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Published in IET Control Theory and ApplicationsReceived on 19th December 2010Revised on 2nd May 2011doi: 10.1049/iet-cta.2010.0746

ISSN 1751-8644

Note on fractional-order proportional–integral–differential controller designC. Yeroglu1 N. Tan2

1Computer Engineering Department, Inonu University, Malatya 44280, Turkey2Electrical and Electronics Engineering Department, Inonu University, Malatya 44280, TurkeyE-mail: [email protected]

Abstract: This study deals with the design of fractional-order proportional–integral–differential (PID) controllers. Two designtechniques are presented for tuning the parameters of the controller. The first method uses the idea of the Ziegler–Nichols and theAstrom–Hagglund methods. In order to achieve required performances, two non-linear equations are derived and solved to obtainthe fractional orders of the integral term and the derivative term of the fractional-order PID controller. Then, an optimisationstrategy is applied to obtain new values of the controller parameters, which give improved step response. The second methodis related with the robust fractional-order PID controllers. A design procedure is given using the Bode envelopes of thecontrol systems with parametric uncertainty. Five non-linear equations are derived using the worst-case values obtained fromthe Bode envelopes. Robust fractional-order PID controller is designed from the solution of these equations. Simulationexamples are provided to show the benefits of the methods presented.

1 Introduction

In recent studies, many researchers use fractional calculus inthe control system applications [1–5]. The significance offractional-order representation comes from its nature. Thefractional-order differential equations can describe real-world systems more adequately [6]. Furthermore, theimplementations of fractional-order differential equations[7, 8] have brought in new horizons in control engineering.Consequently, many studies have been done for thefractional-order control systems (FOCSs) [9–14].

Widespread usage of the PID controllers motivated manyresearchers to look for better design methods or alternativecontrollers [15, 16]. For example, the fractional-orderalgorithm for the control of dynamic systems has beenintroduced and the performance of CRONE (Frenchabbreviation for Commande Robuste d’Ordre Non Entier),over the PID controller, has been demonstrated byOustaloup [17, 18]. Podlubny has proposed a generalisationof the PID controller as PIlDm controller. He alsodemonstrated that the fractional-order PID (PIlDm)controller has better response than classical PID controller[1, 19]. Also, many valuable studies have been done forfractional-order controllers and their implementations [13,20–25]. Tuning of the PIlDm controller using thefrequency-domain approaches are studied in many papers.For example, [26] proposes a method based on optimisationstrategies. Tuning of H1 controllers for fractional single-input single-output (SISO) system was suggested in [23]. Anew design method for PIa controller is given in [27].Some tuning rules for robustness to plant uncertainty forPIl controller are given in [28]. However, in order to

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achieve better results, there are still needs for new methodsto obtain the parameters of PIlDm controllers.

On the other hand, it is known that the parameters ofphysical systems cannot be expressed precisely and containuncertainty due to the tolerance values of elements, non-linear effects or environmental conditions. Therefore theparameter uncertainty is inevitable in the systems [5]. Ingeneral, system uncertainties are analysed in two groupssuch as parameter uncertainty and model uncertainty. Incontrol theory, robust control methods have been developedfor the analysis and design of uncertain systems. Theseissues are addressed under robust control [29]. Thecomputation of the frequency responses of uncertaintransfer functions plays an important role in the applicationof frequency-domain methods for the analysis and design ofrobust control systems [30, 31]. Many studies on thecomputation of the frequency responses of the integer ordercontrol systems with parameter uncertainty structure can befound in the literature [29, 32, 33]. However, in order toapply classical controller design method to FOCSs withparameter uncertainty structure, it is necessary to computethe frequency responses of a given fractional-order intervaltransfer function (FOITF). The procedures for thecomputation of the Bode and Nyquist envelopes of FOITFcan be found in [6, 34]. The structures of PIlDm controllerhave been widely used in recent papers. However, there isno enough study for parameter uncertainty and robustcontrol design issues [13, 28]. Therefore the resultsobtained in this study provide an important contribution tothis field.

In this paper, two methods have been proposed for tuningof the PIlDmcontroller. In the first method, a tuning technique

IET Control Theory Appl., 2011, Vol. 5, Iss. 17, pp. 1978–1989doi: 10.1049/iet-cta.2010.0746

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for a PIlDm controller, which is inspired from the classicalZiegler–Nichols and Astrom–Hagglund tuning methods, isintroduced. The proposed method uses the classicalZiegler–Nichols tuning rules to obtain the values of kp andki. The value of kd is obtained using the Astrom–Hagglundmethod. In order to achieve specified phase margin, twonon-linear equations have been obtained using the idea ofthe Astrom–Hagglund tuning method. The values of l andm are obtained from these two equations. In the secondmethod, a tuning strategy, which is based on the Bodeenvelopes of the FOITF, is introduced for robust PIlDm

controller to control the first-order and first-order plus deadtime (FOPDT) systems with the parametric uncertaintystructure. In this method, the Bode envelopes of the systemare successfully combined with five design criteria, whichMonje–Vinagre et al. have used in their papers [13, 21,28], to obtain new robust PIlDm controller that make thegiven plant robust under parameter uncertainties. Thus, thenovelty of the results obtained in this paper is thedevelopment of a new tuning method for PIlDm controllerand the presentation of a new method to design a robustPIlDm controller. The method presented for the robustPIlDm controller is an extension of the Monje–Vinagreet al. method, which is given in Section 4.1. Theimprovement over those in [13, 21, 28] is that the parameteruncertainty has been considered. Then an improved robustmethod is obtained. Examples are provided to illustrate theresults.

The paper is organised as follows: Mathematicalbackground of fractional-order representation is given inSection 2. In Section 3, a tuning method for PIlDm

controller is introduced. Design of robust PIlDm controllersfor first-order and FOPDT systems with parametricuncertainty structure is provided in Section 4. Section 5includes concluding remarks.

2 Mathematical background

Fractional calculus can be considered to be generalisation ofintegration and differentiation of the integer orderexpressions to the non-integer order one. The mostfrequently used integro-differential definitions areGrunwald–Letnikov, Riemann–Liouville (RL) and Caputoexpressions. The Grunwald–Letnikov definition of thefractional-order derivative is given by the followingequation [35]

aDrt f (t) = lim

h�0h−r

∑[t−a/h]

j=0

(−1)j rj

( )f (t − jh) (1)

where (−1)j rj

( )are the binomial coefficients c(r)

j , ( j ¼ 0,

1, . . .). Following expressions can be used to obtain thecoefficients [36]

c(r)0 = 1, c(r)

j = 1 − 1 + r

j

( )c(r)

j−1 (2)

Riemann–Liouville definition can be given as

aDrt f (t) = 1

G(n − r)

dn

dtn

∫t

a

f (t)

(t − t)r−n+1 dt (3)

where n 2 1 , r , n and G(.) is a Gamma function.


Fractional-order Caputo expression can be given as

aDrt f (t) = 1

G(r − n)

∫t

a

f (n)(t)

(t − t)r−n+1 dt (4)

where n 2 1 , r , n. The Gamma function G(m) can bedefined for a positive real m as follows [35]

G(m) =∫1

0

e−uum−1 du (5)

Numerical solutions for Grunwald–Letnikov, Riemann–Liouville and Caputo expressions can be obtained usingdefinitions given in [1, 35–37].

Generally, dynamic behaviours of the systems can beanalysed using transfer function of the control system.Thus, the Laplace transformations of the integro-differential expressions for FOCSs are important.Fortunately, there is not big difference between the Laplacetransformation of the fractional-order expression and that ofthe integer order. The most general formula for the Laplacetransformations of the integro-differential expressions canbe given as [37]

Ldmf (t)

dtm

{ }= smL{f (t)} −

∑n−1

k=0

sk dm−1−k f (t)

dtm−1−k

[ ]t=0

(6)

where n is an integer number and m satisfies, n 2 1 , m , n.The above expression is simplified as follows if all thederivatives of f (t) are zero

Ldmf (t)

dtm

{ }= smL{f (t)} (7)

Consider a SISO control system. Let y(t) be the output andx(t) be the input of the system. The relation between inputand output of the system can be defined as

an

dan y(t)

dtan+ an−1

dan−1 y(t)

dtan−1+ · · · + a0

da0 y(t)

dta0

= bm

dbm x(t)

dtbm+ bm−1

dbm−1 x(t)

dtbm−1+ · · · + b0

db0 x(t)

dtb0(8)

Transfer function of the system can be obtained as follows bytaking the Laplace transform of the above equation [6]

G(s) = Y (s)

X (s)= bmsbm + bm−1sbm−1 + · · · + b0sb0

ansan + an−1san−1 + · · · + a0sa0(9)

where an . an21 . . . . . a0 ≥ 0, bm . bm21 . . . . .b0 ≥ 0, ak (k ¼ 0, 1, 2, . . . , n) and bl (l ¼ 0, 1, 2, . . . , m)are constants [6, pp. 285–290]. Substituting s ¼ jv in thetransfer function of the control system, frequency-domainanalysis of the FOCS can be studied. Since sm ¼ ( jv)m, theexpression for ( jv)m can be given as [4]

(jv)m = vm cosp

2m+ j sin

p

2m

( )(10)

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3 Tuning method for PIlDm controller

This section presents the development of a tuning method forthe PIlDm controller, which is inspired from the Ziegler–Nichols and Astrom–Hagglund tuning methods for thesystems which have integer order transfer functions. Theproposed method uses the Ziegler–Nichols tuning rules toobtain the values of kp and ki. The initial value of kd isobtained using the Astrom–Hagglund method. In order toachieve specified phase margin, two non-linear equationshave been obtained using critical point information, namelycritical frequency vc and critical gain kc, in the similarmanner as the Astrom–Hagglund tuning method. Finetuning has been done for kd to achieve the best numericalsolutions of these two equations. The values of l and m areobtained from these equations using the ‘fsolve’optimisation toolbox of MATLAB. Fine tuning of thecontroller parameters may be required to achieve better stepresponse of the system. In that case, an optimisation model,which has been developed using Simulink MATLAB, isused. The controller parameters obtained by the proposedmethod are chosen as initial values for optimisation. Thenthe new values for the controller parameters are producedusing the optimisation model. Preliminary study of thissection has been presented in the conference [38].

3.1 Computation of PIlDm controller parameters

Consider the negative unity feedback control system shown inFig. 1. Transfer function of the plant is an integer order.However, the controller of the system is a PIlDm controllerof the form

C(s) = kp +ki

sl+ kdsm (11)

The proportional gain constant kp, the constant of integralterm ki, the constant of derivative term kd, the fractional-order of differentiator l and the fractional order ofintegrator m of the controller C(s) can be obtained using theproposed method.

Let fpm be the required phase margin and vcp be thefrequency of the critical point on the Nyquist curve of G(s)at which

arg(G(jvcp)) = −1808 (12)

and define gain margin as

gm = 1

|G(jvcp)| = kc (13)

Then, in order to make the phase margin of the system equalto fpm and |C( jvcp)G( jvcp)| ¼ 1, the following equationmust be satisfied

C(jvcp) = 1

|G(jvcp)| e jfpm = kc cos fpm + jkc sin fpm (14)

Fig. 1 Negative unity feedback system

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One can write C( jvcp) using (11) and (14) as

C(jvcp) = kp + kiv−lcp cos

p

2l

( )+ kdv

mcp cos

p

2m

( )+ j −kiv

−lcp sin

p

2l

( )+ kdv

mcp sin

p

2m

( )[ ](15)

Considering (14) and (15), one can calculate the followingequations

f1(l, m) = kp + kiv−lcp cos

p

2l

( )+ kdv

mcp cos

p

2m

( )− kc( cos fpm)

= 0 (16)

f2(l, m) = −kiv−lcp sin

p

2l

( )+ kdv

mcp sin

p

2m

( )− kc( sin fpm)

= 0 (17)

The numerical solutions for l and m can be found using (16)and (17).

3.2 Tuning method for PIlDm controller

All the parameters of the PIlDm controller, which are given in(11), can be obtained by using the following procedure:

† Specify the value of required phase margin fpm.† Obtain kp and ki from the Ziegler–Nichols tuning rules.† Obtain (16) and (17).† Specify the initial value for kd using the Astrom–Hagglund method.† Simulation results show that especially variation in kd

affects the numerical solution of the equations seriously.Therefore fine tuning can be required for kd to achieve thebest numerical solution for (16) and (17).† Find the numerical solutions for l and m from (16) and(17), considering new value of kd.† If the step response of the system is not satisfactoryenough, an optimisation can be done by using theoptimisation model to obtain better values for the controllerparameters.

3.3 Application of the proposed tuning method

Example 1: Consider the negative unity feedback system inFig. 1. The transfer function of the system is given as

G1(s) = 1

s(s + 3)(s + 4)(18)

The phase crossover frequency and the gain margin of thesystem can be obtained, respectively, as vcp ¼

p12 and

kc ¼ 84. Constants of proportional, integral and derivativeterms of the controller are obtained by using the Ziegler–Nichols rules as kp ¼ 50.40, ki ¼ 55.60 and kd ¼ 10.96. LetPID controller obtained from Ziegler–Nichols method beC1ZN(s) which is defined as follows

C1ZN (s) = 50.40 + 55.60

s+ 10.96 s (19)

Using the Astrom–Hagglund method, the values of PIDcontroller parameters have been calculated for specified


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phase margins which are shown in Table 1. Let C1AH(s)shows the PID controller obtained from the Astrom–Hagglund method for fpm ¼ 308, which can be writtenfrom Table 1 as

C1AH (s) = 72.7461 + 72.7461

s+ 18.1865 s (20)

The proposed method takes the values of kp and ki fromZiegler–Nichols method. The initial values for derivativeterm kd have been found by using the Astrom–Hagglundmethod for the specified phase margins. Fine tuning hasbeen done for the term kd to achieve the best numericalsolution of (16) and (17) for each specified phase margin.These two equations have been solved by usingoptimisation toolbox ‘fsolve’ of the MATLAB to obtainnumerical values of l and m by considering the new valueof kd for each specified phase margin. Table 2 shows all thevalues of kp, ki, kd, l and m obtained by the proposedmethod for each of the specified phase margin. Let C1(s)shows the desired PIlDm controller designed by theproposed method, which can be written from Table 2 forfpm ¼ 308 as

C1(s) = 50.4 + 55.6

s0.7569+ 22 s0.8564 (21)

In order to obtain better step response, an optimisation modelhas been developed using Simulink by considering leastsquare method for optimisation. This optimisation modelhas been used to obtain new optimised values for theparameters kp, ki, kd, l and m. Table 3 shows theoptimisation results for different initial values, which aretaken from Table 2. Let C1opt(s) shows the PIlDm controllerwith optimised values, which can be written as

C1 opt(s) = 42.4580 + 87.2733

s0.5030+ 53.1352s0.9623 (22)

Consequently, four types of controllers, namely C1ZN(s),

Table 1 Values of PID controller parameters calculated by

Astrom–Hagglund method for fpm ¼ 308, 408, 508 and 608

fpm Astrom-Hagglund

kp ki kd

308 72.7461 72.7461 18.1865

408 64.3477 51.9716 19.9177

508 53.9942 34.0387 21.4122

608 42.0000 19.4923 22.6244

Table 2 Values of PIlDm controller parameters calculated by

proposed method for fpm ¼ 308, 408, 508 and 608

fpm Proposed method

kp ki kd l m

308 50.4 55.6 22.000 0.7569 0.8564

408 50.4 55.6 24.000 0.9714 0.8823

508 50.4 55.6 24.050 0.9762 0.9766

608 50.4 55.6 23.935 0.9208 1.0744


C1AH(s), C1(s) and C1opt(s), have been designed for thegiven plant as follows:

† Parameters of C1ZN(s) are calculated using the Ziegler–Nichols method, such as kp ¼ 50.40, ki ¼ 55.60 andkd ¼ 10.96 for all specified phase margin.† Parameters of C1AH(s) are calculated using the Astrom–Hagglund method, for specified phase margins as shown inTable 1.† Parameters of C1(s) are obtained using the proposedmethod for the specified phase margins as shown in Table 2.† Optimisation model has been used to obtain better stepresponse for the controller C1(s). The controller C1opt(s) isobtained with the new optimised values of kp, ki, kd, l andm as given in Table 3.

It is known that the step response of a system givesvaluable information, such as maximum overshoot, risetime, peak time and settling time. The step responses of thesystem for the controllers C1ZN(s), C1AH(s), C1(s) andC1opt(s) in (19)–(22) are obtained using the ‘nintblocks’ ofMATLAB, which is developed by Duarte Valerio [39] asshown in Fig. 2. The performance specifications for thesecontrollers are given in Table 4. One can conclude fromFig. 2 and Table 4 that the performance specifications ofthe proposed method are much better than the Ziegler–Nichols and Astrom–Hagglund tuning methods.

Bode plots and Nyquist plots of the system for the C1(s) forphase margins fpm ¼ 308, 408, 508, 608 and Nyquist plot ofthe system for C1opt(s) are given in Figs. 3–5, respectively.Looking at the Figs. 3 and 4, it can be observed that thesystem satisfies each of the specified phase margin forC1(s). Fig. 5 shows that the values of the gain and phasemargins of the system for C1opt(s) are suitable.

Table 3 Optimised values of PIlDm controller parameters where

initial values are taken from Table 2 for fpm ¼ 308, 408, 508 and 608

Initial values

form Table 2

Proposed method with optimised values

kp ki kd l m

for fpm ¼ 308 42.4580 87.2733 53.1352 0.5030 0.9623

for fpm ¼ 408 41.2422 89.5344 54.6636 0.5003 0.9570

for fpm ¼ 508 41.0156 95.4288 55.3521 0.4975 0.9639

for fpm ¼ 608 41.7490 88.8996 53.8169 0.4975 0.9623

Fig. 2 Step responses of the system for C1ZN(s), C1AH(s), C1(s) andC1opt(s)

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Table 4 Step response specifications of C1ZN(s), C1AH(s) and C1opt(s)

Step response

specifications

Ziegler–Nichols

PID

Astrom–Hagglund

PID

Proposed

fractional PID

Proposed fractional

PID with optimised values

max. overshoot, % 73.5 59.0 52.5 31.5

peak time, s 2.35 2.10 1.83 1.47

rise time, s 1.73 1.65 1.47 1.31

settling time (%5) 6.02 3.20 3.75 2.60

settling time (%2) 7.30 4.25 3.98 2.83

4 Design of robust PIlDm controller

This section presents the development of a tuning method forrobust PIlDm controller for the first-order and FOPDTsystems with parameter uncertainty structure. All parametersof the PIlDm controller are calculated to satisfy robustperformances of the plant. Five unknown parameters of thePIlDm controller are estimated solving five non-linearequations that satisfy five design criteria. The proposedmethod is an extension of Monje–Vinagre et al. tuningmethod [13, 21, 28] for the first-order and FOPDT systems

Fig. 3 Bode plots of the system controlled with C1(s) for the phasemargins fpm ¼ 308, 408, 508 and 608

Fig. 4 Nyquist plots of the system controlled with C1(s) for thephase margins fpm ¼ 308, 408, 508 and 608

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with the parameter uncertainty structure. Bode envelopes ofthe first-order and FOPDT systems with parameteruncertainty structure are successfully combined with fivedesign criteria to obtain the robust PIlDm controller.

As known, FOPDT systems provide simplecharacterisation of a process and give valuable informationabout dynamics of many applications in process controlindustry. Since the plants are commonly modelled withFOPDT transfer functions in the process industry, most ofthe engineers are familiar with the parameter of FOPDTmodel [40]. A FOPDT system can be representedmathematically as follows

G(s) = k

t s + ae−Ls (23)

where k is the steady-state gain, L represents the process delaytime, t . 0 is the time constant. Sign and magnitude of adetermines the open-loop stability and steady-state gain ofthe process, respectively. In (23), parameters of the plant k,t and L might be uncertain parameters of the system. Exactvalues of these parameters may not be known. But theseparameters can be estimated at certain intervals. Thereforemodelling of this system as an interval time delay system isa realistic approach. Parameters of FOPDT system withparametric uncertainty structure can be defined as

k [ [k, k], t [ [t, t], L [ [L, L] (24)

where k, t and L are lower limits, k, t and L are upperlimits of the parameters, respectively. The fractional-ordercontroller is designed to obtain the desired performances for

Fig. 5 Nyquist plot of the system for C1opt(s)


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the given interval system. Preliminary study of this sectionhas been presented in the conference [41].

4.1 Monje–Vinagre et al. method

Owing to the tuning method, which Monje, Vinagre and theircolleagues have used in their papers [13, 21, 28], we preferredto use ‘Monje–Vinagre et al. method’ as a title of this section.Monje–Vinagre et al. proposed a PIlDm controller tuningalgorithm for the system to satisfy five design criteria, suchas magnitude at gain crossover frequency, phase margin,robustness to plant uncertainties, high-frequency noiseattenuation and sensitivity functions. Five design criteria ofthe Monje–Vinagre et al. method are given as follows.

4.1.1 Phase margin and gain crossover frequency:The gain and phase margins are two important frequency-domain specifications and two important measures ofrobustness. The phase margin is related to the damping ofthe system. Thus, the following equations should be satisfied

|C(jvcg)G(jvcg)|dB = 0 dB (25)

(Arg(C(jvcg)G(jvcg)) = −p+ fpm (26)

where vcg is the gain crossover frequency and fpm is therequired phase margin.

4.1.2 Robustness to variation in the gain of theplant: Satisfying the following constraint [42]

d(Arg(C(jv)G(jv)))

dv

( )v=vcg

= 0 (27)

the phase is forced to be flat at vcg and the phase plot is almostconstant within the interval around vcg. Consequently, thephase plot around the specified frequency vcg is locally flat,which implies that the system will be more robust to gainvariation and overshoots of the step responses are almostconstant within the interval.

4.1.3 High-frequency noise rejection: To satisfy therobustness due to high-frequency noise, the followingcondition must be fulfilled

T (jv) = C(jv) G(jv)

1 + C(jv) G(jv)

∣∣∣∣∣∣∣∣dB

≤ A dB (28)

where A is the desired value of the noise attenuation for thefrequency v ≥ vt rad/s.

4.1.4 Good output disturbance rejection: To ensure agood output disturbance rejection, the following constraintmust be satisfied

S(jv) = 1

1 + C(jv) G(jv)

∣∣∣∣∣∣∣∣dB

≤ B dB (29)

where B is the desired value of sensitivity function for thefrequency v ≤ vs rad/s.

Application of Monje–Vinagre et al. method: Anapplication of the Monje–Vinagre et al. method is given


for the following plant

Gp(s) = k

t s + 1(30)

Parameters of the plant Gp(s) are taken as t ¼ 80 and1 ≤ k ≤ 5. Design specifications for the plant are given asfollows: gain crossover frequency vcg ¼ 0.675 rad/s, phasemargin fpm ¼ 608, robustness to variation in the gain of theplant must be fulfilled, desired value of the noiseattenuation is A ¼ 220 dB for the frequencyv ≥ vt ¼ 10 rad/s and desired value of sensitivity functionis B ¼ 220 dB for the frequency v ≥ v2 ¼ 0.01 rad/s. Fivenon-linear equations are obtained satisfying all of thesespecifications. These five non-linear equations are solvedusing ‘fmincon’ optimisation toolbox of MATLAB, toobtain all the parameters of the PIlDm controller as follows

CMV (s) = 4.8694 + 43.0382

s0.3776+ 0.0637s0.9899 (31)

Step responses, Bode plot, magnitude plot of T (s) and S(s) ofthe system CMV(s) Gp(s) are given in Figs. 6–9, respectively.The step responses in Fig. 6 demonstrate that the stability ofthe system CMV(s)Gp(s) is satisfied and overshoots of the stepresponses are almost constant for the values of 1 ≤ k ≤ 5. InFig. 7, phase plot shows that the system is robust to gainvariation within the interval around the gain crossoverfrequency v ¼ vcg. Figs. 8 and 9 show, respectively, thatthe system CMV(s)Gp(s) satisfies the desired noiseattenuation and the desired value of sensitivity function atthe specified frequencies.

4.2 Design of robust PIlDm controller

This section proposes a design procedure for robust PIlDm

controller to control first-order and FOPDT system withparametric uncertainty structure. The FOPDT system withparametric uncertainty can be represented as

G(s) = [k, k]

[t, t]s + 1e−[L,L]s (32)

The proposed method uses the Bode envelopes of the FOPDTplant given in Fig. 10, to satisfy robust performance of thesystem. Consider the Bode envelopes given in Fig. 10.Minimum and maximum plots of the gain are obtained bythe following transfer functions, respectively

GR1(s) = k

ts + 1e−Ls and GR2(s) = k

ts + 1e−Ls (33)

Time delay L does not have any effect on the gain plot of theplant. Similarly, minimum and maximum plots of phase areobtained by the following transfer functions, respectively.

GR3(s) = k

ts + 1e−Ls and GR4(s) = k

ts + 1e−Ls (34)

Steady-state gain k does not have any effect on phase plot ofthe plant. In order to design robust PIlDm controller, (25)should be satisfied with the transfer function GR2(s), namelyvcg must be taken at the point ‘a’ in Fig. 10. Equation (26)should be satisfied with the transfer function GR4(s), namely

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the point ‘b’ in Fig. 10 should be the minimum phase marginof the system. Similarly, (27) should be satisfied with thetransfer function GR4(s) for minimum phase margin. GR1(s)and GR2(s) minimise (28) and (29), respectively, to satisfythe specified conditions. According to the explanationsgiven above, the following equations can be obtained from

(25) to (29) as follows

|C(jvcg)GR2(jvcg)|dB = 0 dB (35)

(Arg(C(jvcg)GR4(jvcg)) = −p+ fm (36)Fig. 7 Bode plots of CMV(s)Gp(s)

Fig. 8 Magnitude of T(s) for CMV(s)Gp(s)

Fig. 9 Magnitude of S(s) for CMV(s)Gp(s)

Fig. 6 Step response of CMV(s)Gp(s) for 1 ≤ k ≤ 5

Fig. 10 Bode envelopes of a FOPDT plant

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d(Arg(C(jv)GR4(jv)))

dv

( )v=vcg

= 0 (37)

T (jv) = C(jv) GR1(jv)

1 + C(jv) GR1(jv)

∣∣∣∣∣∣∣∣dB

≤ A dB (38)

S(jv) = 1

1 + C(jv) GR2(jv)

∣∣∣∣∣∣∣∣dB

≤ B dB (39)

In order to satisfy the given specifications for the system andfulfil five design criteria, the following equations can bederived with five unknown parameters (kp, ki, kd, l and m),using (33) and (34) in (35)–(39)

k��((t · vcg)2 + 1)

√⎛⎜⎝

⎞⎟⎠ ��

(r)2 + (s)2√( )∣∣∣∣∣∣∣

∣∣∣∣∣∣∣dB

= 0 dB (40)

a tans

r

[ ]− a tan (t · vcg) − vcg · L = −p+ fpm (41)

1

1 + (s/r)2 ·(su · r − s · ru)

(r)2 − t

1 + (t · vcg)2 − L = 0 (42)

k��(rt)2 + (st)2

√��(1 + k · rt)2 + (t · vt + k · st)2

√⎛⎜⎝

⎞⎟⎠

∣∣∣∣∣∣∣∣∣∣∣∣∣∣dB

≤ −20 dB (43)

��(t · vs)

2 + 1√

��(1 + k · rs)2 + (t · vs + k · ss)2

√⎛⎜⎝

⎞⎟⎠

∣∣∣∣∣∣∣∣∣∣∣∣∣∣dB

≤ −20 dB (44)

where

r = kp + kiv−lcg cos

p

2l

( )+ kdv

mcg cos

p

2m

( )s = −kiv

−lcg sin

p

2l

( )+ kdv

mcg sin

p

2m

( )ru = −kilv

−l−1cg cos

p

2l

( )+ kdmv

m−1cg cos

p

2m

( )su = −kilv

−l−1cg sin

p

2l

( )+ kdmv

m−1cg sin

p

2m

( )rt = kp + kiv

−lt cos

p

2l

( )+ kdv

mt cos

p

2m

( )st = −kiv

−lt sin

p

2l

( )+ kdv

mt sin

p

2m

( )rs = kp + kiv

−ls cos

p

2l

( )+ kdv

ms cos

p

2m

( )ss = −kiv

−ls sin

p

2l

( )+ kdv

ms sin

p

2m

( )

Equations (40)–(44) with five unknown parameters (kp, ki,kd, l and m) can be solved to obtain the parameters ofPIlDm for the robust stability of the given plant.

The following procedure can be applied to design robustPIlDm controller for FOPDT system:

† Specify the plant with parametric uncertainty in (23).† Find the transfer functions, which will give the gain andphase extremums of the Bode envelopes.† Obtain Bode envelopes of system using (33) and (34).


† Use five design criteria, which are given in Section 2,with the transfer functions given in (33) and (34) andobtain (35)–(39).† Derive the (40)–(44) using the transfer functions given in(35)–(39) with the desired design specification.† Solve (40)–(44) for five unknown parameters (kp, ki, kd, land m) using ‘fmincon’ optimisation toolbox of MATLABand obtain robust controller C(s).

4.3 Application of the proposed method to designrobust PIlDm controller

Example 2: Consider the following first-order interval plant

G2(s) = [k, k]

[t, t]s + 1(45)

where k [ [k, k] = [2, 4] and t [ [t, t] = [60, 80]. In orderto calculate the Bode envelopes, four different transferfunctions are obtained using gain and phase extremums ofthe plant given in (45) as follows

G21(s) = 2

80 s + 1and G22(s) = 4

60 s + 1(46)

G23(s) = 4

60 s + 1and G24(s) = 2

80 s + 1(47)

Design specifications for the system are given as follows: gaincrossover frequency vcg ¼ 2.1 rad/s, phase marginfpm ¼ 808, robustness to variation in the gain of the plantmust be fulfilled, desired noise attenuation A ¼ 220 dB forthe frequency v ≥ vt ¼ 10 rad/s and desired value ofsensitivity function B ¼ 220 dB for the frequencyv ≤ vs ¼ 0.01 rad/s. In order to satisfy the givenspecifications for the system and fulfil five design criteria,(46) and (47) are used in (40)–(44). FMINCONoptimisation toolbox of MATLAB is used for solution offive non-linear equations. Equation (40) is taken as a mainfunction and the other equations are taken as non-linearconstraints for optimisation. Values of the five unknownparameters (kp, ki, kd, l and m) are calculated. Then, PIlDm

controller to control G2(s) is obtained as

C2(s) = 6.7441 + 27.7461

s0.1461+ 0.0063s0.7874 (48)

Step responses of C2(s)G2(s) for 25 different values of theparameters of G2(s) are obtained by using the ‘nintblocks’of MATLAB [39], as shown in Fig. 11. The step responsesof the system C2(s)G2(s) show that the system is morerobust to gain changes and overshoot of the step responsesis almost constant within the interval of uncertain parameters.

Bode plots, magnitude plots of T2(s) and S2(s) of thesystem C2(s)G2(s) for 100 different values of the uncertainparameters by taking 10-point for each interval of theuncertain parameters of the G2(s) are obtained, respectively,as in Figs. 12–14 using the toolbox developed inMATLAB [43]. Fig. 12 shows that the phase envelopes ofthe system are almost flat and almost constant within aninterval around vcg ¼ 2.1 rad/s with the specifiedconstraints. Magnitude plots of the high-frequency noiseattenuation function and sensitivity function show,respectively, that the system satisfies desired noiseattenuation and the desired value of sensitivity in Figs. 13and 14. One can conclude from the results of Figs. 11–14

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Fig. 11 Step responses of the C2(s)G2(s)

Fig. 12 Bode plots of C2(s)G2(s)

Fig. 13 Magnitudes of T2(s) for C2(s)G2(s)

Fig. 14 Magnitudes of S2(s) for C2(s)G2(s)

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that the controller satisfies the robust performance of thesystem.

Example 3: Now, consider the plant in Example 2 with timedelay as follows

G3(s) = [k, k]

[t, t]s + 1e−[L,L]s (49)

where [k, k] = [2, 4], [t, t] = [60, 80] and [L, L] =[0.5, 1]. Gain crossover frequency is vcg ¼ 0.16 rad/s. Thegain and phase envelopes can be obtained from thefollowing equations, respectively

G31(s) = 2

80 s + 1e−1s and G32(s) = 4

60 s + 1e−1s (50)

G33(s) = 4

60 s + 1e−0.5s and G34(s) = 2

80 s + 1e−1s

(51)

Controller C3(s) can be obtained using the same procedure inExample 2 as follows

C3(s) = 2.8664 + 0.0956

s0.8483+ 1.2878 s0.9721 (52)

The overshoots of the step responses of the systemC3(s)G3(s), which are obtained for 125 different transferfunctions by taking 5-point from each uncertain parametersof the plant C3(s), are almost the same as given in Fig. 15.Bode plots, magnitude of T3(s) and magnitude of S3(s) ofthe system C3(s)G3(s) are obtained in Figs. 16–18,respectively, for 125 different values of the uncertainparameters of the G3(s). The phase envelopes of the systemare almost flat and almost constant within an intervalaround vcg ¼ 0.16 rad/s with the specified constraints.Magnitude plots of the high-frequency noise attenuationfunction and sensitivity function show that the systemsatisfies the desired noise attenuation and the desired valueof sensitivity.

Fig. 15 Step responses of the system controlled with C3(s)


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Example 4: Consider the following real liquid-level systemgiven in [13]

G4(s) = k

t s + 1e−Ls = 3.13

433.33 s + 1e−50s (53)

Interval representation of this plant is a realistic approach. Theparameters of this real plant can be considered as[k, k] = [2.7, 3.5], [t, t] = [423, 443] and [L, L] =[45, 55], which are interval representations of theparameters. In order to calculate the Bode envelopes, fourdifferent transfer functions are obtained using gain and





phase extremums of the plant given in (53) as follows

G41(s) = 2.7

443 s + 1e−55s, G42(s) = 3.5

423 s + 1e−55s (54)

G43(s) = 3.5

423 s + 1e−45s, G44(s) = 2.7

443 s + 1e−55s (55)

Design specifications for the system are given as follows: gaincrossover frequency vcg ¼ 0.008 rad/s, phase marginfpm ¼ 608, robustness to variation in the gain of the plantmust be fulfilled, desired noise attenuation A ¼ 220 dB forthe frequency v ≥ vt ¼ 1 rad/s and desired value ofsensitivity function B ¼ 220 dB for the frequencyv ≤ vs ¼ 0.001 rad/s. The controller C4(s) to control G4(s)is calculated as in Example 2 as follows

C4(s) = 0.5775 + 0.0047

s0.9733+ 4.3865s0.5253 (56)

Step responses of the system controlled with C4(s) for 125different values of the parameters of the G4(s) are obtainedas in Fig. 19. Overshoots of the step responses are almostconstant within the interval of uncertain parameters and the

Fig. 19 Step responses of the system controlled with C4(s)

Final value of the reference input step ¼ 0.47 as in [13]


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controller satisfies the robust stability of the FOPDT system,which is more robust to gain changes. Bode plots, magnitudeof T4(s) and magnitude of S4(s) of the system C4(s)G4(s) for125 different values of the uncertain parameters of theG4(s) are shown, respectively, in Figs. 20–22. The Bodeplots of the system shows that the phase of the system arealmost flat and almost constant within an interval aroundvcg ¼ 0.008 rad/s with the specified constraints. Magnitudeplots of the high-frequency noise attenuation function andsensitivity function show that the system satisfies desirednoise attenuation for v ≥ vt ¼ 1 rad/s and the desired valueof sensitivity for v ≤ v2 ¼ 0.001 rad/s. One can concludefrom Figs. 19–22 that the controller satisfies the robustperformance of the system.

5 Conclusions

In this paper, two methods for tuning of PIlDm controller havebeen proposed. The first method is based on the idea of usingthe Ziegler–Nichols and Astrom–Hagglund methodtogether. Values of the kp and ki parameters of PIlDm

controller have been computed from the Ziegler–Nicholsmethod and the remaining parameters kd, l and m havebeen found from the Astrom–Hagglund method usingcritical point information. Values of the controllerparameters are optimised to achieve better step response.The simulation results demonstrated that the PIlDm

controller has better response than the classical PIDcontrollers. In the second method, we proposed a newrobust tuning method for a PIlDm controller to control first-order systems with parameter uncertainty structure. Theproposed method benefits from design specifications givenin Monje–Vinagre et al. tuning method. Five designspecifications of the Monje–Vinagre et al. method are usedto derive five non-linear equations. Values of the fiveunknown parameters of PIlDm controller are obtained from



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these non-linear equations. Gain and phase extremums ofBode envelopes of the plant is used to satisfy robustperformances of the system. As seen from the results of thenumerical example, the PIlDm controller satisfies robustperformance of the system.

It is necessary to point out that there are many other tuningmethods in the literature for PID controllers, which may givebetter results than the Ziegler–Nichols and Astrom–Hagglund methods for some cases. Some tuning methodsfor PIlDm controller are also proposed in recent years. Thecomparison studies of the proposed methods for tuning ofPIlDm controllers certainly will be very important. On theother hand, the robust PIlDm controller design for differenttypes of plants with parameter uncertainty structure mightbe a subject of future work.

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