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Chapter 2

Controller Architecture

2.1 Introduction

The ideal continuous time domain PID controller for a SISO process is expressedin the Laplace domain as follows:

)s(E)s(G)s(U c= (2.1)

with

)sTsT

11(K)s(G d

icc ++= (2.2)

and with cK = proportional gain, iT = integral time constant and dT =derivative time constant. If =iT and 0Td= (i.e. P control), then it is clearthat the closed loop measured value y will always be less than the desired value r(for processes without an integrator term, as a positive error is necessary to keep

the measured value constant, and less than the desired value). The introduction ofintegral action facilitates the achievement of equality between the measuredvalue and the desired value, as a constant error produces an increasing controlleroutput. The introduction of derivative action means that changes in the desiredvalue may be anticipated, and thus an appropriate correction may be added priorto the actual change. Thus, in simplified terms, the PID controller allowscontributions from present controller inputs, past controller inputs and futurecontroller inputs.

Many variations of the PID controller structure have been proposed (indeed,the PI controller structure is itself a subset of the PID controller structure). AsTan et al. (1999a) suggest, one important reason for the non-standard structuresis due to the transition of the controllers from pneumatic implementation throughelectronic implementation to the present microprocessor implementation. Asubstantial number of the variations in the controller structures used may be

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Chapter 2: Controller Architecture 5

summarised by controller structure supersets. In this book, nine such supersetsare specified, which allows a sensible restriction on the number of tables that

need to be detailed. The controller structures specified are detailed below.

1. The ideal PI controller structure:

+=

sT

11K)s(G

icc (2.3)

2. The ideal PID controller structure:

++= sT

sT

11K)s(G d

icc (2.4)

This controller structure has also been labelled the non-interacting controller(McMillan, 1994), the ISA algorithm (Gerry and Hansen, 1987) or theparallel non-interacting controller (Visioli, 2001). A variation of the

controller is labelled the parallel controller structure (McMillan, 1994).

sTsT

1K)s(G d

icc ++= (2.5)

This variation has also been labelled the ideal parallel, non-interacting,independent or gain independent algorithm.The ideal PID controller structure is used in the following products:

(a) Allen Bradley PLC5 product (McMillan, 1994)(b) Bailey FC19 PID algorithm (EZYtune, 2003)(c) Fanuc Series 9030 and 9070 Independent Form PID algorithm

(EZYtune, 2003)(d) Intellution FIX products (McMillan, 1994)(e) Honeywell TDC3000 Process Manager Type A, non-interactive mode

product (ISMC, 1999)

(f) Leeds and Northrup Electromax 5 product (strm and Hgglund,1988)

(g) Yokogawa Field Control Station (FCS) PID algorithm (EZYtune,2003).

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Handbook of PI and PID Controller Tuning Rules6

3. Ideal controller in series with a first order lag:

1sT1sT

sT11K)s(G

fd

icc

+

++= (2.6)

4. Controller with filtered derivative:

+

++=

NTs1

sT

sT

11K)s(G

d

d

icc

(2.7)

This structure is used in the following products:

(a) Bailey Net 90 PID error input product with N = 10 (McMillan, 1994) andFC156 Independent Form PID algorithm (EZYtune, 2003)

(b) Concept PIDP1 and PID1 PID algorithms (EZYtune, 2003)(c) Fischer and Porter DCU 3200 CON PID algorithm with N = 8 (EZYtune,

2003)(d) Foxboro EXACT I/A series PIDA product (in which it is an option

labelled ideal PID) (Foxboro, 1994)(e) Hartmann and Braun Freelance 2000 PID algorithm (EZYtune, 2003)(f) Modicon 984 product with 30N2 (McMillan, 1994; EZYtune,

2003)

(g) Siemens Teleperm/PSC7 ContC/PCS7 CTRL PID products with N = 10(ISMC, 1999) and the S7 FB41 CONT_C PID product (EZYtune, 2003).

5. Classical controller: This controller is also labelled the cascade controller(Witt and Waggoner, 1990), the interacting or series controller (Poulin andPomerleau, 1996), the interactive controller (Tsang and Rad, 1995), therate-before-reset controller (Smith and Corripio, 1997), the analogcontroller (St. Clair, 2000) or the commercial controller (Luyben, 2001).

N

Ts1

sT1

sT

11K)s(G

d

d

icc

+

+

+= (2.8)

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Chapter 2: Controller Architecture 7

The structure is used in the following products:

(a) Honeywell TDC Basic/Extended/Multifunction Types A and B productswith N = 8 (McMillan, 1994)(b) Toshiba TOSDIC 200 product with 10N33.3 (McMillan, 1994)(c) Foxboro EXACT Model 761 product with N = 10 (McMillan, 1994)(d) Honeywell UDC6000 product with N = 8 (strm and Hgglund, 1995)(e) Honeywell TDC3000 Process Manager product Type A, interactive

mode with N = 10 (ISMC, 1999)(f) Honeywell TDC3000 Universal, Multifunction and Advanced

Multifunction products with N = 8 (ISMC, 1999)(g) Foxboro EXACT I/A Series PIDA product (in which it is an option

labelled series PID) (Foxboro, 1994).

A subset of the classical PID controller is the so-called series controllerstructure, also labelled the interacting controller or the analog algorithm(McMillan, 1994) or the dependent controller (EZYtune, 2003).

( )di

cc sT1sT

11K)s(G +

+= (2.9)

The structure is used in the following products:

(a) Turnbull TCS6000 series product (McMillan, 1994)(b) Alfa-Laval Automation ECA400 product (strm and Hgglund, 1995)

(c) Foxboro EXACT 760/761 product (strm and Hgglund, 1995).

A further related structure is one labelled the interacting controller (Fertik,1975).

+

+

+=

N

Ts1

sT1

sT

11K)s(G

d

d

icc (2.10)

The structure is used in the following products:

(a) Bailey FC156 Classical Form PID product (EZYtune, 2003)(b) Fischer and Porter DCI 4000 PID algorithm (EZYtune, 2003).

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Handbook of PI and PID Controller Tuning Rules8

6. Generalised classical controller:

++

++

+

++=2

2f1f

22f1f0f

d

d

icc

sasa1

sbsbb

sN

T1

sT

sT

11K)s(G (2.11)

7. Two degree of freedom controller 1:

[ ] [ ] )s(Rs

N

T1

sT1sT

11K)s(Ud

d

ic

+++= )s(Y

sN

T1

sTsT

11Kd

d

ic

+++ (2.12)

This controller is also labelled the m-PID controller (Huang et al., 2000), theISA-PID controller (Leva and Colombo, 2001) and the P-I-PD (only P isDOF) incomplete 2DOF algorithm (Mizutani and Hiroi, 1991).

This structure is used in the following products:

(a) Bailey Net 90 PID PV and SP product (McMillan, 1994)(b) Yokogawa SLPC products with 1= , 1= , 10N= (McMillan,

1994)(c) Omron E5CK digital controller with 1= and 3N= (ISMC, 1999).

Notable subsets of this controller structure are:

)s(Y

sN

T1

sT

sT

11K)s(R

sT

11K)s(U

d

d

ic

ic

+

++

+= (2.13)

which is used in the following products:

(a) Allen Bradley SLC5/02, SLC5/03, SLC5/04, PLC5 and Logix5550products (EZYtune, 2003)

(b) Modcomp product with N = 10 (McMillan, 1994).

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Chapter 2: Controller Architecture 9

)s(YsTsT

11K)s(R

sT

11K)s(U d

ic

ic

++

+= (2.14)

Also labelled the PI+D controller structure (Chen, 1996) or thedependent, ideal, non-interacting controller structure (Cooper, 2006a), itis used in the following products:

(a) ABB 53SL6000 product (ABB, 2001)(b) Genesis product (McMillan, 1994)(c) Honeywell TDC3000 Process Manager Type B, non-interactive

mode product (ISMC, 1999)(d) Square D PIDR PID product (EZYtune, 2003).

)s(YsTsT

11K)s(R

sT

1K)s(U d

ic

ic

++

= (2.15)

which is used in the following products:

(a) Toshiba AdTune TOSDIC 211D8 product (Shigemasa et al., 1987)(b) Honeywell TDC3000 Process Manager Type C non-interactive

mode product (ISMC, 1999).

8. Two degree of freedom controller 2:

( ) )s(YK)s(Y)s(R)s(Fsa1sb1

sN

T1

sTsT

11K)s(U 01f

1f

d

d

ic

++

+

++= ,

45f

34f

23f2f

45f

34f

23f2f

sasasasa1

sbsbsbsb1)s(F

++++

++++= (2.16)

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Handbook of PI and PID Controller Tuning Rules10

9. Two degree of freedom controller 3:

)s(Y)s(F)s(R)s(F)s(U21

= ,

[ ] [ ] ( )

+

+

+

+

+=

sN

T1

sT1

sT1sT

11K)s(F

d

d

iic1 2

2f1f

1f

sasa1

sb1

++

+,

[ ] [ ] [ ]

+

+

+=

sN

T1

sT1

sT

11K)s(F

d

d

i

c2 2

4f3f

2f

sasa1

sb1

++

+

( )

sa1

sbbK

5f

4f3f0

+

+ (2.17)

Notable subsets of this controller structure are:

( )

++

+= )s(YsNT1

sT

1)s(RsT

1

1K)s(U d

d

ic (2.18)

Also labelled the reset-feedback controller structure (Huang et al., 1996),it is used in the following products:

(a) Bailey Fisher and Porter 53SL6000 and 53MC5000 products (ISMC,1999)

(b) Moore Model 352 Single-Loop Controller product (Wade, 1994).

( )

+

+

+= )s(Y

sNT1

sT1)s(R

sT

11K)s(U

d

d

ic (2.19)

Also labelled the industrial controller structure (Kaya and Scheib,1988), it is used in the following products:

(a) Fisher-Rosemount Provox product with 8N= (ISMC, 1999;McMillan, 1994)

(b) Foxboro Model 761 product with 10N= (McMillan, 1994)(c) Fischer-Porter Micro DCI product (McMillan, 1994)

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Chapter 2: Controller Architecture 11

(d) Moore Products Type 352 controller with 30N1 (McMillan,1994)

(e) SATT Instruments EAC400 product with 33.8N= (McMillan, 1994)(f) Taylor Mod 30 ESPO product with 7.16N= (McMillan, 1994)(g) Honeywell TDC3000 Process Manager Type B, interactive mode

product with 10N= (ISMC, 1999).

)s(Y

N

sT1

sT1

sT

11K)s(R

sT

K)s(U

d

d

i

c

i

c

+

+

+= (2.20)

which is used in the following products:

(a) Honeywell TDC3000 Process Manager Type C, interactive modeproduct with N = 10 (ISMC, 1999)

(b) Honeywell TDC3000 Universal, Multifunction and AdvancedMultifunction products with N = 8 (ISMC, 1999).

2.2 Comments on the PID Controller Structures

In some cases, one controller structure may be transformed into another; clearly,the ideal and parallel controller structures (Equations 2.4 and 2.5) are veryclosely related. It is shown by McMillan (1994), among others, that the

parameters of the ideal PID controller may be worked out from the parameters ofthe series PID controller, and vice versa. The ideal PID controller is given inEquation (2.22) and the series PID controller is given in equation (2.23).

++= sT

sT

11K)s(G dp

ipcpcp (2.22)

( )dsis

cscs sT1sT

11K)s(G +

+= (2.23)

Then, it may be shown that

csis

dscp K

T

T1K

+= , ( )dsisip TTT += , ds

dsis

isdp T

TT

TT

+= .

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Handbook of PI and PID Controller Tuning Rules12

Similarly, it may be shown that, provided dpip T4T > ,

+=

ip

dpcpcs

TT411K5.0K ,

+=

ip

dpipis

TT411T5.0T ,

=

ip

dpdpds

T

T411T5.0T .

strm and Hgglund (1996) point out that the ideal controller admits complexzeroes and is thus a more flexible controller structure than the series controller,which has real zeroes; however, in the frequency domain, the series controllerhas the interesting interpretation that the zeroes of the closed loop transferfunction are the inverse values of isT and dsT . ODwyer (2001b) developed acomprehensive set of tuning rules for the series PID controller, based on the idealPID controller; as these are not strictly original tuning rules, only representativeexamples are included in the relevant tables.

In a similar manner, it is straightforward to show that controller structures

(2.6), (2.7) and (2.8) are subsets of a more general controller structure, andcontroller parameters may be transformed readily from one structure to another.

2.3 Process Modelling

Processes with time delay may be modelled in a variety of ways. The modellingstrategy used will influence the value of the model parameters, which will in turnaffect the controller values determined from the tuning rules. The modelling

strategy used in association with each tuning rule, as described in the originalpapers, is indicated in the tables (see Chapters 3 and 4). These modellingstrategies are outlined in Appendix 2. Process models may be classified as self-regulating or non-self-regulating, and the models that fall into these categoriesare now detailed.

2.3.1Self-regulating process models

1. Delay model: msmm eK)s(G =

2. Delay model with a zero:

ms3mmm e)sT1(K)s(G

+= or ms4mmm e)sT1(K)s(G =

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Chapter 2: Controller Architecture 13

3. First order lag plus time delay (FOLPD) model:m

sm

msT1

eK)s(G

m

+=

4. FOLPD model with a zero:

1m

s3mm

msT1

e)sT1(K)s(G

m

+

+=

or1m

s4mm

msT1

e)sT1(K)s(G

m

+

=

5. Second order system plus time delay (SOSPD) model:

=)s(G m 1sT2sT

eK

1mm22

1m

s

m

m

++

or ( )( )sT1sT1

eK)s(G

2m1m

s

mm

m

++=

6. SOSPD model with a zero:

1sT2sT

e)sT1(K)s(G

1mm22

1m

s3mm

m

m

++

+=

or)sT1)(sT1(

e)sT1(K)s(G

2m1m

s3mm

m

m

++

+=

( )( )( )2m1m

s4mm

msT1sT1esT1K)s(G

m

++

=

or

1sT2sT

e)sT1(K)s(G1mm

221m

s4mm

m

m

++

=

7. Third order system plus time delay (TOSPD) model:

( )33221

s33

221m

msasasa1

e)sbsbsb1(K)s(G

m

+++

+++=

or

=)s(Gm ( )( )( )3m2m1m

sm

sT1sT1sT1

eK m

+++

or

( )( )1sT1sT2sTe)1sT(K

)s(G2m1mm

221m

s3mm

m

m

+++

+=

8. Fifth order system plus delay model:

( )554433221

s5s

44

33

221m

msasasasasa1

e)sbsbsbsbsb1(K)s(G

m

+++++

+++++=

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Handbook of PI and PID Controller Tuning Rules14

9. General model.10. Non-model specific. Corresponding tuning rules may also apply to non-self-

regulating processes.

2.3.2Non-self-regulating process models

1. Integral plus time delay (IPD) model:s

eK)s(G

msm

m

=

2. IPD model with a zero:

( )s

esT1K)s(Gm

s3mm

m

+= or ( )s

esT1K)s(Gm

s4mm

m

=

3. First order lag plus integral plus time delay (FOLIPD) model:

( )m

sm

msT1s

eK)s(G

m

+=

4. FOLIPD model with a zero:

( )

( )1m

s3mm

msT1s

esT1K)s(G

m

+

+=

or( )

( )1m

s4mm

msT1s

esT1K)s(G

m

+

=

5. Integral squared plus time delay ( PDI2 ) model: =)s(G m 2

sm

s

eK m

6. Second order system plus integral plus time delay (SOSIPD) model:

=)s(Gm( )21m

sm

sT1s

eK m

+

or)1sT2sT(s

eK)s(G

1mm22

1m

sm

m

m

++=

7. SOSIPD model with a zero:( )

( )( )2m1m

s3mm

msT1sT1s

esT1K)s(G

m

++

+=

8. Third order system plus integral plus time delay (TOSIPD) model:

( )( )( )3m2m1m

sm

msT1sT1sT1s

eK)s(Gm

+++=

or( )31m

smm

sT1seK)s(G

m

+=

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Chapter 2: Controller Architecture 15

9. General model with integrator:

( )n1m

s

mmsT1seK)s(G

m

+=

or

( )

( ) ( )m

4d43

2n21

s

i iimim

22imim

iimim

22im

iim

mm e

1sT2sT1sT

1sT2sT1sT

s

K)s(G

+++

+++

=

10. Unstable FOLPD model: =)s(Gm1sT

eKm

sm m

11. Unstable FOLPD model with a zero:

( )

1sT

esT1K)s(G

1m

s3mm

m

m

+=

or =)s(Gm1sT

e)sT1(K

m

s4mm

m

12. Unstable SOSPD model (one unstable pole):

=)s(Gm ( )( )2m1m

sm

sT11sT

eK m

+

13. Unstable SOSPD model (two unstable poles):

=)s(Gm ( )( )1sT1sT

eK

2m1m

sm

m

14. Unstable SOSPD model with a zero:

=)s(G m( )

1sT2sT

esT1K

1mm22

1m

s3mm

m

+

+ or

( )

1sT2sT

esT1K

1mm22

1m

s3mm

m

++

+ or

=)s(Gm1sT2sT

e)sT1(K

1mm221m

s4mm

m

+

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Handbook of PI and PID Controller Tuning Rules16

Table 1 shows the number of tuning rules defined for each PI/PID controllerstructure and types of process model. The following data is key to the process

model type:

Model 1: Stable FOLPD model (with or without a zero)Model 2: Stable SOSPD model (with or without a zero)Model 3: Other stable modelsModel 4: Non-model specificModel 5: Models with an integratorModel 6: Open loop unstable models

Table 1: PI/PID controller structure and tuning rules a summary

Process model

ControllerEquation

1 2 3 4 5 6 Total

(2.3) 261 63 58 59 90 32 563(2.4) 140 82 20 63 66 35 406(2.6) 20 23 3 9 16 17 88

(2.7) 36 9 5 11 6 0 67(2.8) 74 53 1 12 26 17 183

(2.11) 7 7 1 4 5 6 30(2.12) 74 36 14 9 106 37 276(2.16) 7 3 0 0 16 8 34(2.17) 28 15 1 2 8 30 84Total 649 291 103 169 339 182 1731

Table 1 shows that for the tuning rules defined, 60% are based on a self-

regulating (or stable) model, 10% are non-model specific, with the remaining30% based on a non-self-regulating model.

2.4 Organisation of the Tuning Rules

The tuning rules are organised in tabular form in Chapters 3 and 4. Within eachtable, the tuning rules are classified further; the main subdivisions made are asfollows:

(i) Tuning rules based on a measured step response (also called process reactioncurve methods).

(ii) Tuning rules based on minimising an appropriate performance criterion,either for optimum regulator or optimum servo action.

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Chapter 2: Controller Architecture 17

(iii) Tuning rules that give a specified closed loop response (direct synthesistuning rules). Such rules may be defined by specifying the desired poles of

the closed loop response, for instance, though more generally, the desiredclosed loop transfer function may be specified. The definition may beexpanded to cover techniques that allow the achievement of a specified gainmargin and/or phase margin.

(iv) Robust tuning rules, with an explicit robust stability and robust performancecriterion built into the design process.

(v) Tuning rules based on recording appropriate parameters at the ultimatefrequency (also called ultimate cycling methods).

(vi) Other tuning rules, such as tuning rules that depend on the proportional gainrequired to achieve a quarter decay ratio or to achieve magnitude andfrequency information at a particular phase lag.

Some tuning rules could be considered to belong to more than one subdivision,so the subdivisions cannot be considered to be mutually exclusive; nevertheless,they provide a convenient way to classify the rules. All symbols used in thetables are defined in Appendix 1.