1-s2.0-S0959152411002198-main

Journal of Process Control 22 (2012) 11 25

Contents lists available at SciVerse ScienceDirect

Journal of Process Control

jo u rn al hom epa ge: ww w.elsev ier .com

Extending the Symmetrical Optimum criterion toloops

Konstanta ABB Switzerla itzerlb Aristotle Univ essalon

a r t i c l

Article history:Received 17 JuReceived in reAccepted 23 OAvailable onlin

Keywords:PID controlDesignTuningOptimizationProcess controControl engineIndustrial control

timumops a

PID tyf typed ver and

ordeant tiThe ncer

lts for

2011 Elsevier Ltd. All rights reserved.

1. Introduction

The needbeen challeety [1,2]. Froloops have order to achbased on thinvolves thfunction. Hecontrollers of both the

Throughlaw offers to many reof industriaalgorithms,designing hresearchersschemes su

CorresponE-mail add

kpapadop@en1 Tel.: +30 22 Internal M

[27]. Main drawback of the above approaches is that the proposed

0959-1524/$ doi:10.1016/j. to design higher order type control loops has alwaysnging and critical over the academic and industry soci-m the theory, it is known that higher order type controlthe advantage of tracking fast reference signals [1]. Inieve this target, and if the design of the control loop ise frequency domain, the design of a type-p control loope existence of p integrators in the open-loop transfernce, if for the design of a type-p control loop, PID typeare adopted, then the sum of the pure free integratorsprocess and the controller should be equal to p.out the literature, it is evident that the PID controlthe simplest, feasible and yet most efcient solutional-world control problems, see [522]. More than 90%l controllers still implemented, are based around PID

see [1723]. However, the demanding problem ofigher order type control loops has been treated by many

after employing or modifying well established controlch as the IMC2 principle [1,25,26] or the Smith predictor

ding author. Tel.: +41 58 5 893242; fax: +41 58 5 892580.resses: [email protected],g.auth.gr (K.G. Papadopoulos), [email protected] (N.I. Margaris).310 9 96283; fax: +30 2310 9 96447.odel Control.

control laws are restricted to the design of type-II control loopsand whatsmore, for verifying their control laws potential, simpleprocess models are employed [9,24,32] (rst order plus dead timeprocess, rst order reduced integrating process). Over the litera-ture, a rst attempt of designing type-III control loops can be foundin [3,4]. There, it is shown that by applying the principle of Symmet-rical Optimum criterion along with the use of PID type controllers,robust type-III control loops can be designed. In similar fashion andsince the aim of this work is to present a feasible control designmethod that can be applied in many industry applications, oncemore the simplicity and widespread application of the PID con-trol law will be exploited. Therefore, the approach proposed inthat work, is faced with the following problem. Tune a PID typecontroller, such that the output of the nal closed-loop control sys-tem eliminates higher order steady state errors, position, velocity,acceleration etc.

For developing the proposed theory, the principle of Symmet-rical Optimum criterion will be adopted. In the sequel, it will beshown that for achieving the design of type-p control loops alongwith the aid of the Symmetrical Optimum criterion, the conven-tional design principle for tuning PID type controllers has to beadopted. This conventional principle implies that exact pole-zerocancellation has to be achieved between the process poles andthe controllers zeros. Therefore, for applying the proposed theoryall dominant time constants of the process have to be measuredaccurately. The Symmetrical Optimum criterion is an extension of

see front matter 2011 Elsevier Ltd. All rights reserved.jprocont.2011.10.014inos G. Papadopoulosa,, Nikolaos I. Margarisb,1

nd Ltd., Department of Power Electronics & Medium Voltage Drives, CH-5300 Turgi, Swersity of Thessaloniki, Department of Electrical & Computer Engineering, GR-54124 Th

e i n f o

ly 2011vised form 23 October 2011ctober 2011e 23 November 2011

lering

a b s t r a c t

An extension of the Symmetrical Opsystems is proposed. Type-p control loloop transfer function. For designing aand so on, if the controlled process is oachieves zero steady state position anposition, velocity and acceleration erroreference signals and eliminate highera transfer function containing dominconsidered in the frequency domain. both dominant dynamics and model uthe proposed theory, simulation resuare presented./ locate / jprocont

the design of PID type-p control

andiki, Greece

criterion for the design of PID type-p closed- loop controlre characterized by the presence of p integrators in the open-pe-p control loop there should exist an PIpD, or PI(p1)D, or PID-0 or type-1 or type-(p 1) respectively. A type-II control looplocity error, a type-III control loop achieves zero steady state

therefore a type-p control loop is expected to track both fasterr errors at steady state. For deriving the proposed control law,me constants and the plants unmodelled dynamics has beennal control law consists of analytical expressions that involvetainty of the controlled process. For justifying the potential of

representative processes met in many industry applications

12 K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11 25

the Magnitude Optimum criterion, introduced by Oldenbourg andSartorius [28], and is based on the idea of designing a controllerwhich will render the magnitude of the closed-loop frequencyresponse as close as possible to unity in the widest possible fre-quency range. In succession, Kessler suggested the SymmetricalOptimum criterion [29], which in reality is the application of theMagnitude Optimum criterion in type-II control systems. The nameof the Symmexhibited bity, the Symbut the appcontrol syst

The desimum and Sand Kesslertages: (1) ththe setpoinHowever, etude Optim

For the dclear presencontrollers Section 3. Btype-III contheory is exknowledge is provided 6), represenare employedominant t

2. Denitio

Accordine(s) = r(s) function T(s

T(s) = bmsm

ans

then the res

e(s) =(

ans

an

where cj = aand if e(s) is

e() = lims0

s

If r(s) = 1/s t

e() = lims0

which becoS(s) = y(s)/dby

T(s) = smbmsnan

S(s) = s [ans b2) +

3 S(s) standsby S(s) = y(s)/d

respectively. If (5) and (6) hold by, the closed-loop control systemis said to be of type-I. In similar fashion, if, then the velocity erroris equal to,

e() = lims0

(ansn + + cmsm + + c1s + c0ansn + an1sn1 + + a1s + a0

)1s

(7)

becoerror

s(t) =

com-loop-loop

smbmsna

s2an

ing tf type fo

(ansn

bmsm

ansn

tivelteriz6), (1he adate h

con

us nos), ythe inance

ry ap

Tms(

Tm iant t26]. L

vectnsta

is thrquere, t

f the nt. Fior co, kp s

: statetrical Optimum criterion comes from the symmetryy the open-loop frequency response [29,30]. In real-metrical Optimum criterion is not something different,lication of the Magnitude Optimum criterion to type-IIems.gn of control systems both with the Magnitude Opti-ymmetrical Optimum criteria of Oldenbourg-Sartorius

respectively presents at least two important advan-ey do not require the complete plant model [30] and (2)t response of the closed-loop system is satisfactory [2].xcluding the German bibliography [3336], the Magni-um criterion is rarely referred today [30].evelopment of the proposed theory and for the sake of atation of this work, the conventional tuning of PID typevia the Symmetrical Optimum criterion is presented inased on this principle, in Section 4 the design of PIDtrol loops is presented. Finally, in Section 5 the proposedtended to the design of type-p control loops. Preliminaryregarding the denition of the type of the control loopsin Section 2. For testing the proposed theory (see Sectiontative processes met over the literature and industryd, processes with time delay, processes with equivalent

ime constants and non-minimum phase processes.

ns and preliminaries

g to Fig. 1 the error e(s) is given byy(s) = [1 T(s)]r(s) = S(s)r(s) . If the closed-loop transfer) = y(s)/r(s) is dened by

+ bm1sm1 + + b1s + b0n + an1sn1 + + a1s + a0

(1)

ulting error e(s) is dened by

n + + cmsm + + c1s + c0sn + an1sn1 + + a1s + a0

)r(s) (2)

j bj (j = 0, . . ., m). According to the nal value theorem stable, e() is equal to(

ansn + + c2s2 + c1s + c0ansn + an1sn1 + + a1s + a0

)r(s). (3)

hen(c0a0

), (4)

mes zero when c0 = 0, or when a0 = b0 . Hence, sensitivityo(s)3 and closed-loop transfer function T(s) are dened

+ + s2b2 + sb1 + a0+ + s2a2 + sa1 + a0

, (5)

n1 + an1sn2 + + (am bm)sm1 + (am1 bm1)sm2 + s(a2 snan + sn1an1 + + s2a2 + sa1 + a0

for the sensitivity of the closed-loop control system and is denedo(s) when r(s) = nr(s) = di(s) = nr(s) = 0, Fig. 1.

whichof the

limt

evs

and beclosedclosedtively,

T(s) =

S(s) =

Accordto be ohave th

S(s) = sp

and

T(s) =

respeccharacS, see (have telimin

3. The

Let e(s), u(error, disturbindust

G(s) =

wheredominstant [used intime cowhichload tothermoloop oconstaIf vectdrives)

4 SFOC a1 b1] (6)

mes nite if c0 = 0 or a0 = b0. As a result the nal value is given by

lims0

(c1a0

)= lim

s0

(a1 b1

a0

)(8)

es zero when c1 = 0 or when a1 = b1. In that case, the control system is said to be of type-II. Sensitivity S and transfer function T take the following forms respec-

+ sm1bm1 + + sa1 + a0n + sn1an1 + + sa1 + a0

, (9)

sn2 bmsm2 bm1sm3 + a2 b2snan + sn1an1 + + s2a2 + sa1 + a0

. (10)

o the above analysis, a closed-loop control system is saide-p when sensitivity S and complementary sensitivity Tllowing formp + an1sn1p bmsmp bm1sm1p + ap bp)

snan + sn1an1 + + s2a2 + sa1 + a0(11)

+ + apsp + ap1sp1 + + a1s + a0+ + apsp + ap1sp1 + + a1s + a0

(12)

y. Also, one could argue that type-p control loops areed by the order of zeros at s = 0 in the sensitivity function0) and (11). According to the above, type-p control loopsvantage of tracking fast reference signals, since theyigher order errors.

ventional Symmetrical Optimum design criterion

w consider the closed-loop system of Fig. 1, where r(s),(s), do(s) and di(s) are the reference input, the controlput and output of the plant, the output and the inputs respectively. An integrating process met in manyplications can be dened by (13)

11 + Tp1s)(1 + Tps)

, (13)

s the integrators plant time constant, Tp1 the plantsime constant and Tp the process parasitic time con-et it be noted that such type of modelling is frequentlyor controlled induction motor drives. More specically,nt Tm stands for the mechanical subsystem of the motore mechanism that involves the electromagnetic and, the difference of which, makes the shaft rotating. Fur-ime constant Tp1 is involved in the inner current controlelectrical drive and represents the stator winding timenally, Tp stands for the motors unmodelled dynamics.ntrol4 is to be followed (control of induction motortands for the pulse width modulators gain (kPWM) which

or eld oriented control; RFOC: rotor eld oriented control.

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11 25 13

Fig. 1. Block d the care the output ls at thplants dc gain

is supposed(0 1 p.u.)back path othe sequel,

Back to F

C(s) = (1 +Tis

is adopted. design metcontrollersapplied, becble. This is jform

C(s) =Tis(1

is applied, t

T(s) =TiTm

where TpT

T(s) =TiTm

From (17), missing, see

C(s) = 1 Tis(1

is employethe convencellation mconstant Tp

T(s) =TiTm

which is uAppendix B

Assuminrately measby (14), T

5 In many in

etricaes th

TiTm

gnit

=

nom

(TiT

+[(

com

Tn

23) a

8T3

s

lly a in

8s3

specta) ansed-in thiagram of the closed-loop control system. G(s) is the plant transfer function, C(s) is and input disturbance signals respectively and nr(s) and no(s) are the noise signa

and kh is the feedback path.

to remain constant all over the whole operating range regarding output frequency.5 Parameter kh is the feed-f the output measurement and as it will be proved inkh should satisfy condition kh = 1.ig. 1, for controlling (13), the PID controller dened by

Tns)(1 + Tvs)(1 + Tcs)

(14)

For its tuning, the conventional Symmetrical Optimumhod is employed. Time constant Tc stands for the

parasitic dynamics. If Tn = Tv = 0, I control cannot beause the closed-loop transfer function becomes unsta-ustied as follows. If for controlling (13), I control of the

1 + Tcs)

(15)

hen the closed-loop transfer function is given by

kps2(1 + Tp1s)(1 + Ts) + khkp

(16)

c 0 and T = Tp + Tc. From (16) it is evident that

kpTp1Ts

4 + TiTm(Tp1 + T)s3 + TiTms2 + khkp. (17)

it is clear that T(s) is unstable since the term of s is Appendix B. In similar fashion, if PI control of the form

+ Tns + Tcs)

(18)

d, then for determining controller parameter Tn viational Symmetrical Optimum criterion, pole-zero can-

Symmbecom

T(s) =

The ma

|T(j)|

The de

D() =

and bewhen

kh = 1,

Using (

T(s) =

or naresults

T(s) =

The reFig. 2(the clo43.4% ust take place, Tn = Tp1 . Therefore, the dominant time1 has to be evaluated and in that case, T(s) becomes

kpTs4 + TiTmTs3 + TiTms2 + khkp

, (19)

nstable again for the same reason as for (17), see.g again that the dominant time constant Tp1 is accu-ured and considering a PID controller as that describedv = Tp1 is set (pole-zero cancellation, conventional

dustry applications kp stands for the plants dc gain at steady state.

frequency dfrequency rfrequency

try of the

where its slc = 1/(2T

Fol(s) = 1

8s

In order toence input The great ozero of the ontroller transfer function, r(s) is the reference signal, do(s) and di(s)e reference input and process output respectively. kp stands for the

l Optimum design). The closed-loop transfer functionen equal to

kpTns + kpTs3 + TiTms2 + khkpTns + khkp

. (20)

ude of (20) is given by

kpkp[1 + (Tn)2](kpkh TiTm2)2 + 2(kpkhTn TiTmT2)2

. (21)

inator of (21) is equal to

p1T)26 + TiTp1 (TiTp1 2kpkhTnT)4

kpkhTn)2 2kpkhTiTp1 ]2 + k2pk2h (22)

es minimum, see [30,31], in the lower frequency range

= 4T, Ti = 8kpkhT2

Tm, Tv = Tp1 . (23)

long with (20) results in

1 + 4Ts + 8T2

s + 4Ts + 1

(24)

fter normalizing the frequency by substituting s = Ts

1 + 4s+ 8s2 + 4s + 1 . (25)

ive step and frequency response of (25) are shown ind (b). From there, it is clear that the step response ofloop control system exhibits an undesired overshoot ofe time domain Fig. 2(a), and a peak overshoot in the

omain Fig. 2(b). This is also justied by the open-loopesponse Fig. 3 where the phase margin in the crossoverc = 1/(2T) is m 35 < 45. Note also the symme-critical frequencies

(1

4T, 1T

)exhibited by |Fol(j)|

ope is equal to 1/deg around the crossover frequency), Fig. 3. The open loop transfer function is given by

+ 4s2(1 + s)

. (26)

overcome the obstacle of 43.4% overshoot, the refer-is ltered by adding an external controller Cex(s), Fig. 4.vershoot of the step response in (24) is owed to thetransfer function, N(s) = 1 + 4s. This can be removed by


Fig. 2. Type-II closed-loop control system. (a) The effect of the two degrees of freedom controller to the step response of the closed-loop control system. Step response (solidblack) and ltered step response (dotted black). (b) The effect of the two degrees of freedom controller to the frequency response of the closed-loop control system.

Fig. 3. Ty

including thexternal lt

Cex(s) = r(r(

is chosen, tnoted that tdynamics, cin the refer

4. Extenditype-III con

Accordintems, a simcontrol sys

integrating process of the form (13) is assumed, where Tp1 standsfor the dominant time constant of the process and Tm, Tp stand forthe integrators time constant and the unmodelled plant dynam-

pectited, t

(1 +Tis2(

Tc1red ting t

the c

TiTm

T =

Fig. 4. Two dethe closed-loope-II closed-loop control system. Open loop frequency response.

ics resevalua

C(s) =

wherecompaassumtion of

T(s) =

whereat zero as a pole in the reference lter. In that, if aner of the form

s)s)

= 11 + 4s (27)

he overshoot decreases from 43.4% to 8.1%. Let it behe rise time increases from trt = 3.1T to trt = 6.6T. Suchan for sure be improved by adding additional dynamicsence lter.

ng the Symmetrical Optimum design criterion totrol loops

g to the design of type-II closed-loop control sys-ilar methodology for the design of type-III closed-looptems will be proposed. For the following analysis, an

|T(j)| =

where A0 =dened by

D() = (TiT+ k

+ (k

One way toj = 2, 4, 6, . .

grees of freedom controller. Controller Cex(s) lters the reference input so that the undesirp transfer function T(s) and not the output and input disturbance transfer functions So(s)vely. Supposing that the dominant time constant Tp1 ishe proposed I-PID controller is dened by,

Tns)(1 + Tvs)(1 + Txs)1 + Tc1 s)(1 + Tc2 s)

(28)

, Tc2are known and sufciently small time constants

o Tp1 . By setting Tx = Tp1 (pole-zero cancellation) andhat Tc = Tc1 + Tc2 , Tc1 Tc2 0, the transfer func-losed-loop control system is equal to

kpTnTvs2 + kp(Tn + Tv)s + kpTs4 + TiTms3 + kpkhTnTvs2 + kpkh(Tn + Tv)s + kpkh

,

(29)

Tc + Tp . The magnitude of (29) is given by

k2p(1 TnTv2)2 + k2p(Tn + Tv)22[TiTmT4 + kpkh(1 TnTv2)]2 + A02

(30)

[kpkh(Tn + Tv) TiTm2]2. The denominator of (30) is

mT)28 + TiTm(TiTm 2kpkhTnTvT)6

pkh[2TiTmT 2(Tn + Tv)TiTm + kpkhT2i T2m]4

pkh)2(T2n + T2v )2 + (kpkh)2. (31)

optimize the magnitude of (30) is to set the terms of j, ., in (31), equal to zero, starting again from the lowered overshoot at the output y(s) is diminished. Controller Cex(s) affects = y(s)/do(s), Si(s) = y(s)/di(s).


Fig. m. (b)

Fig. 6. Ope

frequency rleads to

Ti =2kpkhT

Tm

In similar fathe aid of (3

4T2 4(TnIf Tv = nT

Tn = 4(n n 4

Proper selefeasible I-Pclosed-loop

T(s) =8n(n

Normalizinwhich nal

T(s) =8n(n

Note that tterms of s

j,2. The respeferent value

6 thn is

4n

m Fintaryeter and m

everncy r). Sinin th), whex(s)

endi con5. (a) Step response and disturbance rejection of type-III closed-loop control syste

n-loop frequency response of type-III closed-loop control system.

6

in Fig. functio

Fol(s) =

Froplemeparamn = 4.1 by. Forfreque(50%shoot Fig. 5(alter C

5. Exttype-pange. Setting kh = 1 and the term of equal to zero

nTvT . (32)

shion, setting the term of 4 equal to zero along with2), leads to

+ Tv)T + TnTv = 0. (33)

is chosen, then (33) becomes

1)T. (34)

ction of parameter n (n > 4 must hold by) leads to aID control law. Substituting Eqs. (32) and (34) into the

transfer function results in

4n(n 1)T2s2 + (n2 4)Ts + n 4

1)T4s4 + 8n(n 1)T3

s3 + 4n(n 1)T2

s2 + (n2 4)Ts + (n 4)

.

(35)

g again the time by setting s = sT where s = ju, (s = j)ly leads to u = T, (35) becomes equal to

4n(n 1)s2 + (n2 4)s + (n 4) 1)s4 + 8n(n 1)s3 + 4n(n 1)s2 + (n2 4)s + (n 4)

. (36)

he control loop dened in (36) is of type-III, since the j = 0, 1, 2, are equal, a0 = b0, a1 = b1, a2 = b2, see Sectionctive step and frequency responses of (36) for two dif-s of parameter n, are presented in Fig. 5. In addition,

Accordinsis for tuninregarding tp stands forTherefore, l

G(s) =Tmsq

consisting constant. Adened by Tics by Tsk (kgenerality w

nsk=1

(1 + Tsk s

where Tstime constacontrol systin Section 4

C(s) =nm

j=

Thus, accorcontroller hconstants (achieved. Min order thtion is a ful Frequency response of type-III closed-loop control system.

e open-loop frequency response is shown. Its transfergiven by

(n 1)s2 + (n2 4)s + n 48n(n 1)s3(1 + s)

. (37)

g. 6 it is concluded that the magnitude of the com- sensitivity |T(ju)| is practically independent of then. Moreover, sensitivity |S(ju)| becomes maximum ifinimum, if n = 7.46. In that case (n = 7.46), Tn = Tv holds

y other value of parameter n, the shape of the open-loopesponse is preserved exactly as presented in Fig. 5(a),ce the phase margin is m = 35 < 45, an undesired over-e step response of the closed-loop system is expected,ich can be decreased along with the aid of an external

as mentioned in Section 3.

ng the Symmetrical Optimum design criterion totrol loops

g to the analysis presented in Section 4 a similar analy-g the PID type controllers parameters will be presentedhe design of type-p control loops. Note that parameter

the free integrators of the open-loop transfer function.et the process be dened by

1nmj=1(1 + Tmj s)

nsk=1(1 + Tsk s)

, (38)

of q integrators and Tm one of the integrators timessuming that the plants dominant time constants aremj (j = 1, 2, . . ., nm) and the process unmodelled dynam-

= 1, 2, . . ., ns) we can substitute in (38), without loss ofith the approximation

) = 1 + Tss (39)

=ns

k=1Tsk stands for the process small unmodelled

nts. Since the target of the design is the nal closed-loopem to be of type-p, according to the analysis presented, the proposed PID type controller is given by

1(1 + Tmj s)p1

r=1 (1 + Tnr s)Tispq

ncz=1(1 + Tcz s)

. (40)

ding the design of type-II, III control loops, the PID typeas to contain nm zeros equal to the Tmj dominant timej = 1, 2, . . ., nm) so that exact pole-zero cancellation isoreover, it is proved after some calculus in T(s), that

e denominator of the nal closed-loop transfer func-l polynomial in terms of the sj coefcients, p 1 zeros


must exist. Furthermore, the controller must introduce pq integra-tors, so that the nal closed-loop is of type-p. Finally, in order thecontroller transfer function is strictly causal, denominators ordermust be greater or equal to p 1 + nm. The unmodelled controllersdynamics a

ncz=1

(1 + Tcz s

where

Tc =nc

z=1T

In that case

Fol(s) = kpk

or by substi

Fol(s) = kpk

where T =transfer fun

T(s) =TiTm

After some

T(s) = bap+1

where

bp1 =p1j=1

T

b2 = kpp1

i /= j=

and

ap+1 = TiTm

a3 = kpkhi /=

a1 = kpkhpi=

According t

kh = 1. Since the athe magnituFor every o

Ti = 2kpkhT

T

This can beconstant de

loop is of type-II p = 2, the PID type controller (according to theSymmetrical Optimum criterion) is given by

C(s) = (1 + Tn1s)(1 + Tn2s)T

, (54)

ich T, the

kpk

e clo

TiTm

ing tonsta

a1a3

as a]. Aft

pkhT

T

Tn1 =ordinenl loop

(1 +

ing p)(1ecom

kpk

ore t

TiTm

ing tted v

a2a4.

, aftent is

pkhT

T

imilad by , the

(1 +

ing ting t

2kpkre represented by

) = 1 + Tc s (41)

cz . (42)

, the open-loop transfer function becomes

hG(s)C(s) = kpkhp1

r=1 (1 + Tnr s)TiTmsp

nsk=1(1 + Tsk s)

ncz=1(1 + Tcz s)

(43)

tuting (39), (40), (41) and (42) results in

h

p1r=1 (1 + Tnr s)

TiTmsp(1 + Ts)(44)

Ts + Tc and TsTc 0. Finally, the closed-loopction is equal to

kpp1

r=1 (1 + Tnr s)Tsp+1 + TiTmsp + kpkh

p1r=1 (1 + Tnr s)

. (45)

calculus in (45) it is concluded that

p1sp1 + bp2sp2 + + b3s3 + b2s2 + b1s + b0sp+1 + apsp + ap1sp1 + + a3s3 + a2s2 + a1s + a0

(46)

pj = Tp1Tp2 . . . Tpp1 , b3 = kpp1

i /= j /= k=1TniTnj Tnk , (47)

1

TniTnj , b1 = kpp1i=1

Tni , b0 = kp, (48)

T, ap = TiTm, (49)p1

j /= k=1TniTnj Tnk , a2 = kpkh

p1i /= j=1

TniTnj , (50)

1

1

Tni , a0 = kpkh. (51)

o (A.9), if a0 = b0 then

(52)

im is to determine parameters Ti, Tnr (r = 1, . . . , p 1)de of (46) will be optimized according to Appendix A.

rder p, the optimal integral gain is given by

m

p1r=1

Tnr . (53)

proved as follows. For a process of one dominant timened by (13) where (q = 1), then in order the nal control

for whIn that

Fol(s) =

and th

T(s) =

Accordtime c

a22 = 2

is set, [30,31

Ti = 2k

and if Acc

stant dcontro

C(s) =

Assum(1 + sTFol(s) b

Fol(s) =

Theref

T(s) =

Accordcalcula

a23 = 2

Finallyconsta

Ti = 2k

In sdenetype-p

C(s) =

Accordregard

Tik1 =is(1 + Tcs)

n2 = Tp1 and (1 + sTp)(1 + sTc) 1 + sT have been set. open-loop transfer function is given by

h(1 + Tn1s)

TiTms2(1 + Ts)(55)

sed-loop transfer function is then given by

kp(1 + Tn1s)s2(1 + Ts) + kpkh(1 + Tn1s)

. (56)

o the analysis presented in Section 3, the integratorsnt is calculated if

(57)

nother means of optimizing the magnitude of (56),er some calculus it is obtained

mTn1 (58)

4T then Ti = 8kpkh(T2/Tm), see Section 3.g to Section 4, for a process of one dominant time con-

ed again by (13) where (q = 1) then in order the nal is of type-III p = 3, the PID type controller is given by

Tn1s)(1 + Tn2s)(1 + Tn3s)Tis2(1 + Tcs)

. (59)

again pole-zero cancellation, Tn3 = Tp1 and + sTc) 1 + sT the open-loop transfer functiones

h(1 + Tn1s)(1 + Tn2s)

TiTms3(1 + Ts). (60)

he closed-loop transfer function is equal to

kp(1 + Tn1s)(1 + Tn2s)s3(1 + Ts) + kpkh(1 + Tn1s)(1 + sTn2 )

. (61)

o (57) and since n = 2, the integrators time constant isia

(62)

r some calculus it was shown that the integrator timeequal to

mTn1Tn2 . (63)

r fashion, for a process of one dominant time constant(13) and if n = k 1, in order the nal control loop is ofPID type controller is given by

Tn1s)(1 + Tn2s) . . . (1 + Tnk s)Tisk1(1 + Tcs)

. (64)

o the analysis presented previously, it can be claimedhe integrators time constant Tik1 , that

hTTm

k1j=1

Tnj . (65)


Therefore, for n = k, it has to be proved that

Tik = 2kpkhTTm

kTnj

= 2kpk

According ttroller is giv

C(s) = (1 +

for which Tcellation. Sclosed-loop

Fol(s) = kpk

T(s) =TikTm

or

T(s) =Tik TmT

respectively

a2k+1 = 2ak+or

(TikTm)2 = 2

or TikTm = obtained

Tik = 2kpk

= 2kpk

which is eqresults in

T(s) =2kpkhT

For determithe magnituconstants T

4T2 p3

i=1

This is justing parameresults in

k2pT2n1

= 2kpor nally

Tn1 4T =

In similar fashion, in type-III control loops for determining parame-ters Tn1 , Tn2 we make use of a

22 2a3a1 + 2a4a0 = 0, see (A.11). This

results in

2 Tn2 2 2

lly,

4Ting t

is of follo

2ak

ak+1a

es thes.reforsed-

iTmT

it waain

2rkhich

4Tr

then

rk

then2kpkhe obt

2rk

r som

2 4

p2i=1

ove e

lly

pi=

is truiousetersditio

e-V c

n2 + T

n1Tn

Tn2Tnj=1

hTTm

k1

j=1Tnj

Tnk = Tik1Tnk

. (66)

o the design of type-p control loops, the PID type con-en by

Tn1s)(1 + Tn2s) . . . (1 + Tnk s)(1 + Tnk+1s)Tisk(1 + Tcs)

(67)

nk+1 = Tp1 is set, assuming design via pole-zero can-ince again (1 + sTp)(1 + sTc) 1 + sT, the open and

transfer functions are given by

h

kj=1(1 + Tnj s)

TikTmsk+1(1 + Ts)

, (68)

kpk

j=1(1 + Tnj s)sk+1(1 + Ts) + kpkh

kj=1(1 + Tnj s)

, (69)

kp(rksk + rk1sk1 + + r2s2 + r1s + 1)sk+2 + Tik Tmsk+1 + kpkh(rksk + rk1sk1 + r2s2 + r1s + 1)

(70)

. Then, according to (57), Ti is calculated by

2ak (71)

kpkhTikTmTrk (72)

2kpkhTrk. Finally, along with the aid of (70), it is

hTTm

rk = 2kpkhTTm

kj=1

Tnj

hTTm

k1

j=1Tnj

Tnk = Tik1Tnk

(73)

ual to (66). In that case, if (73) is substituted into (69),

kpp1

r=1 (1 + Tnr s)2p1

r=1Tnr sp+1 + 2kpkhT

p1r=1Tnr s

p + kpkhp1

r=1 (1 + Tnr s). (74)

ning now parameters Tnr , it will be shown that in orderde of (74) satises condition |T(j) 1|, controller time

nr must satisfy condition

Tni 4T p2

i=1Tni +

p1i=1

Tni = 0. (75)

ied as follows. In type-II control loops for determin-ter Tn1 we make use of a

21 2a2a0 = 0 (see (A.10)). This

(2kpTn1T) (76)

0. (77)

4TTn1

or na

Tn1Tn2

Accordsystemk), the

a2k1 =

a2k = 2the onquenci

Thethe clo

T(s) =T

In (72)we obt

k2pr2k1

from w

rk1 If n = k,

rk 4TIf n = k TiTm = (70) w

k2pr2k

or afte

4T2rk

or

4T2

The ab

4T2Tnp

or na[4T2

which

Obvparaming conin

Typ

4(Tn1T

4(T+ Tn1 4TTn1Tn2 (Tn1 + Tn2 ) + Tn1Tn2 = 0 (78)

(Tn1 + Tn2 ) + 4T2 = 0. (79)o the above, and based on (65) if the closed-loop control

type-p, then for determining parameters Tnj (j = 1, 2, . . .,wing optimization conditions are claimed to be,

2ak 2ak3ak+1, (80)

k1 (81)

at satisfy condition |T(j) 1| in a wide range of fre-

e, if n = k 1 then controller C(s) is dened by (64) andloop transfer function is given by

kp(rk1sk1 + rk2sk2 + + r2s2 + r1s + 1)sk+1 + TiTmsk + kpkh(rk1sk1 + rk2sk2 + + r2s2 + r1s + 1)

. (82)

s shown that TiTm = 2kpkhTrk. By applying (80) to (82)

2kp(2kpTrk1) + 2kprk3(2kpTrk1)T = 0, (83) after some calculus results in

k2 + 4T2rk3 = 0. (84) we are going to show that

1 + 4T2rk2 = 0. (85) the closed-loop transfer function is given by (70). SinceTrk then by applying a2k = 2ak1ak+1 2ak2ak+2 toain

1kp(2kpTrk) + 2kprk2(2kpTrk)T = 0 (86)e calculus

Trk1 + rk = 0 (87)

Tni 4T p1

i=1Tni +

pi=1

Tni = 0. (88)

quation is rewritten in the form of

p3i=1

Tni 4TTnp p2

i=1Tni + Tnp

p1i=1

Tni = 0 (89)

3

1

Tni 4T p2

i=1Tni +

p1i=1

Tni

]Tnp = 0 (90)

e, since (83) holds by.ly, the number of combinations of the Tni optimal

that satisfy (90) is innite. More specically, by apply-n (90) for the design of up to type-V control loops results

ontrol loops:

n1Tn3 + Tn1Tn4 + Tn2Tn3 + Tn2Tn4 + Tn3Tn4 )T22Tn3 + Tn1Tn2Tn4 + Tn2Tn3Tn4 + Tn1Tn3Tn4 )T3Tn4 = 0. (91)


ed-lo

Type-IV

4(Tn1 + Tn2= 0.Type-III

4T2 4(Tn1Type-II c

Tn1 = 4T.Note that (In similar fsimplicity o

Tn1 = Tn2 =the respectare given by

Fol(s) 2np

and

T(s) 2np

The optimaloop we waconsequent

Type-V c

n2(n2 16nType-IV

nn2 12n +Type-III

(n2 8n + 4With respecPID type conTherefore, i

Tn1 = Tn2 =according t

Tn3 =4n(n

n2 Based on thtions are gi

[4n3(n

a5s5

n3(n

n2(nn

n3(n

n3T3

n2(nn

ing aj = bl sysntroncy re typeter mete

freqncy re th

S ben2 =s zerlar responFig. 7. Type-IV control loop. (a) Step and (b) frequency response of the nal clos

control loops:

+ Tn3 )T2 4(Tn1Tn2 + Tn2Tn3 + Tn1Tn3 )T + Tn1Tn2Tn3(92)

control loops:

+ Tn2 )T + Tn1Tn2 = 0. (93)ontrol loops:

(94)

93) and (94) are equal to (18) and (33) respectively.ashion with type-III control loops and for the sake off the analysis, if we choose

= Tnp1 = nT (95)ive open Fol(s) and closed-loop T(s) transfer functions

(1 + nTs)p11Tp

sp(1 + Ts)

(96)

(1 + nTs)p11Tp+1

sp+1 + 2np1Tp

sp + (1 + nTs)p1

. (97)

l value of parameter n depends on the type of the controlnt to design. If we substitute (95) into (92)-(94), we havely,ontrol loops:

+ 24)T4 = 0 nopt = 14.32. (98)control loops:

12)T3 = 0 nopt = 10.89. (99)control loops:

)T2 = 0 nopt = 7.46. (100)

Fol(s) =

T(s) =

where

b3 = 4

b1 = 2

and

a5 = 8

a3 = 4

a1 = 2

Accordsince, controtype cofrequeT of thparamof paraS in thefreque

NotsitivityTn1 = Ttrollern. Simistep re

t to the above, for the design of a type-IV control loop, atroller of three zeros in its transfer function is required.

f we chose

nT (101)

o (95), then from (92) it is obtained

2)8n + 4T =

4n(n 2)(n 0.536)(n 7.464)T. (102)

e above, the corresponding Fol(s) and T(s) transfer func-ven by

an oversho( = 32 < 45For decreastem, the twbe exploiteterms of tim

T(s) =D1(s

where

N1(s) = (1 +op control system for various values of parameter n.

2)T3s3 + n2(n2 12)T2

s22n2(n 6)Ts + (n 0.536)(n 7.464)]

8n3(n 2)T5s5 + 8n3(n 2)T4

s4

.

(103)

b3s3 + b2s2 + b1s + b0+ a4s4 + a3s3 + a2s2 + a1s + a0

(104)

2)T3, b2 = n2(n2 12)

n 2 T2 (105)

6) 2 T, b0 =

(n 0.536)(n 7.464)n 2 (106)

2)T5, a4 = 8n3T4 (107)

, a2 = n2(n2 12)

n 2 T2 (108)

6) 2 T, a0 =

(n 0.536)(n 7.464)n 2 . (109)

to (104), the closed-loop control system is of type-IVj, j = 0, 1, 2, 3, see Section 2. If n < 7.464 the closed-looptem is unstable. As a result, for having a feasible PIDl law, n > 7.464 has to hold by, see (105). In Fig. 7(b) theesponse of sensitivity S and complementary sensitivitye-IV closed-loop is presented for several variations ofn, n [7.5, ). From there, it is apparent that variationsr n do not lead to critical variations of both functions T,uency domain. Sensitivity S is affected only in the loweregion.at, in similar fashion with type-III control loops, sen-comes minimum when all controller zeros are equal,

Tn3 , n = 10.89, Fig. 10. There, it is shown how the con-os are affected in case of variations in design parametersults are also observed in the time domain, Fig. 7(a). These of the type-IV closed-loop control system exhibits

ot of 50%, which is justied by the phase margin) of the open-loop Fol(s) frequency response, Fig. 8.ing the overshoot of the nal closed-loop control sys-o degrees of freedom controller structure will again

d. If n = 10.89, then the closed-loop transfer function ine constants form is given by

N1(s))D2(s)D3(s)

(110)

10.89Ts)3, D1(s) = (1 + 2.3Ts) (111)


Fig. 8. Open-lparameter n.

Fig. 9. The effresponse of th

Fig. 10. Variatn.

D2(s) = (2.2

D3(s) = (14Thus, by ch

Cex(s) =(1

overshoot i

6. Simulation results

For justifying the controls law potential simulation examplesof type-II, III, IV, V control loops are presented. According to the

l lawlling

(1 +Tis

hreents ont Tcontrocontro

C(s) =

In all tconstaconstaoop frequency response of a type-IV control loop for various values of

ect of the two degrees of freedom controller structure to the stepe type-IV closed-loop control system.

ions of parameters Tn1 , Tn2 , Tn3 according to variations of parameter

74)2T2s2 + 0.99(2.274)Ts + 1 (112)

.75)2T2s2 + 1.9(14.75)Ts + 1. (113)

oosing an external controller of the form

+ 2.3Ts)[(14.75)2T2s2 + 1.9(14.75)Ts + 1](1 + 10.89Ts)3(1 + Ts)

(114)

s reduced to 14.75%, Fig. 9.

plants and loops are den has been is calculatethree cases

6.1. Process

The proc

G(s) =(1 +

is considereloop controis decreaseCex1 (s), Fig.Note that dexternal coof the contrif Cex2 (s) =the overshosystem is oboth ramp

6.2. Process

A delay

G(s) =(1 +

is assumed not take intthis exampis also testthe controluse of bothrespectivelyCex2 (s) is ofcontrol signFigs. 11(b) a

6.3. A non-

Althougexistence oprocess of t

G(s) =(1 + presented in Section 4 the I-I-PID type controller for a type-0 process is given by

Tn1s)(1 + Tn2s)(1 + Tn3s)3(1 + Tc1 s)(1 + Tc2 s)

. (115)

examples, it is assumed that the sum T of all timef the controlled process is accurately measured. Time=kj=1Tpj + Tc and Tc = Tc1 + Tc2 includes both

controllers unmodelled dynamics. Since type-III controlsigned Tn1 = Tp1 , Tn2 = 4(n1)n4 T, Tn3 = nT. Parameterchosen equal to n = 7.46. The integrators time constantd through Ti = 2kpkh TTm

p1r=1Tnr = 2kpkhTn2Tn3T. In all

Tm = 1 has been set.

with dominant time constants

ess described by

2 s)(1 + 0.84s)(1 + 0.78s)(1 + 0.57s)(1 + 0.28s)

(116)

d. From Fig. 11(a) it is apparent that the type-III closed-l system exhibits an undesired overshoot of 87.4% whichd by ltering the reference with an external controller

11. Settling time remains almost unaltered, tss = 143.isturbance rejection has remained the same since thentroller Cex1 (s) acts only at the reference signal outsideol loop. For manipulating the overshoot of the output,

1(tn2 tn3 )s

2+(tn2+tn3 )s+1reference lter is to be used, then

ot is decreased to 6.2%. Since the closed-loop controlf type-III, the output of the process can track perfectlyand parabolic reference signals, Fig. 12.

with time delay

process of the form

2 s)(1 + 0.99s)(1 + 0.57s)(1 + 0.28s)(1 + 0.1s) e

s

(117)

in this example. Note that the proposed control law doeso account the effect of the time delay and therefore inle the robustness of the method to model uncertaintiesed. If no external lter is used for reference tracking,

loop exhibits an overshoot of 100.4%, Fig. 13(a). The Cex1 (s), Cex2 (s) eliminates the overshoot to 9.4% and 0%, Fig. 13(a). Disturbance rejection remains unaltered.

the same form as in the previous example. Note thatal u() is improved in case the reference is ltered,nd 13(b).

minimum phase process

h the proposed theory does not take into account thef zeros in the process model, a non-minimum phasehe form

1.34(1 0.771s) s)(1 + 0.33s)(1 + 0.12s)(1 + 0.056s)(1 + 0.038s)

(118)


Fig. 11. Type- em an = 250. Extern ersho

and output dis

Fig. 12.

Fig. 13. Type- = 250. Extern

and output dis

is adopted The step reFig. 15(a) acontrol signis used, theundesirableis reduced ttion remaininto Si(s) = y

6.4. Contro

For testieter uncertaIII closed-loop control system. (a) Step response of the output of the control systal lter of the form Cex(s) = 1(0.45tn2 tn3 )s2+(tn2 +0.45tn3 )s+1 is used for decreasing the ovturbance di(s) = 0.1r(s) is applied at = 500.Type-III closed-loop control system. (a) Ramp response of the closed loop control system

III closed-loop control system. (a) Step response of the output of the control system anal lter of the form Cex1 (s) = 1(0.45tn2 tn3 )s2+(tn2 +0.45tn3 )s+1 is used for decreasing the overshoturbance di(s) = 0.1r(s) is applied at = 500.

for testing the robustness of the proposed control law.sponse of (118) is presented in Fig. 14. In addition, innd (b) the step response of the output y() and theal u() are presented respectively. If no external lter

overshoot of the step response is 59.9%. Since this is, if r(s) is ltered by Cex1 (s), Cex2 (s) then the overshooto 0% in both cases. Output and input disturbance rejec-

unaltered since the external lter does not participate(s)/di(s), So(s) = y(s)/di(s) respectively.

ller tuning without pole zero cancellation

ng the robustness of the proposed control law to param-inties, a type-III closed loop control system is designed

where the Therefore, pa is the erro

G(s) =(1 +

From Fig. 1measuring of the closedisturbanced (b) control signal u(). Output disturbance rejection is applied atot of the output. Input disturbance di(s) = 0.1r(s) is applied at = 250 and (b) parabolic response of the closed-loop control system.

d (b) control signal u(). Output disturbance rejection is applied atot of the output. Input disturbance di(s) = 0.1r(s) is applied at = 250

PID controller does not achieve pole-zero cancellation.arameter Tn1 is determined by Tn1 = (1 + a)Tp1 wherer when measuring Tp1 . The process is given by

1.23 s)(1 + 0.872s)(1 + 0.367s)(1 + 0.287s)(1 + 0.11s) .

(119)

6(a) and (b) it is apparent that if an error of 30% whenTp1 occurs, a small change is observed in the overshootd loop control system. In addition, both input and output

rejection remain almost unaltered.


Fig. 14. Step response of the non-minimum phase process dened by (118).

6.5. Comparison between a type-I and a type-III control loop

For showing the advantages of designing a higher order fastercontrol loop, the following process

G(s) = 1.23(1 + s)(1 + 0.992s)(1 + 0.692s)(1 + 0.139s)(1 + 0.107s)

(120)

is adopted. For this process, a type-I, III closed control loop will bedesigned. For designing the PID type-I control loop the conventionalMagnitude Optimum criterion (see Appendix D) is employed. Notethat for determining controllers zeros, exact pole zero cancellationhas to take place (see Appendix D) [28]. From Fig. 17 it is apparent

that the type-I control loop fails to track both the ramp and theparabolic reference signal achieving constant non-zero steady statevelocity and acceleration error.

6.6. A type-IV and a type-V control loop

From the Laplace transformation it is known that if r(t) = tn thenL{y(t)} = n!/sn+1. Specically, if n = 1 then L{r(t)} = 1/s2 and the sys-tem is of type-II, or if n = 2 then L{r(t)} = 2/s3 and the system is oftype-III. For a type-IV and type-V control loop the Laplace transfor-mation of the reference signal is given if n = 3 and n = 4 for whichwe have L{r(t)} = 3 !/s3+1 and L{r(t)} = 4 !/s4+1 respectively. Accord-ing to the proposed theory for a type-IV, V control loop the proposedPID type controllers are given by

C(s) = (1 + Tn1s)(1 + Tn2s)(1 + Tn3s)(1 + Tn4s)Tis4(1 + Tc1 s)(1 + Tc2 s)

. (121)

C(s) = (1 + Tn1s)(1 + Tn2s)(1 + Tn3s)(1 + Tn4s)(1 + Tn5s)Tis5(1 + Tc1 s)(1 + Tc2 s)

(122)

respectively. For determining parameters Tn1 , Tn2 , Tn3 , Tn4 , Ti in(121) according to the proposed theory, we set Tn4 = Tp1 and Tn1 =Tn2 = nT according to (95). For that reason, (92) becomes4(2nT + Tn3 ) 4(n2T + 2nTn3 ) + n2Tn3 = 0 (123)or nally

Tn3 =4n(n 2)

(n2 8n + 4)T. (124)

Integrators time constant for the type-IV control loop is equal to

Ti = 2kpkhTn1Tn2Tn3T. (125)

Fig. 15. Type- se of disturbance re

tn3 )s+1di(s) = 0.1r(s) is

Fig. 16. Type-cancellation a III closed-loop control system for a non-minimum phase process. (a) Step responjection is applied at = 200. External lter of the form Cex1 (s) = 1(0.45tn2 tn3 )s2+(tn2 +0.45

applied at = 200 and output disturbance di(s) = 0.1r(s) is applied at = 300.III closed-loop control system. The PID controller is tuned without pole zero cancellation= 0.the output of the control system and (b) control signal u(). Outputis used for decreasing the overshoot of the output. Input disturbance

: a = 0.3 and a = 0.3. The PID controller is tuned via exact pole-zero


Fig. 17. Comp the rasteady state ve

Fig. 18. (a) Re en chosen equal to n = 10.89 according to (99). (b) Response of the type-Vcontrol loop to g to (99).

In similar fcontrol loonT. Accord

4(3n2T2 + 3

and after so

Tn4 =4n

n(n2

Integrators

Ti = 2kpkhTThe controrespective rtype-IV and

6.7. Effect operformanc

The effethis examp

G(s) =(1 +

is adopted. depends onmodels thefar from thTp is the pthe processdominant tarison between a type-I, III PID control loop. The type-I control loop fails to track locity and acceleration error is observed.

sponse of the type-IV control loop to reference signal r(t) = t3; parameter n has be reference signal r(t) = t4; parameter n has been chosen equal to n = 14.32 accordin

ashion, for the (122) PID type controller and since thep is of type-V, we set Tn5 = Tp1 and Tn1 = Tn2 = Tn3 =ingly, (91) becomes

3 3 2 2 3 2nTTn4 )T 4(n T + 3n TTn4 ) + n TTn4 = 0(126)

me calculus results in2(n 3)

12n + 12)T =4n(n 3)

n2 12n + 12T. (127)

time constant for the type-V control loop is equal to

n1Tn2Tn3Tn4T. (128)

lled process in this example is dened by (120). Theesponse to r(t) = t3 and r(t) = t4 reference signals for the

the type-V control loop are presented in Fig. 18.

f the process unmodelled dynamics to the controle

ct of the process unmodelled dynamics is discussed inle. The process dened by

1 s)(1 + as)(1 + a2s)(1 + a3s)(1 + a4s) (129)

As proved in Sections 4 and 5 the proposed control law pole-zero cancellation and time constant T which

process unmodelled dynamics (poles of the processe origin), see (32) and (34) where T = Tc + Tp androcess parasitic time constant and Tc Tp . In Fig. 19

is modelled by a = 0.15 containing a relatively largeime constant and in the next case where a = 0.6 the

Fig. 19. Step rprocess dene

parasitic tim

nant time c

a = 0.15 the

Tp = Tp1

is that thezeros, time(T Tpj ), tof the outp

7. Conclus

The Symdesign of tmp r() = and the parabolic r() = 2 reference signal since constantesponse of the PID type-III control loop when a = 0.15 and a = 0.6 for ad by (129).

e constant of the process is comparable to its domi-

onstant. SinceTpTp1

=4

j=1aj , it is apparent that when

n Tp = Tp14

j=1aj = 0.1764Tp1 and when a = 0.6 then

4j=1a

j = 1.3056Tp1 . The conclusion according to Fig. 18 less accurate the model of the process in terms of

delay, poles compared to the dominant time constanthe poorer the performance becomes (see settling timeut and input disturbance rejection Fig. 19).

ions and discussion

metrical Optimum criterion has been extended for theype-p control loop. Based on the conventional tuning


for PID type controllers via the Symmetrical Optimum principle, asimilar design technique for type-III control loops was proposed. Itwas shown that type-III control loop achieve zero steady state posi-tion, velocity and acceleration error and therefore they are able totrack faster reference signals than type-I or II control loops. Basedon this technique, the proposed control law was extended for tun-ing PID typsignals is acarried out the processunmodelleddelay constsince nowafor most indated for theapplicationpromising rthe control and process

Acknowled

The auththree anonypeer review

Appendix A

Let the c

T(s) = bmsm

ans

where m (A.1) we wiThus, by set

|T(j)|2 = ||

or

T(j) = N(jD(j

Polynomial

N(j) + b0 + j(

and

D(j) + a55

or

|D(j)|2 (

+ (a25 + 2a

2a1a7

+ (a22 + 2a

6 Time delay

series esd =

and

|N(j)|2 (b28)16 + (b27 b8b6)14 + (b26 + 2b4b8 2b5b7)12

+ (b25 + 2b b 2b b 2b b )10 + (b2 + 2b b + 2b b

2b1b7

+ 2b, |T(j

|2 =

king 2, |Nesul

2a0

3a1

a1a5

a0a8

+ 2b

dix B via

the i

sTm(

Tm, T cont

sTi(1

ied, t

s2TiT

TpT

s4TiT

ing tissi

1 sTi(1

loyedtionust t

s4TiT

is un, PIDoles come-p control loops so that tracking of faster referencechieved. The development of the proposed control isin the frequency domain where the transfer function of

involves the dominant time constants and the plants dynamics. Future work deals with introducing the time

ant6 as one more parameter in the proposed control law,days the time delay is straightforward to be measuredustrial processes. The proposed theory has been evalu-

control of representative plants met in many industrys. The robustness of the proposed control law achievesesults also for the control of processes with parameterslaw disregards, such as non-minimum phase processeses with time delay.

gements

ors would like to express their greatful thanks to themous reviewers for their valuable feedback during the

process.

. Optimization conditions

losed-loop transfer function be dened by (A.1),

+ bm1sm1 + + b2s2 + b1s + b0n + an1sn1 + + a2s2 + a1s + a0

= N(s)D(s)

(A.1)

n. By applying the Symmetrical Optimum criterion toll force |T(s)| 1 in the wider possible frequency range.ting s = j into (A.1) and squaring |T(j)| leads to

N(j)|2

D(j)|2(A.2)

))

= (j)mbm + + (j)2b2 + (j)b1 + b0

(j)nan + + (j)2a2 + (j)a1 + a0. (A.3)

s N(j) and D(j) are rewritten as follows

+ b88 b66 + b44 b22

b77 + b55 b33 + b1) (A.4)

+ a88 a66 + a44 a22 + a0 + j( a77

a33 + a1) (A.5)

a28)16 + (a27 a8a6)14 + (a26 + 2a4a8 2a5a7)12

3a7 2a2a8 2a4a6)10 + (a24 + 2a0a8 + 2a2a6 2a3a5)8 + (a23 + 2a1a5 2a6a0 2a2a4)6

0a4 2a1a3)4 + (a21 + 2a0a2)2 + (a0)0 (A.6)

constant can be introduced in the process model (38) by the Taylork=0

(1k! s

kdk).

+ (b22Finally

|T(j)

By ma|D(j)|range r

a0 = b0a21 2aa22 2aa23 + 2

a24 + 2= b24

=

Appentuning

Let

G(s) =

where(B.1), I

C(s) =

is appl

T(s) =

where

T(s) =

Accordof s is m

C(s) =

is empconvention m

T(s) =

which Finallyplex pT(s) be3 7 2 8 4 6 4 0 8 2 6

2b3b5)8 + (b23 + 2b1b5 2b6b0 2b2b4)6

0b4 2b1b3)4 + (b21 + 2b0b2)2 + (b0)0. (A.7))|2 is equal to

|N(j)|2

|D(j)|2= + B4

8 + B36 + B24 + B12 + B0 + A48 + A36 + A24 + A12 + A0

.

(A.8)

equal the terms of j (j = 1, 2, . . ., n) in polynomials(j)|2 so that |T(s)| 1 in the wider possible frequencyts in

(A.9)

= b21 2b2b0 (A.10)+ 2a4a0 = b22 2b3b1 + 2b4b0 (A.11) 2a6a0 2a4a2 = b23 + 2b1b5 2b6b0 2b4b2 (A.12)

+ 2a6a2 2a1a7 2a3a5

0b8 + 2b6b2 2b1b7 2b3b5 (A.13)

. Instability of the PI control conventionalthe Symmetrical Optimum criterion

ntegrating process be dened by

11 + sTp1 )(1 + sTp)

, (B.1)

p1 , Tp have been dened in Section 3. If for controllingrol of the form

1 + sTc)

, (B.2)

hen the closed loop transfer function is given by

kp

m(1 + sTp1 )(1 + sT) + khkp(B.3)

c 0 and T = Tp + Tc. From (B.3) it is evidentkp

mTp1T + s3TiTm(Tp1 + T) + s2TiTm + khkp. (B.4)

o (B.4), it is evident that T(s) is unstable since the termng. In similar fashion, if PI control of the form

+ sTn + sTc)

(B.5)

, then for determining controller parameter Tn via theal Symmetrical Optimum criterion, pole-zero cancella-ake place, Tn = Tp1 . Therefore, T(s) becomes

kp

mTp1T + s3TiTm(T + Tp1 ) + s2TiTm + khkp, (B.6)

stable again for the same reason as stated for (B.4). control by cancelling two real or conjugate com-of G(s) cannot be applied, since it is proved that

es unstable for the same reason as for (B.4). This is


justied by the Routh theorem. For a polynomial of the formD(s) = ansn + an1sn1 + + a1s + a0, necessary and sufcient condi-tion for D(s) to be stable is aj > 0, j = 0, 1, 2, . . .. Since both in (19),(16) and (B.6), a3 = 0 and a1 = 0 then according the Routh theorem,D(s) is unst

Appendix C

In SectioIV, III contro

4(Tn1Tn2

4(Tn1+Tn1Tn2T

4(Tn1 + T+Tn1Tn

4T2 4(Tn1By substitu

4(n2T2 + n2

+ n3T3 +

4(nT + nT

4T2 4(nTor

24n2T4 1

12nT3 12

4T2 8nT2Since T /=respectively

Appendix Dcriterion

For cont

G(s) =(1 +

where T2pics, the PID

C(s) = (1 +T

is adopted. terion paramthe widest p

T2 = T2psystem acco

T(s) =sTi(1

By forcing p

Tn1 = Tp1 ,

By substituting (D.4) into (D.3) and calculating |T(j)| results in

|T(j)|

k2p

T2T 4 + (T 2kpk T )T 2 + k2k2. (D.5)

ore, c

, Ti

pkh(T

stitu

2T22

rmal

2s2

nces

Morar1989.

stront Soc

Marg. Papallers fferen. Ang,E Tran

Ziegles of t. Cohtrol, Tourtio

ves, Au Dwyss, 20

stro001) oron, omatkoczo

the ro(6) (20koges, Journ

stroomat

stro for P714uo, Y.

DCD722andyode conIndustummst co328. ZuriompaDC Bnsactiim, R

tem, Iummst co328B. Shin

variab (2012

strok, IEE

Isaksstem, Aaya, C

phasable.

. Proof of the parameter n

n 5 it was shown that zeros of the controller for type-V,l loops are given by

+ Tn1Tn3 + Tn1Tn4 + Tn2Tn3 + Tn2Tn4 + Tn3Tn4 )T2Tn2Tn3 + Tn1Tn2Tn4 + Tn2Tn3Tn4 + Tn1Tn3Tn4 )Tn3Tn4 = 0

(C.1)

n2 + Tn3 )T2 4(Tn1Tn2 + Tn2Tn3 + Tn1Tn3 )T2Tn3 = 0.

(C.2)

+ Tn2 )T + Tn1Tn2 = 0. (C.3)ting (95) into (C.1)(C.3) results in

T2 + n2T2 + n2T2 + n2T2 + n2T2)T2 4(n3T3 n3T3 + n3T3)T + n4T4 = 0, (C.4)

+ nT)T2 4(n2T2 + n2T2 + n2T2)T + n3T3 = 0,(C.5)

+ nT)T + n2T2 = 0 (C.6)

6n3T4 + n4T4 = 0, (C.7)

n2T3 + n3T3 = 0, (C.8)

+ n2T2 = 0. (C.9) 0, from (C.6)(C.9) we obtain (98), (99) and (100).

. The conventional Magnitude Optimum

rolling the process dened by

1 sTp1 )(1 + sTp2 )(1 + sT2p )

, (D.1)

=n

i=3Tpi stands for the process unmodelled dynam-controller of the form

Tn1s)(1 + Tn2s)is(1 + Tc s)

(D.2)

According to the conventional Magnitude Optimum cri-eters Tn1 , Tn2 , Ti will be determined so that |T(j)| 1 in

ossible frequency range. Assuming that Tc T2p and+ Tc , the transfer function of the closed loop controlrding to Fig. 1 is equal to

kp(1 + sTn1 )(1 + sTn2 ) + sTp1 )(1 + sTp2 )(1 + sT2 ) + khkp(1 + sTn1 )(1 + sTn2 )

.

(D.3)

ole zero cancellation according to

Tn2 = Tp2 . (D.4)

Theref

kh = 1= 2k

By sub

T(s) =

and no

T(s) =

Refere

[1] M. NJ,

[2] K.J.me

[3] N.I.[4] K.G

troCon

[5] K.HIEE

[6] J.G.tion

[7] G.Hcon

[8] B. Cdri

[9] A. OPre

[10] K.J.9 (2

[11] L. LAut

[12] S. Sing52

[13] S. Sing

[14] K.J.Aut

[15] K.J.tion699

[16] L. Gfor223

[17] B. Bmoon

[18] V. Mboo288

[19] E.WA cDCTra

[20] K. Ksys

[21] V. Mboo288

[22] H.-for(3)

[23] K.J.boo

[24] A.J.sys

[25] I. Kandi 2 i h 2 i p h

ondition |T(j)| 1 is satised when= 2kpkhT2 = 2kpkh(T Tp1 Tp2 )

Tn1 Tn2 ). (D.6)ting (D.4) and (D.6) into (D.3) results nally

1

s2 + 2T2s + 1(D.7)

izing the time by setting s = sT2 leads to1

+ 2s + 1 . (D.8)

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Extending the Symmetrical Optimum criterion to the design of PID type-p control loops1 Introduction2 Definitions and preliminaries3 The conventional Symmetrical Optimum design criterion4 Extending the Symmetrical Optimum design criterion to type-III control loops5 Extending the Symmetrical Optimum design criterion to type-p control loops6 Simulation results6.1 Process with dominant time constants6.2 Process with time delay6.3 A non-minimum phase process6.4 Controller tuning without pole zero cancellation6.5 Comparison between a type-I and a type-III control loop6.6 A type-IV and a type-V control loop6.7 Effect of the process unmodelled dynamics to the control performance

7 Conclusions and discussionAcknowledgementsAppendix A Optimization conditionsAppendix B Instability of the PI control conventional tuning via the Symmetrical Optimum criterionAppendix C Proof of the parameter nAppendix D The conventional Magnitude Optimum criterionReferences

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