drives in particular are widely used for this application, where a motor controlled forclockwise or a nticlockwise rota tion d rives a wo rm g ea r to operate a valve, throttle, va ria ble trans -

former or simila r device . B oth DC a nd 3-phas e moto rs a re use d, w ith single pha se moto rs us ed forsma ller drives, sw itched by c onta cto rs or rela ys (se e C hapte r 1.7).

These drives sta nd o ut from the c ontroller a pplica tions a lrea dy disc uss ed in one pa rticula r wa y.When the heating is switched on in a furnace, it operates immediately at full power, and when it issw itched off, the supply of pow er stops immediately. In co ntra st, a ctua tors req uire a ce rta in time toreach the maximum manipulating variable (valve opening etc.). In addition, an electrical actuatorholds the p os ition it has rea ched , even w hen there is no s igna l from the c ontroller. The a ct ua torca n, for exa mple, s ta y a t 60 % open, e ven though it is not ope ra ted b y the c ontroller at this time.The c ontroller must ta ke these prop erties into a ccount.

Modulating controllers are used for this type of actuator drive.

The mo dulating controller co nsists of a co ntinuous c ontroller (P or P D) a nd a sw itc hing e leme nt. Ifwe reg a rd the va lve po sition a s the ma nipula ting varia ble, the c omb ination of mod ula ting co ntrollera nd reg ula ting va lve exhibit P I or P ID ac tion.

To understand the operation of the modulating controller take a look at Fig. 61:

Fig. 61: The modulating controller, with a regulating valve in the control loop

The modulating controller show n co ntrols the tempera ture in a furna ce via a reg ula ting va lve in thega s flow. The s witching s ta ge provides two rela y outputs w hich d rive the valve op en a nd c los edover the range 0 to 100%.

The controller, the sw itching e leme nt and the regulating va lve must now b e thoug ht of a s a s ing le

unit. The mo dulating c ontroller (mea ning here the c ontinuous controller a nd the sw itching element)can be configured for PI or PID action. If a control deviation occurs, the valve will exhibit the corre-sponding PI or PID action. If then, for example, PI action is set on the controller, then the combinedmodulating controller and regulating valve (with the valve opening as manipulating variable) willhave P I a ction.

Example: The mo dulating c ontroller of Fig . 61 w a s configured a s a P I controller. The propo rtiona lband XP was set to 25°C, and the reset time Tn to 120 sec onds . The a ss oc ia ted regula ting va lvehas a n ac tua tor stroke time Ty (the time required by the actuator to travel from 0 to 100% or from100 to 0% manipula ting va ria ble) of 60 sec onds . Fig. 63 show s the ste p res pons e o f the sys tem.





Fig. 62: Step response of the modulating controller and regulating valve system

Fig. 62 show s the step res pons e of a P I co ntroller with the parame ters XP = 25 ° C an d Tn = 120sec.The step c hang e in control de via tion oc curs a t t = 60 se c a nd a mounts to 10° C. The mod ula tingcontroller has the following settings: XP = 25 ° C , Tn = 120s ec, a nd the ac tuator stroke time Ty =60seconds. By operating via the “Open” output of the modulating controller, the PI action of thecomb ined mod ula ting c ontroller and regulating va lve w ill be implemented . The opening o f the va lvewill, of course, lag behind the manipulating variable of a PI controller, as it has a stroke time of 60seconds.

The mod ula ting c ontroller rec eives no indica tion of the exa c t pos ition of the va lve. It a s sume s tha tthe valve ope ns a nd c los es a t exa ctly the sa me s peed . The mod ula ting c ontroller ca lcula tes thetime for which the “ Open” co ntac t must be close d, until, in theory, the va lve pos ition c orres pondsto the manipulating variable of the corresponding PI controller. For this to work, the modulatingco ntroller must ha ve knowledg e o f the a ctua tor stroke time.

The mod ula ting c ontroller also has its c onta c t spa cing s et so a s to lie s ymmetrica lly a bo ut the s et-point. Within the co ntac t spa cing, no c ontrol opera tion oc curs on the a ctua tor, w hich me a ns tha t ifthe proces s varia ble e nters the c onta ct spa cing, the va lve w ill rema in in its old pos ition.

With modulating controllers, a minimum pulse duration TMmin ca n be ta ken into a cc ount. This ma ybe necessary because of minimum switch-on times of the actuator drive (e.g. play in the gears).With a microprocessor controller, however, it is at least the sampling time or cycle time of the con-

troller. The minimum p ulse d ura tion TMmin ca n be se t direc tly on ma ny co ntrollers.The minimum pulse dura tion ha s a direc t influence on the positioning accura cy of the ac tua tor, a ndco nseq uently on the expected control ac cura cy.

The fo llow ing relationship genera lly a pplies for linea r proc es ses :

∆x : control a ccura cyXMa x : maximum proces s value

TMmin : minimum pulse lengthTy : ac tuator stroke time

∆x XMa x

TMminTy

----------------•=



97



It is important that the contact spacing XS h of a modulating controller is not set smaller than thecontrol accuracy ∆x, calculated from the minimum pulse duration. Choosing a contact spacingsma ller than this value w ill res ult in permanent fluctuations o f the proce ss varia ble a s the a ctua torco ntinually cha nges over from c loc kwise to a nticlockwise rota tion, ma king exc es sive de ma nds onthe actuator.

If the co rrec t co ntac t spa cing is cho se n, the true co ntrol devia tion w ill be sma ller tha n the se t co n-tac t spa cing, be ca use the final pulse runs the ac tua tor into the conta ct s pac ing a nd thereby redu-ce s the c ontrol devia tion.

As the a ctua tor drive ha s the sa me c hara cte ristic for cloc kwise a nd a nticlockwise rotation, there isonly one s etting ea ch for XP , Tn and Td . The s etting pa ra meters a re then a s follow s:

Table 11: Setting parameters with the modulating controller

Dynamic action PI PID

Setting parameters XP XP

Tn Tn

- Td

Ty Ty

XS h XS h





5.7 Continuous controller with integral motor actuator driver

A “co ntinuous controller with integ ra l moto r ac tua tor driver” o r, for sho rt, an a c tua ting c ontroller, ismuch mo re s uita ble for operating a motorized a ctua tor than is a mod ula ting co ntroller. It forms atype of ca sc a de s tructure (se e C ha pter 6.6), a nd c ons ists of a c ontinuous co ntroller a nd a sub ordi-

nate actuating controller (Fig. 63).

Fig. 63: The actuating controller with a regulating valve in the control loop

The continuous co ntroller outputs the ma nipula ting va ria ble, ba se d on the co urse of the co ntrol de -via tion a nd the p a rame ters s et o n the c ontroller (Fig . 63). The us ua l control struct ures , i.e. P, P I, I,P D, P ID ca n be s et for the co ntinuous c ontroller. The duty o f the act ua ting c ontroller is no w to re-gula te this ma nipula ting va ria ble on the reg ula ting va lve. The a ctua ting c ontroller opera tes the a c-tuator via two switching outputs, and receives an actuator retransmission signal (usually a stan-da rd signa l 0/4 — 20mA, 0/2 — 10V etc.), w hich feed s the a ctua tor position b a ck to the c ontroller.

Example: The c ontinuous controller det ermines a ma nipula ting va ria ble of 20% from the c ourse of

the c ontrol de via tion. The a c tua ting co ntroller now controls the va lve a t 20% ope ning. The va lveprovides a 0 — 10V a ctuator retra nsmiss ion signa l that co rrespo nds to 0 — 100% opening of thevalve. If the actuating controller has controlled the valve to 20% opening, the actuator retransmis-s ion signa l would b e 2V.

An actuator stroke time must also be fed into the actuating controller, which the controller thenuses to o ptimize its co ntrol paramete rs.

Where a mo tor is b eing ope rated w hich has a n apprec ia ble overrun (poo r bra king a c tion), judd eringof the ac tua tor motor ca n be a voided by increa sing the co ntact spa cing (XS h).

Advantages of the actuating controller in comparison with the modulating controller:

Unlike the modulating controller, the actuating controller offers the advantage of a subordinate

controller structure. If a control deviation occurs, the actuating controller ensures that the motor isdriven direc tly to a new position. This is a chieved b y co mpa ring the a c tua tor pos ition w ith the ma-nipulating va ria ble (yR) of the continuous controller.

A mod ula ting c ontroller does not rec eive a n ac tuator retransmiss ion s igna l a nd must a lwa ys a ss u-me a linea r ac tuator a ction. If the a ctua tor has a non-linea rity, or pla y is pres ent in the a ctua tor me-cha nism, this a ss umption will only b e a n a pproxima tion.



99



An actuating controller offers the following setting parameters for the corresponding control action:

Table 12: Setting parameters with the actuating controller

Fig. 64: Example of an application for an actuating controller

Example:

The a ctua ting c ontroller des cribed is used to c ontrol the o utflow temperature o f a hea ting s ystem .The ma in element is a mixer valve w hose cha mbers “ C” a nd “ W” for co ld a nd w a rm wa ter are lin-ked through piping to the w a ter return pipe. The mix temperature is mea sured by a P t100 a nd c a nbe varied by a djusting the pos ition of the s lider “ S ”, which is operate d b y a n a ctua tor motor. Theinput variable of the actuator motor is in the form of switching pulses for opening and closing theoutflow opening.

Dynamic

action

PD PDD PI PID PD/PID

Settingparameters

XP XP - XP XP

- - Tn Tn Tn

- Td - - Td

Ty Ty Ty Ty Ty

XS h XS h XS h XS h XS h








6 Improved control quality through special controls

S o far, w e ha ve o nly c ons idered s ingle-loo p c ontrol circuits, where c ontroller a nd proce ss form aclosed signal loop. However, when using such single-loop control circuits, there are limits to theco ntrol q uality which c a n be a chieved in certain co ntrol proc es se s. It is po ss ible to g o b eyond thecontrol qua lity limits imposed by the s ingle-loo p c ontrol circuit b y us ing multi-loo p c ontrol circuits,or by s w itching a uxilia ry va ria bles o n a nd o ff. To s ome extent, rela tively s imple so lutions ca n lea d

to considerable improvements in control quality.

6.1 Base load

With a b a se loa d s etting , the c ontroller only influences pa rt of the to ta l ma nipula ting va ria ble, a nd afixed proportion is continuously supplied to the process (combination of control and operation). Itco uld then be the ca se that, for example in a n elec trica lly hea ted furnac e, one s ec tion of the hea t-ing e lements is c ontrolled by the controller, w hereas a nother sec tion is s upplied a t full supply volt-a ge (see Fig . 65).

Fig. 65: Base load setting

Ess entia lly, ba se loa d s etting offers the follow ing a dva ntag es :

- The a ctuator, e.g . a thyristor controller, ca n be mo re c ompa ct a nd less expensive, as it onlyneeds to c ontrol low pow er.

- Loa d fluctua tions on the supply network, as a result of power consumption in bursts ca used b ythe s w itching controller, a re s imila rly reduc ed .

- If the c ontroller fa ils, the proce ss ca n still be o perated with the ba se loa d c omponent of the totalpower.

Furnace

R1

K1

N

L1

x

t

100 % powe r

with ba se load

without ba se loa d

R2





Aga inst the adva ntage s s hown, there a re a lso a number of disa dvanta ges :

- The dyna mic c ontrol a ction is impa ired, es pec ia lly with reg a rd to disturba nces . As the co ntrollernow no longe r provides the full ma nipula ting va ria ble, the c oo ling curve, for insta nce, is not o nlyshifted by the amount of the base load setting, but is also clearly flatter (see Fig. 65). If, for anyrea so n, the pow er req uirement sudd enly bec omes les s tha n the ba se load se tting, the co ntrolleris helpless in this situation, as it cannot reduce the manipulating variable below the value of thebase load .

- In ad dition, the ba se load s etting must also be ma tched to the setpoint. If the setpoint ischa nged dow nwa rds , for instanc e, the excess powe r could s uddenly be too la rge; w ith anupward cha nge, the excess powe r co uld be too s mall. In such ca ses , the bas e loa d s hould becha nged a t the sa me time a s the setpoint.



103



6.2 Power switching

If a proc es s is being ope ra ted w ith different s etpoints or wo rking p oints, it is be tter to sw itch theapplied power rather than use a base load. In an electrically heated furnace, this can be achievedby sw itching pa rt of the hea ting elements through a limit value s w itch (limit c ompa ra tor as a n a d-

vance contact), in order to facilitate furnace operation at full power in the upper temperature range(see Fig. 66).

This gives the follow ing a dva ntag es :

- At any one time, the proc ess ca n be operated in the upper third of the cha rac teristic valid a t thattime (se e Fig. 66). In this wa y, the exc es s pow er at s ma ll values of the proc es s varia ble ca n beminimized.

- The dynamic c ontrol a ction is rather better here w hen compared with the ba se load method, asin this ca se the c ontrol powe r ca n be reduc ed to z ero (a fter falling b elow the c hang eover point).

There c a n be a disa dva ntag e w ith this c ircuit if it operate s w ith a s etpoint close to the c hang eoverpoint, as the proc es s ha s tw o d ifferent values of proc es s g a in here.

Fig. 66: Power switching





6.3 Switched disturbance correction

The effec t of a d is turba nce ca n often be predic ted w ithin ce rtain limits . For exa mple, o pening a fur-na ce do or lea ds to a fall in temperature of 30° C. Inste a d of first w a iting for the proc es s to res pondto this disturbanc e a nd then for the co ntroller to ta ke c orrec tive a ction, the disturbanc e c a n be re-

spond ed to direc tly. To d o this, the furnac e do or is fitted w ith a po s ition sw itch that increa se s thema nipula ting va ria ble (e.g . hea ting po we r) by s everal perce nt when the furna ce do or opens .

This principle is known a s sw itched disturba nce co rrec tion. It is useful when the ca use a nd e ffec tof a disturba nce a re known, a nd w here the disturba nce oc curs freq uently a nd reprod ucibly. Thedisturbance is quickly compensated by the rapid response made possible without time delaysca used by the c ontroller and proc ess .

We w ill now loo k at three different po ss ibilities of s w itched disturba nce correc tion:

Maintaining the disturbance constant

The e ffec t o f the d isturba nce on the proc es s varia ble is eliminated by ma intaining the d isturbanc eco nsta nt by mea ns o f an a uxilia ry control loo p (se e Fig 67 a). Mainta ining the disturba nce co nsta nt

should only be used when suitable technology is available to measure disturbances and maintainthem cons tant.

An example of this is the temperature control of a gas-fired annealing furnace. Here, the main dis-turba nce, ga s pres sure, c a n be ma intained co nsta nt by a n in-line press ure c ontroller, w hich a t thesa me time ca n a lso reduc e the higher supply pres sure to the lowe r burner pres sure. The b loc k dia -gra m of this me thod c a n be a pplied to o ur ow n example:The c ontroller has the job of bringing the proc es s va ria ble x of the proc es s (the tem pera ture o f theannealing furnace) to the setpoint w, by giving out the manipulating variable y. If the disturbance z(the ga s pres sure) is not ma intained co nsta nt, then, when the ga s p res sure fluctuate s, the co ntrol-ler has to cha nge its output repea ted ly, if it is to ho ld the s a me s etpo int. The a uxilia ry co ntroller (thepres sure controller) now ma intains a co nsta nt ga s pres sure, so that this disturba nce no longer in-

fluences the a nnealing furnac e.



105



Fig. 67: Forms of switched disturbance correction

Additive/multiplicative switched disturbance correction

With both these methods, when the disturbance changes, the manipulating variable y of the con-troller is manipulated to counteract the effect of the disturbance (see Fig. 67 b, c).

With additive switched disturbance correction (Fig. 67 b) the manipulating variable (y) is in-crea se d b y a n amo unt propo rtiona l to the disturba nce. In other wo rds , this type of sw itched distur-bance correction takes into account any offset shifts in the process. Controllers that allow such aswitched disturbance correction to be implemented (compact controllers), normally provide an in-put for the switching signal. A signal proportional to the disturbance is applied to the controller in-put, w hich influences the ma nipula ting va ria ble in a cc orda nce w ith the s etting. To illustra te this, w e

can take the example above where the furnace door is opened. When the door is opened, the ma-nipula ting varia ble is increa se d by a fixed a mount.

a) Maintaining the disturbance constant

b) Additive switched disturbance correction

Controller Process

Auxiliarycontroller

yw

x

z

Controller Processy

w

x

z

zy

yz

c) Multiplicative switched disturbance correction

Controller Process

w

x

z

z

0—100%

KP





A multiplicative switched disturbance correction exerts an influence on the controller gain KP . Asthe measured disturbance changes its value, so the value of KP se t a t the c ontroller is cha nged inthe s a me ratio, in the ra nge 0 — 100% (se e Fig. 67 c). This method is suita ble for use in proc es se swhere the ma nipula ting s igna l (co ntroller output) must b e c hang ed to the s a me extent a s a ny dis-turba nce w hich ma y oc cur.

Fig. 68: Neutralization plant

As a n exa mple, a neutra liza tion plant c ould be q uoted, in which a lkaline w a ste wa ter is neutra lizedw ith a c id (see Fig. 68). The proces s va ria ble is the pH va lue, w hich sho uld b e in the neutra l rang e.The c ontroller exerts a n influence o n the pH va lue b y c ha nging the inflow of a cid (y). Firs t o f a ll, letus consider how the plant operates without multiplicative switched disturbance correction. As-sume that the c ontroller has sta bilized a t a defined flow ra te w ith, sa y, 30% ma nipula ting va ria ble.Now, the d isturbanc e (flow ) cha nges , a nd the q uantity of wa ste wa ter per unit time is no w twice a sla rge. The pH va lue w ill now increa se, a nd t he c ontroller w ill increa se its ma nipula ting va ria ble untilthe proces s varia ble rea ches the s etpoint a ga in. This will be the c a se with 60% manipula ting vari-a ble (do uble the q uantity of a cid). We c a n s ee that the ma nipula ting varia ble must b e kept propor-tional to the disturbance to maintain the same setpoint, other conditions remaining unchanged.This c a n be a chieved by me a suring the d is turba nce (flow ) a nd a pplying multiplic a tive sw itching.The d isturba nce is sc a led a t the c ontroller over the ra nge from zero to the ma ximum disturbanc evalue which could occur; the controller now changes its proportional action to the same extent,over the range 0 — 100%.

If we now look at our example again:

Assume here that the controller has stabilized again with, say, 30% manipulating variable. Now thedisturbance (flow) changes to twice the value. Likewise, through the multiplicative switched distur-

ba nce co rrec tion, the propo rtiona l ga in (that c orres ponds to the overall ga in, se e a lso Fig. 41) is se tto d oub le its va lue. The ma nipula ting va ria ble of the c ontroller immed ia tely inc rea ses to 60 % a ndthere are no larger control deviations.

The e xamples of s witched disturba nce co rrec tion s how n here a pply to disco ntinuous c ontrollerswith 2-sta te a ction a nd c ontinuous c ontrollers. The rela tions hips for 3-sta te, mo dula ting a nd a ctu-a ting co ntrollers a re mo re c omplex, a nd w ill not b e d isc uss ed here.



107



6.4 Switched auxiliary process variable correction

Where a disturba nce ca nnot be mea sured or loc a lized , it is pos sible to derive a n a uxilia ry proc es svariable Xa ux from the proces s, w here Xa ux ha s a sho rter time de la y than the ma in proc es s va ria blex, a nd a pply it to the c ontroller input, a fter suita ble conversion (see Fig . 69). In this w a y, the distur-

ba nces a t the proce ss input (e.g . s upply d isturbanc es ) a re q uickly reported to the c ontroller.How ever, Xa ux is norma lly a pplied through a n a da ptive timing element, s o tha t the proce ss varia bleis not distorted under stabilized conditions. With this arrangement, two control loops, each with itsow n complete signal path, a re c oupled to ge ther. It should b e noted that the c ontrol loo p ca n poss i-bly beco me unsta ble a s a res ult of overly strong s witching o f the a uxilia ry proc es s va ria ble and a nunsuita ble c ontroller se tting.

Fig. 69: Switched auxiliary process variable correction

6.5 Coarse/fine control

Two co ntrol loop s in se ries a re use d to ma intain some pa ra meter of a ma ss flow or energy flow cons ta nt. The residua l de via tion from the firs t c ontroller, the c oa rse c ontroller (C 1), is correc ted bythe s ec ond , fine c ontroller (C2) – see Fig. 70.

Fig. 70: Coarse/fine control





Here again we can use as an example a pH control system for neutralizing industrial waste water.B ec a use of the large varia tions in inflow norma lly present, a nd the c hang ing co mpos ition, it is oftena ppropria te to co nnect tw o c ontrol loo ps in s eries , s o tha t the va ria tions in pH value a re ma intainedwithin the permissible tolerances.

6.6 Cascade control

Ca sca de control c a n signific a ntly improve the control q ua lity. This a pplies in pa rticula r to the d y-na mic a ction of the c ontrol loop , in other wo rds , the trans ition o f the proc es s varia ble follow ing s et-point changes or disturbances. Processes with a Tg /Tu ratio less than 2 or 3 can be difficult tocontrol with a simple control system; because of the relatively long delay time, the controller doesnot b ec ome a w a re o f how it s hould respo nd until a very la te s ta ge . We therefore try to s plit the c on-trol loop into several partial loops (usually two), which are controlled separately. Control of thesepa rtia l loo ps is much ea sier, a s ea ch ha s only a fra ction of the overall delay time. This a rra ngeme ntis also known as multi-loop or networked control.

Fig. 71 shows the block dia gram for ca sc a de control.

Fig. 71: Cascade control

An a uxilia ry proc es s va ria ble xa ux is d erived from the proc es s a nd a pplied to the input of a n auxilia -

ry co ntroller, the output o f w hich c ontrols the m a nipula ting va ria ble y. The s etpo int w 1 of the auxil-iary controller is determined by the manipulating variable of the main controller, such that the pro-ce ss varia ble rea ches the s et va lue. The a uxilia ry c ontrol loo p c a n be se t to res pond more ra pidly,a nd q uickly eliminate s a ll disturba nces a t the input to the proce ss .

The subo rdinate a uxilia ry controller is c ons truc ted in the sa me w a y a s a n ordina ry controller. How -ever, it must have an input for an electrical setpoint signal, as its setpoint is set by the supervisoryco ntroller. In other res pec ts, it must b e ma tched to the dema nds of its duty, w ith reg a rd to input,output e tc . The a uxilia ry controller has the job of c ha nging the a uxilia ry proces s va ria ble veryq uickly, in proportion to the ma nipula ting va ria ble of the m a in controller; henc e P or P D c ontrollersa re no rmally used for this a pplica tion, or a lso, les s freq uently, a P I controller. The ma st er co ntroller,set for setpoint response, is usually a PI or PID controller.



109



For ca sc a de c ontrol, it is important tha t the subordinate loop is a t lea st 2 — 3 times fa ster than theouter loop, as otherwise the overall control loop will tend to oscillate. One advantage of cascadeco ntrol is that the dyna mic res pons e o f the co ntrol loo p is much improved. Another ad vanta ge isthat the c ontrollers a re muc h ea sier to a djust. The ma ste r controller is sw itched to ma nual mode,a nd the slave controller is o ptimized . Then the ma s ter controller is o ptimized , w ith the s la ve c on-

troller kept in auto ma tic mod e.

An example of ca sc a de c ontrol is the tempera ture c ontrol of a furnac e hea ted b y a ga s b urner (se eFig. 72).

Fig. 72: Cascade temperature control for a burner

The ma st er co ntroller outputs a ma nipula ting va ria ble y1 in the rang e 0 — 100%, on the ba sis of

the c ontrol differenc e a pplied to it. The s la ve c ontroller now rec eives this ma nipula ting va ria ble a sits se tpoint, but o nly a fter the s igna l is normalized : on the ba sis of the normaliza tion, the s etpoint ofthe s la ve c ont roller (w 1) a mounts to 0 — “ma ximum ga s flow ”, co rresponding to 0 — 100% manip-

ula ting va ria ble o f the m a s ter controller. With its ma nipula ting va ria ble, the ma s ter controller prac ti-ca lly prese ts the d es ired ga s q uantity pe r unit time. The s la ve c ontroller has the job of c ontrollingthe g a s flow a cc ura tely. The s la ve c ontroller now ta kes over part of the timing e lements a nd c or-rec ts disturba nces a t the input to the proces s, for example, fluctua tions in the g a s pres sure. Theco ntrol a ction is improved o n this b a sis, a nd, in certain ca se s, proc es se s c a n only be c ontrolled b yintroducing cascade control.





6.7 Ratio control

Ra tio controllers a re us ed in b urner co ntrols (co ntrol of the ga s/a ir mixture ra tio), a na lytica l tech-niq ues (mixing of rea gents ) a nd in proc es s eng ineering (prepa ra tion of m ixtures ). Thes e controllersha ve tw o proc es s va lue inputs. The ratio of the two input varia bles is the real proc es s va ria ble. The

va lue req uired for this ra tio is s et a s the s etpo int, d irec tly a t the c ontroller.A ratio co ntroller is freq uently use d a s a slave controller. Here, the c ontroller has the ta sk of c ontrol-ling the q uantities o f two s ubs ta nces in such a wa y that the mixing ratio s ta ys c ons ta nt when differ-ing total quantities of the mixture are required. With this kind of slave control, there are two set-points: the mixing ratio and the total quantity. Accordingly, two controllers are used, one of whichcontrols the to ta l q uantity o f the mixture p er unit time, w hils t the other influenc es the mixing ratio,by a djusting the do sa ge of the s epa ra te c ompo nents. As the tota l q uantity per unit time is the ulti-mately decisive setpoint, this controller is designated as master controller, whilst the subordinatecontroller controls the substance mixing ratio to meet the requirements of the master controller.

Fig. 73: Ratio control

An example of this is the mixture control shown in Fig. 73: two substances have to be mixed in afixed ra tio to ea ch o ther, w hilst the de ma nd for the q uantity of the mixture fluctua tes a cc ording toproduc tion req uireme nts. Tw o c ontrol c ircuits a re req uired for this , one to control the to ta l q uantityof both substances after mixing, the other to control the mixing ratio. In controlling the total quanti-ty, it is sufficient to influence only one component, since the other is made to follow according tothe set ratio. However, the mixing ratio is controlled independently of the master controller, so thatthe master controller and its associated valve have been fitted purely to control the air flow andhence the total quantity. Without the master controller, only the mixing ratio remains constant,wherea s the tota l q uantity of the two s ubsta nces is d isrega rded .A ratio controller is a standard controller whose input stage has two inputs to suit this modifiedspecification. With regard to the dynamic action, all the variations of the standard controller could



111



co nceiva bly be used . B ec a use o f the na ture of the proc es s, the c ontrollers a re usua lly co ntinuous,modulating or actuating controllers, with PI or PID action. With microprocessor controllers, func-tions such as ratio control can usually be configured directly.

6.8 Multi-component controlIn a multi-component control system, various process-dependent variables produce the processva lue for the co ntroller and de termine the c ontrol de via tion, a s in a s tea m/feed w a ter co ntrol, for in-s ta nce (se e Fig . 74).

Fig. 74: Multi-component control

In this case, the individual process values can each be allocated a different weighting factor, sothat they affect the control deviation to different extents; the main process variable is normally allo-ca ted the highes t weighting fa cto r.

In the exa mple given, the follow ing relationship might a pply:

x3 = steam flow

x2 = feedwa ter flow

x1 = wa ter level

If we now co nsider the rela tions hip in the a bo ve formula without the c onsta nts, w e ha ve:

x x1 2b x2 2a x3 c+( )–( )+=

x x1 x2 x3–( )+=





We ca n see tha t when the stea m draw -off x3 is e q ual to the c old w a ter supplied x2, the expres sionin bra ckets b ec omes zero, and the proc es s va lue x now d epend s o nly on the wa ter level x1. Such amulti-component controller must have suitable number of inputs, and, if necessary, a facility forcombining the individual signals via certain computations. In the example shown, there are threeco mponents, henc e this c ircuit has bee n given the na me “3-compo nent control”.




7 Special controller functions

This c ha pter introd uce s a numbe r of a dd itiona l functions, i.e prope rties which a re s pec ific to pa r-ticula r controllers a nd a re req uired for certa in co ntrol tas ks. With mod ern microproc es sor-ba sed in-struments, such options c a n be implemented us ing s uita ble softwa re.

7.1 Control station / manual modeWith a control or auto-ma nua l s ta tion, the ma nipula ting va ria ble ca n be a ltered d irec tly. The c ontrolloo p is interrupted so tha t the proc es s va ria ble has no s ignific a nce . The co ntroller now a c ts purelya s a n ope ra ting device (se e Fig. 75).

Fig. 75: Use of a control station

This feature is use ful whe re va lves or a ctua tors ha ve to be fully op ened or closed for clea ning, a sthis would not otherwise be possible. Control stations are also invaluable for trials and test runs,whe n the ma nipula ting va ria ble has to b e o perated in ma nual mode, i.e. without a utoma tic c ontrol.They c a n be integrated in the co ntroller or a rra nged a s a se pa ra te instrument.

Nowa da ys, with many microproc es so r controllers, the function o f a co ntrol sta tion is provided bythe manual mode setting. If a controller is switched from automatic mode (where the controller istrying to c ontrol the proces s value a t the s etpoint) to ma nual mode, the c ontrol function is disa bled .Now the actuator, the thyristor power regulator, the cooling etc., can be controlled by setting thema nipula ting va ria ble ma nua lly. Manual ope ra tion is po ss ible w ith a ll types of c ontroller.

In manual mode, the operator could set a value for the actuator very different to the value of thecurrent ma nipula ting varia ble se t b y the co ntroller. How ever, s uch a n extreme cha nge co uld have ade structive effec t on the ac tua tor. This problem is o vercome b y the provision of bumpless transferfrom a utoma tic to ma nual mode, where the ma nipula ting varia ble rema ins a t its current va lue, a ndca n then be cha nged manually.

In ca se of a broken probe, c a used, for example, by a ca ble fault or mecha nica l da mag e to the se n-so r, a utoma tic co ntrol is no longer pos sible, a nd the co ntroller sw itches off the e nergy supply forsa fety rea so ns. In certain proce ss es this co uld d estroy the ba tch or ca use a lengthy los s of pro-duction, owing to the long warm-up times or similar such conditions. Here, controllers offeringmanual operation have the a dvanta ge that, in such ca ses , the proce ss ca n still be o pera ted ma nu-a lly, a lbeit with reduc ed a cc ura cy, a nd b rought to co mpletion.





7.2 Ramp function

Ra mp functions a re often used in proc es se s w ith delica te prod ucts o r to protec t heating e lements,as they ensure that the process variable approaches the setpoint slowly. In processes with largeexce ss pow er, it a lso ma kes se nse to limit the rate of rise of the proces s varia ble. With this func-

tion, the us er sets a ma ximum ra te o f cha nge (e.g . in ° C/min, ° C/hr etc .) a t w hich a ra mp s etpointtra vels tow a rds the ma in se tpoint. The proces s varia ble co ntinuously trac ks this cha nging value(see Fig. 76).

Fig. 76: Diagram of the ramp function

When the ramp function is activated, the ramp setpoint is normally made equal to the current pro-cess value, and then altered towards the main setpoint at the set gradient. Once the setpoint isreached, the ramp function is terminated, and the instrument controls at the set value until, for in-stance, the main setpoint is changed. If it does change, the newly activated value will once againbe a pproa ched by a ra mp. In this w a y, bo th rising a nd falling ramps ca n be implemented.

7.3 Limiting the manipulating variable

A manipulating variable limit can be used to limit the controller output signal at either a maximum

or a minimum value. One application is where the actuating device fitted (e.g. pump, electric heat-ing etc.) is over-sized; it avoids excess power and its associated problems, such as the processvariable overshooting the setpoint. Further, a minimum manipulating variable limit can be a wiseprecaution in the control of gas burners, for instance. Setting a minimum manipulating variable(e.g . 3%) a voids the g a s s upply be ing interrupted a nd thus the burner going out. The c ontrollerthen only gives out a manipulating variable within the range of the set minimum or maximum val-ues.

In a c ontinuous co ntroller with a 0 — 10V output signa l a nd a 5 — 95% ra nge limita tion, the outputsigna l is restricte d to the rang e 0.5 — 9.5V.



115



7.4 Program controller

S o far, we ha ve alwa ys a ss umed tha t the proc ess varia ble has to be ma intained co nstant a t a fixedse tpoint value. S uch c ontrols a re a lso ca lled fixed s etpoint co ntrols. In co mpa riso n, there a re a lso anumber of manufacturing processes where the setpoint does not represent a fixed value for the

co ntrol sys tem. Instea d it repres ents a pa ra meter which varies w ith time, i.e. a spe cific profile forthe variation of the setpoint with time is required. Controllers used for this application are calledprogra m or profile c ontrollers. They a re often enco untered a s te mpera ture c ontrols, e.g . in annea l-ing furnac es for meta llic ma teria ls , in ce ramic kilns, in chemica l eng ineering for the ba tch manufac -ture o f prod ucts , in clima tic cha mbers e tc.

The s electa ble prog ram time is no rmally of the o rde r of minutes , hours or even d a ys . Fig . 77 show ssuc h a p rofile for the c ourse of the se tpoint. Co nditioned by the proces s, the c ourse of the proc es svaria ble doe s not normally sho w the sa me s harp tra nsitions a s the se tpoint profile.

Fig. 77: Diagram of a setpoint profile

The s etpo int profiles a re usua lly predete rmined by a n externa l prog ramming d evice o r a co mputer;they a re fed into the c ontroller via the interfac e o r a se pa ra te s etpoint input. As the proces s varia blehas to follow the c ontinuously va rying se tpoint, the a rra ngeme nt is a lso ca lled follow er co ntrol.

The rise of the se tpoint profile must no t exce ed the rise of the proc es s c ha rac teris tic , otherw ise theprocess variable no longer follows the program profile, but rises at the maximum rate set by theproce ss c ha ra c teris tic (see Fig. 77). This is pa rticularly impo rta nt in non-linea r proc es ses suc h a stemperature c ontrols, w here the proc es s g a in dec rea se s w ith higher proc es s va lues. This c a nsometimes be avoided by programming the instrument so that the program run is terminated asso on a s a ce rta in devia tion oc curs, i.e. the proc es s va ria ble ca n no long er follow the prog ra m.

w /x

Rise too steep

Setpoint

P roces s va lue200 ° C

150 ° C

100 ° C

50 ° C

2 h 4 h 6 h 8 h 10 h

t





7.5 Self-optimization

Optimum adjustment of a controller to the process can be a time-consuming affair, particularly ifthe process concerned is rather slow. Furthermore, as we saw in Chapter 2.8, the optimal valuesfor XP , Td and Tn a re d epende nt on the w orking po int. S o it is q uite likely tha t s everal sets of c on-

trol parameters w ill have to be found for one proce ss .It se ems a n ob vious ste p to let the co ntroller itself make automa tic a djustments to the c ontrol pa-rame ters. With microproc es sor c ontrollers, the fac ility fo r automa tic controller s etting (self-optimi-za tion) is a va ila ble.

Basically, the choice lies between controllers which determine the control parameters simply onthe basis of user requirements, and adaptive controllers, where the settings are continuallychecked and changed .

The me thod s used fo r self-optimiza tion a re us ua lly b uilt into the controller as softw a re functiona lblocks, and identify the process on the basis of the response of the process variable to stepcha nges in ma nipula ting varia ble, a nd from this dete rmine the be st co ntrol parame ters. The va lues

de termined for XP , Td , Tn etc. can be refined by the user (see Fig. 78).

Fig. 78: Operating principle of self-optimization

In principle, a se lf-optimiza tion c a n be a rra nged , for insta nce, to ta ke plac e a bo ut the se tpoint: ifthe c ontroller has sta bilized the proces s varia ble a t the s etpoint, the s elf-optimiza tion c a n then bes ta rted , a nd the controller outputs 100% and 0% ma nipula ting va ria ble a lterna tely. The c ontrollerdetermines the best parameters by examining the oscillations of the process variable about thesetpoint, and then automatically accepts the parameters. With this method, it should be ensuredthat no dama ge c a n be ca used w hen the proc ess varia ble exceeds the setpoint.



117



Fig . 79 sho ws a nother type of s elf-optimiza tion: in this ca se , the s elf-optimiza tion proce ss is sta rt-ed w hen the proce ss va lue is b elow the s etpo int. The controller dete rmines t he de la y time from theinitia l res pons e o f the proc es s va ria ble. The c ontroller ca lcula tes a sw itching level ba sed jointly onthe de la y time a nd the g ra dient of the res pons e o f the proc es s va ria ble, and if this level is e xcee d-ed it sets the manipulating variable to 0%. With a linear process, interposing the switching level

prevents the process variable from overshooting the setpoint. With non-linear processes, the over-sho ot is no t c ompletely prevented, but is a t lea st reduc ed . In all, the c ontroller outputs 100% ma-nipulating variable twice, interrupted by a 0% manipulating variable output. Afterwards, the con-troller ac ce pts the optimized pa ra meters a nd co ntrols a cc ura tely a t the setpoint.

Fig. 79: Fluctuation of the process variable about the switching lineManufa cturers normally a ss ume a proc es s with self-limita tion a nd w ithout dea d time elements a sthe ba sis for determining the co ntrol parame ters. The c los er the ac tual proc es s co rres ponds to thismod el, the more effective is the s elf-optimiza tion.





7.6 Parameter/structure switching

By switching between parameter sets, it is possible to operate the controller in a control processwhich requires different settings as the conditions vary. A set may comprise various parameters,such a s dyna mic respons e, XP ba nds, c ycle time C y etc. S ome c ontrollers even a llow sw itching b e-

twe en c omplete c ontroller structures , s uch a s sw itching from P D to P ID structure.Switching between sets is initiated either via the controller’s logic inputs or via programmed valueswhich depend on the control deviation or various setpoint ranges (see Fig. 80).

Fig. 80: Schematic diagram of parameter set switching

P a ra meter set s witching o f this type finds a pplica tion, for insta nce, w hen there a re repea ted s ta rt-ups d uring a utoma tic operation, where the s etpoint must be rea ched in minimum time, a nd w herethe proce ss va ria ble x must not overshoo t the se tpo int. This is not norma lly pos s ible, how ever, withthe pa ra meters used for sta ble operation. As a n exa mple, d epend ing o n the co ntrol devia tion, theco ntroller sw itches from P a ction with a s omew hat s ma ller XP ba nd to P I a ction before the setpo intis reached. Another application where switching can help is when running different charges in anindustria l furnac e (a nnealing furna ce ). The c ontrol ac tion o f the proces s cha nges a cc ording to theloa ding (ha lf cha rge, full cha rge), a nd the c ontroller must a da pt to ea ch individua l c a se. Va rious pa -rameter sets can be allocated, based on certain preliminary tests. Moreover, this special functioncan be used to advantage where different working points are operated over the full range of thecharacteristic, and where a non-linear process is concerned, in which variable margins of excess

pow er a re to be expec ted.

7.7 Fuzzy logic

Fuzzy log ic w a s de velope d in the US A by Lotfi A. Zade h in 1965. This ma thema tica l metho d isba se d o n the fuzzy set theo ry, which represents a n extension of B oo lea n algeb ra . This te chnologyhas undergone continuous development, and its a dvantag es were q uickly reco gnized in J apa n,w here they be ga n to a pply the tec hnolog y to a multitude o f control tas ks (fuzzy co ntrol). This a ppli-ca tion o f fuzzy logic to a utoma tion a nd c ontrol engineering repres ents a log ica l extension of tra di-tional control technology.



119



A detailed coverage of this technology is outside the scope of this book; only its application incombination with a PID controller will be illustrated. It should be mentioned briefly that fuzzy logicde a ls w ith the subjec tive uncertainty of expres sions, s uch a s “ temperature too high” , “press ure toolow ” or “humidity too low” , a s we ll as dec ision-ma king ba se d on s uch linguistic va ria bles . In thiswa y, it rec ons tructs the imprec ise co ncepts of human thought.

The follow ing illustra tes how fuzzy log ic a nd a P ID control a lgorithm w ork tog ether to improve thedisturbance and setpoint responses of a control loop. Fig. 81 shows the simplified block diagramof a P ID controller co mbined w ith a fuzzy module. The input va ria bles o f the fuzzy c ontroller a re theco ntrol devia tion a nd the time d erivative o f the proc es s varia ble, a s we ll a s informa tion o n w hetherthe c ontroller should operate fo r se tpo int or dis turba nce res pons e. The output ma nipula ting va ri-a ble of the fuzzy c ontroller is we ighted by a pa ra meter Fc1 a nd a dd ed to the ma nipula ting va ria bleof the PID controller. In this way, the manipulating variable acting on the process is made up of themanipulating variable of the PID controller and that of the fuzzy controller. A second output fromthe fuzzy controller is a control output which influences the PID control parameters according topa ra meter Fc2. Thus, w ith just tw o pa ra meters, Fc1 a nd Fc2, the fuzzy controller ca n be ma tchedto the proc ess .

Fig. 81: Simplified block diagram of a PID controller with fuzzy module

The fuzzy cont roller incorpo rate s ling uis tic control bloc ks ba sed on “ IF-THEN” rules . Thes e linguis -tic rules determine the transient response of the fuzzy controller to setpoint changes and distur-

bances.

The c omb ina tion o f a fuzz y co ntroller in pa rallel with a co nventiona l co ntroller offers se veral ad va n-

tages :

- With a non-linea r proc es s b eha vior or on higher-order proc es se s, the fuzzy controller ca n com-pensate for specific inaccuracies of the PID controller by supportive intervention under criticaloperating co nditions.

This is exemp lified b y sys tems w hos e be ha vior cha nge s o ver the ope rating time. The fuzzy co ntrol-ler is les s se nsitive to p roc es s pa ra meter cha nges than the c ontroller with its fixed pa ra meter set-tings . This mea ns tha t varia ble proc es s cha ra cte ristics a re d ea lt with more effectively by the co m-bination of fuzzy logic a nd the P ID c ontroller.





- For se tpoint res pons e, the integral fuzzy co ntrollers increas e the damping fa cto r of the co ntrolloo p in ce rtain co nditions . This m ea ns tha t a ny tend enc ies to o sc illa te d uring the run-up to thesetpoint are reduced a nd overshoots a re d a mped c onsiderably.

- The problem o f optimizing the co ntroller spec ifica lly for setpoint or disturbanc e respo nse isles se ned b y the fuzzy c ompo nents, a s fuzzy c ontrol comb ines the co ntrol theory of bo th optimi-za tion c riteria .

- When disturbanc es o cc ur, the fuzzy controller res ponds more dynamica lly than the PID co ntrol-ler and henc e s ta bilizes the d isturba nce more rapidly. The c riterion o f co ntrol effec tivenes s isimproved, i.e. the res ulting c ontrol error a rea is sma ller than w ith s ta biliza tion b y the P ID control-ler a lone (see Fig . 82).

Fig. 82: Disturbance response of a third-order process, using a controller with and without

fuzzy module

We c a n s ummarize b y s a ying tha t fuzzy log ic, in its a pplica tion a s fuzzy c ontrol in a utoma tion a ndcontrol engineering, is highly regarded today, despite initial problems in accepting this technology.The fuzzy c onc ept ha s p roved to b e a pow erful too l, pa rticula rly in co mplex control tas ks. Trad i-tiona l control technolog y w ill ce rta inly not be repla ce d by fuzzy logic, b ut technologica l proc es se swill emerge that ca n be co ntrolled more ec onomica lly a nd mo re s a fely using fuzzy log ic. The c om-bination of the fuzzy c ontroller with the P ID controller is a sens ible exa mple of this .

From this we ca n rec ord that the fuzzy co ntroller in ge nera l, a s we ll a s the c omb ination of a tra di-tional conventional controller and fuzzy module, bring about increased flexibility, so that the con-

troller ca n be b etter matc hed to different proc es se s a nd c ontrol objec tives.




8 Standards, symbols, literature references

Standards

S ta nda rds a nd g uide lines a re very important in engineering, including co ntrol engineering. They laydow n co ncepts a nd d esigna tions, compo nent va lues, d imensions, a s well a s important numerica lvalues a nd numerica l ra nges .

From the large number of published standards and guidelines, the following list below has beenlimited to those which are useful for the basic principles of control engineering and have been usedin the preparation of this book.

Symbols

DIN 19 221 1993-05 Me ss en, S te ue rn, Reg eln; Fo rme lze ic hen d erReg elungs- und S teuerungste chnik(Measurement, operation, control; symbols in control engineering)

DIN 19 225 1981-12 Messen, S t euern, Regeln; Bennenung und Einte ilung von Reg le rn(Measurement, operation, control; control engineering concepts)

DIN 19 226 (1-5) Regelungs- und S teuerungs technik;Begriffe und allgemeine Grundlagen

(Cont rol Enginee ring ; Definitions a nd Terms )DIN 19 236 1977-01 Me ss en, S t eue rn, Re ge ln; Optimie rung , B e g riffe

(Measurement, o pera tion, control; o ptimiza tion, de finitions )

G uide line VDI/VDE 2189Sheet 1

B es chreibung und Untersuc hung vonZwei- und Mehrpunktreglern ohne Rückführung(Des cription a nd a nalysis of tw o-sta te a nd multi-sta teco ntrollers without feedb a ck)


B es chreibung und Untersuc hung vonZwei- und Dreipunktreg lern mit Rückführung

(Des cription a nd a nalysis o f two-sta te a nd three-sta teco ntrollers with feedb a ck)


Beschreibung und Untersuchung von Dreipunkt-Schrittreglern(Description and analysis of modulating controllers)


B es chreibung und Untersuc hung digita l a rbeitenderKompaktregler(Des cription a nd a nalysis of d igital compa ct co ntrollers)


B es chreibung und Untersuchung s tetige r Reg el-geräte G rundlag en(Des cription a nd a nalysis of c ontinuous c ontrolinstrument funda menta ls)

C y cyc le time,quasi-continuous controller

Ton switch-on time,quasi-continuous controller

e devia tion Tosc os c illa tion time,discontinuous controller

fsw sw itching freq uency,discontinuous controller

Ts s ta biliza tion time,quasi-continuous controller

KI inte gra l c oe ffic ie nt o f the co ntro lle r Tt dea d time



8 Standards, symbols, literature references


References

- E. SAMAL /W. B ECKER ‘G rundriss der P ra ktisc hen Rege lungs technik 18’Basics of practical control engineering 18 (Oldenburg)

- LENZ /OBERS T /KOEG S T ‘G rundla ge n der Steuerungs - und Reg elungste chnik’Funda menta ls of control engineering (Hüthig Verla g )

- P. BUS CH ‘Elementare Regelungstechnik’

Elementa ry co ntrol engineering (Vog el Verla g )- B ÖTTLE /B OY /G ROTHUS MANN ‘Elektrisc he Mess - und Reg eltechnik’

Elec tric a l mea surement a nd control engineering (Vog el Verla g)

- D. WEBER ‘Reg elungste chnik- Wirkunswe ise und Einsa tz elektronisc her Regler’Control engineering - ope ra tion a nd a pplica tion of e lec tronic c ontrollers (Expert Verla g)

- D. WEBER ‘Elektronisc he Reg ler- G rundlag en, B a uformen und Einst ellkriterien’Elec tronic co ntrollers - funda menta ls , types a nd a djus tment c riteria (J UMO)

- J UMO G mbH &Co. KG ‘Betriebs a nleitung- Universeller Progra mmreglerJ UMO DICON 401/501’Operat ing Ins tructions - universa l profile c on troller J UMO DICON 401/501

KIS tra nsfer co efficient of proce ssw ithout s elf-limita tion

TI integral time

KS transfer coefficient (process gain)of the proc ess

Ty stroke time

KP proport ionality fac tor o f the contro ller Vma x ma ximum rate of rise

R ON-time ra tio ,quasi-continuous controller

w setpoint (S P )

R(%) ON-time ra tio,q uasi-continuous controller in%

x pro c es s va ria b le , p ro c es s va lue (P V)

t time (continuous) XMa x ma ximum proc es s value (proc es s varia ble)

T time cons ta nt of firs t-order process Xo overshoot

Ta a pproa ch time XP proportional band of the controller

TC os c illa tion time a t XP c XP c critical XP , a t w hich the co ntrol loopw ith P controller osc illa tes uniformly

Td deriva tive time of the controller XS d switching differential,discontinuous controller

Tg response time XS h contact s pac ing

TMmin minimum pulse duration,mod ula ting a nd a ctua ting co ntrollers

y ma nip ula ting va ria b le (MV)

Tn reset time of the controller yH manipulation range

Toff sw itch-off time,

quasi-continuous controller

z dis turba nce

Tu delay time



123

Index

A

a ctua ting co ntroller 98

actuator retransmission signal 98

a ctua tor stroke time 95–96, 98

actuators 24– 25

a nalog a nd digita l controllers 18

approach time 11

auxiliary control loop 104

auxiliary controller 108

auxiliary process variable 107

B

bas e load 101

ba se loa d se tting 101

behavior- dynamic 63

- static 63

C

ca sca de c ontrol 108

characteristic 47

- falling 48

- rising 48

Chien, Hrones and Reswick 69

co a rse co ntroller 107

coa rse/fine c ontrol 107

co nstant devia tion 57

cons truc tion o f co ntrollers 12

contact spa cing 91, 96

continuous controller 12, 45

control deviation 9control differenc e 9control error area 71

- linear 71

control loop 10

- close d 9–10- multi-loop 101

- single loo p 101

control qua lity 101, 108

co ntrol sta tion 113

controller- co ntinuous 12, 45

- dis co ntinuous 12, 79, 90

- dynamic action 89, 94

- sw itching 12

controller ad justme nt

- ac co rding to the ra te of rise 75- ac co rding to the trans fer function 72



Index

124

- by the oscillation method 71

controller g a in 71

controller output 10

controller setting 66

controller with dynamic action 85

co urse co ntrol 28cycle time 87 –88

D

D component 60

delay time 41

derivative time 57

digita l and a nalog co ntrollers 18

discontinuous controller 12

disturbance 10, 104

disturba nce respons e 63, 65drive control 27

duty-cycle 86

dynamic a ction 52

dyna mic c harac teristic 31

dynamic respons e 68

F

fine controller 107

fluctua tion ba nd 82, 84

fuzzy c ontroller 119

fuzzy log ic 118

fuzzy module 119

G

gain 45

H

hysteresis 80, 91

I

I controller 53, 62

I proc ess es 33

inflection tangent 41, 74

M

manipulating devices 23ma nipula ting va ria ble 10, 45



125

Index

manipulation range 10

manual mode 113

master controller 108

modulating controller 95

multi-sta te co ntroller 79

O

ON-time ratio 86, 88

ON-time, rela tive 86

operation 26

oscillation method 71

overshoot 11, 67

P

PD controller 57

period of o scilla tion 82, 84

P I controller 54, 60

PID action 62

P ID c ontroller 61–62

point of inflection 74

pos ition co ntrol 28

powe r control 87

pow er sw itching 103

proces s ga in 30

process step response 72

proces s tra nsfer coe fficient 73

proc es s va lue (proc es s va ria ble) 9proces s w ith s elf-limita tion 31

processes 43

- first-order 39

- higher-order 41

- pure d ea d time 35

- sec ond-order 40

- stab le 54

- unstab le 50

- with and w ithout dea d time 35

- with and without self-limitation 34- with de la y 38

- with one d ela y 38

- with two delays 39

- w ithout s elf-limitat ion 33, 85

proportional band 47

proportionality factor 46, 48

P T1 proc es s 39

P T2 proc es s 40



Index

126

Q

quasi-continuous controllers 13

Rramp function 114

ra te of change 60

rate of rise Vma x 75

ra te of rise vmax 42

ra tio co ntrol 110

reset time 55

res pons e time 41– 42

S

self-optimization 116

setpoint 64

setpo int (de sired va lue) 9setpoint response 63, 65

signa l types 18

signals 18

- 3-state 19

- a nalog 19

- binary 19

- dig ita l 15

slave controller 110

speed control 27

s ta biliza tion time 11

start-up value 43

static characteristic 30

sta tic proce ss cha ra cteristic 50

step respo nse 16, 31–32, 41

switched disturbance correction 104

- a dd itive 105

- multiplicative 105

sw itching d ifferential 81–82, 84, 91

sw itching freq uency 82, 86

T

three-state controller 12, 25, 79, 91

tolerance limits 11

transfer coefficient 30, 33, 43

transfer function 38, 72

Tt proc es se s 35



127

Index

W

w orking point 43, 49

X Xp ba nd 47

- asymmetrical 49

- symmetrica l 48

Z

Ziegler and Nichols 69



Index

128





Control Engineering [1].

Documents